RIMS Kôkyûroku Bessatsu
Bx (201x), 000–000

A PROOF OF THE ABC CONJECTURE
AFTER MOCHIZUKI
By

Go Yamashita∗

Abstract
We give a survey of S. Mochizuki’s ingenious inter-universal Teichmüller theory and explain how it gives rise to Diophantine inequalities. The exposition was designed to be as
self-contained as possible.

Contents§ 0. Introduction.
§ 0.1. Un Fil d’Ariane.
§ 0.2. Notation.
§ 1. Reduction Steps via General Arithmetic Geometry.
§ 1.1. Height Functions.
§ 1.2. First Reduction.
§ 1.3. Second Reduction — Log-volume Computations.
§ 1.4. Third Reduction — Choice of Initial Θ-Data.
§ 2. Preliminaries on Anabelian Geometry.
§ 2.1. Some Basics on Galois Groups of Local Fields.
§ 2.2. Arithmetic Quotients.
§ 2.3. Slimness and Commensurable Terminality.
§ 2.4. Characterisation of Cuspidal Decomposition Groups.
§ 3. Mono-anabelian Reconstruction Algorithms.
Received xxxx, 201x. Revised xxxx, 201x.
2010 Mathematics Subject Classification(s):Key Words: inter-universal Teichmüller theory, anabelian geometry, Diophantine inequality,
height function, abc Conjecture, Hodge-Arakelov theory
Supported by Toyota Central R&D Labs., Inc. and JSPS Grant-in-Aid for Scientific Research (C)15K04781∗ RIMS, Kyoto University, Kyoto 606-8502, Japan.
e-mail: gokun@kurims.kyoto-u.ac.jp
c 201x Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

2

Go Yamashita§ 3.1. Some Definitions.
§ 3.2. Belyi and Elliptic Cuspidalisations — Hidden Endomorphisms.
§ 3.2.1. Elliptic Cuspidalisation.
§ 3.2.2. Belyi Cuspidalisation.
§ 3.3. Uchida’s Lemma.
§ 3.4. Mono-anabelian Reconstruction of the Base Field and Function Field.
§ 3.5. On the Philosophy of Mono-analyticity and Arithmetic Holomorphicity.
§ 4. The Archimedean Theory — Formulated Without Reference to a Specific ModelC.
§ 4.1. Aut-Holomorphic Spaces.
§ 4.2. Elliptic Cuspidalisation and Kummer Theory in the Archimedean Theory.
§ 4.3. On the Philosophy of Étale-like and Frobenius-like Objects.
§ 4.4. Mono-anabelian Reconstruction Algorithms in the Archimedean Theory.
§ 5. Log-volumes and Log-shells.
§ 5.1. Non-Archimedean Places.
§ 5.2. Archimedean Places.
§ 6. Preliminaries on Tempered Fundamental Groups.
§ 6.1. Some Definitions.
§ 6.2. Profinite Conjugates vs. Tempered Conjugates.
§ 7. Étale Theta Functions — Three Fundamental Rigidities.
§ 7.1. Theta-related Varieties.
§ 7.2. The Étale Theta Function.
§ 7.3. l-th Root of the Étale Theta Function.
§ 7.4. Three Fundamental Rigidities of Mono-theta Environments.
§ 7.5. Some Analogous Objects at Good Places.
§ 8. Frobenioids.
§ 8.1. Elementary Frobenioids and Model Frobenioids.
§ 8.2. Examples.
§ 8.3. From Tempered Frobenioids to Mono-theta Environments.
§ 9. Preliminaries on the NF Counterpart of Theta Evaluation.
§ 9.1. Pseudo-Monoids of κ-Coric Functions.
§ 9.2. Cyclotomic Rigidity via κ-Coric Functions.
§ 9.3. -Line Bundles and -Line Bundles.
§ 10. Hodge Theatres.
§ 10.1. Initial Θ-Data.
§ 10.2. Model Objects.
§ 10.3. Θ-Hodge Theatres and Prime-strips.

a proof of the abc conjecture after Mochizuki

3§ 10.4. The Multiplicative Symmetry  : ΘNF-Hodge Theatres and NF-, Θ-Bridges.
§ 10.5. The Additive Symmetry  : Θ±ell -Hodge Theatres and Θell -, Θ± -Bridges.
§ 10.6. Θ±ell NF-Hodge Theatres — An Arithmetic Analogue of the Upper Half
Plane.
§ 11. Hodge-Arakelov-theoretic Evaluation Maps.
§ 11.1. Radial Environments.
§ 11.2. Hodge-Arakelov-theoretic Evaluation and Gaussian Monoids at Bad Places.
§ 11.3. Hodge-Arakelov-theoretic Evaluation and Gaussian Monoids at Good Places.
§ 11.4. Hodge-Arakelov-theoretic Evaluation and Gaussian Monoids in the Global
Case.
§ 12. Log-links — An Arithmetic Analogue of Analytic Continuation.
§ 12.1. Log-links and Log-theta-lattices.
§ 12.2. Kummer Compatible Multiradial Theta Monoids.
§ 13. Multiradial Representation Algorithms.
§ 13.1. Local and Global Packets.
§ 13.2. Log-Kummer Correspondences and Multiradial Representation Algorithms.
Appendix A. Motivation of the Definition of the Θ-Link.
§ A.1. The Classical de Rham Comparison Theorem.
§ A.2. p-adic Hodge-theoretic Comparison Theorem.
§ A.3. Hodge-Arakelov-theoretic Comparison Theorem.
§ A.4. Motivation of the Definition of the Θ-Link.
Appendix B. Anabelian Geometry.
Appendix C. Miscellany.
§ C.1. On the Height Function.
§ C.2. Non-critical Belyi Maps.
§ C.3. k-Cores.
§ C.4. On the Prime Number Theorem.
§ C.5. On the Residual Finiteness of Free Groups.
§ C.6. Some Lists on Inter-universal Teichmüller Theory.
References§ 0.

Introduction.

The author once heard the following observation, which was attributed to Grothendieck: There are two ways to crack a nut — one is to crack the nut in a single strokeby using a nutcracker; the other is to soak it in water for an extended period of time

4

Go Yamashita

until its shell dissolves naturally. Grothendieck’s mathematics may be regarded as an
example of the latter approach.
In a similar vein, the author once heard a story about a mathematician who asked
an expert on étale cohomology what the main point was in the `-adic (not the p-adic)
proof of the rationality of the congruence zeta function. The expert was able to recall,on the one hand, that the Lefschetz trace formula was proved by checking various
commutative diagrams and applying various base change theorems (e.g., for proper orsmooth morphisms). On the other hand, neither the commutativity of various diagrams
nor the various base change theorems could be described as the main point of the proof.
Ultimately, the expert was not able to point out precisely what the main point in the
proof was. From the point of view of the author, the main point of the proof seems to lie
in the establishment of a suitable framework (i.e., scheme theory and étale cohomology
theory) in which the Lefschetz trace formula, which was already well known in thefield of algebraic topology, could be formulated and proved even over fields of positive
characteristic.
A similar statement can be made concerning S. Mochizuki’s proof of the abc Conjecture. Indeed, once the reader admits the main results of the preparatory papers(especially [AbsTopIII], [EtTh]), the numerous constructions in the series of papers[IUTchI], [IUTchII], [IUTchIII], [IUTchIV] on inter-universal Teichmüller theory arelikely to strike the reader as being somewhat trivial. On the other hand, the way inwhich the main results of the preparatory papers are interpreted and combined in order to perform these numerous constructions is highly nontrivial and based on very
delicate considerations (cf. Remark 9.6.2 and Remark 12.8.1) concerning, for instance,
the notions of multiradiality and uniradiality (cf. Section 11.1). Moreover, when taken
together, these numerous trivial constructions, whose exposition occupies literally hundreds of pages, allow one to conclude a highly nontrivial consequence (i.e., the desired
Diophantine inequality) practically eﬀortlessly! Again, from the point of view of theauthor, the point of the proof seems to lie in the establishment of a suitable frameworkin which one may deform the structure of a number field by abandoning the framework of conventional scheme theory and working instead in the framework furnished byinter-universal Teichmüller theory (cf. also Remark 1.15.3).
In fact, the main results of the preparatory papers [AbsTopIII], [EtTh], etc. arealso obtained, to a substantial degree, as consequences of numerous constructions thatare not so diﬃcult. On the other hand, the discovery of the ideas and insights that
underlie these constructions may be regarded as highly nontrivial in content. Examples
of such ideas and insights include the “hidden endomorphisms” that play a central rolein the mono-anabelian reconstruction algorithms of Section 3.2, the notions of arithmetically holomorphic structure and mono-analytic structure (cf. Section 3.5), and thea proof of the abc conjecture after Mochizuki

5

distinction between étale-like and Frobenius-like objects (cf. Section 4.3). Thus, in summary, it seems to the author that, if one ignores the delicate considerations that occur
in the course of interpreting and combining the main results of the preparatory papers,
together with the ideas and insights that underlie the theory of these preparatory papers, then, in some sense, the only nontrivial mathematical ingredient in inter-universal
Teichmüller theory is the classical result [pGC], which was already known in the lastcentury!
A more technical introduction to the mathematical content of the main ideas ofinter-universal Teichmüller theory may be found in Appendix A and the discussion at
the beginning of Section 13.
The following results are consequences of inter-universal Teichmüller theory (cf.
Section 1.1 for more details on the notation):
Theorem 0.1.
(Vojta’s Conjecture [Voj] for Curves, [IUTchIV, Corollary 2.3])
Let X be a proper, smooth, geometrically connected curve over a number field; D ⊂ X
a reduced divisor; UX := X \ D. Write ωX for the canonical sheaf on X. Suppose that
UX is a hyperbolic curve, i.e., deg(ωX (D)) > 0. Then for any d ∈ Z>0 and  ∈ R>0 ,we have
htωX (D) . (1 + )(log-diﬀ X + log-condD )on UX (Q)≤d .
Corollary 0.2.
(The abc Conjecture of Masser and Oesterlé [Mass1], [Oes]) For
any  ∈ R>0 , we have
1+∏
max{|a|, |b|, |c|} ≤ 
p
p|abc

for all but finitely many coprime a, b, c ∈ Z with a + b = c.
Proof. We apply Theorem 0.1 in the case where X = P1Q ⊃ D = {0, 1, ∞}, andd = 1. Thus, we have ωP1 (D) = OP1 (1), log-diﬀ P1 (−a/b) = 0, log-cond{0,1,∞} (−a/b) =∑p|a,b,a+b log p, and htOP1 (1) (−a/b) ≈ log max{|a|, |b|} ≈ log max{|a|, |b|, |a + b|} forcoprime a, b ∈ Z with b 6= 0, where the first “≈” follows from [Silv1, Proposition 7.2],
and we apply the inequality |a + b| ≤ 2 max{|a|, |b|}. Now let , 0 ∈ R>0 be such that > 0 . According to Theorem 0.1, there exists C ∈ R such that log max{|a|, |b|, |c|} ≤∑(1+0 ) p|abc log p+C for any coprime a, b, c ∈ Z with a+b = c. Observe that there are1+
only finitely many triples a, b, c ∈ Z with a+b = c such that log max{|a|, |b|, |c|} ≤ −
0 C.
∑
0
−
0
Thus, we have log max{|a|, |b|, |c|} ≤ (1 +  ) p|abc log p + 1+ log max{|a|, |b|, |c|} forall but finitely many coprime triples a, b, c ∈ Z with a + b = c. This completes the proofof Corollary 0.2.

6

Go Yamashita§ 0.1.

Un Fil d’Ariane.

By combining a relative anabelian result (a relative version of the Grothendieck
Conjecture over sub-p-adic fields (Theorem B.1)) and the “hidden endomorphism” diagram (EllCusp) (resp. the “hidden endomorphism” diagram (BelyiCusp)), one obtains
a(n) (absolute) mono-anabelian result, i.e., the elliptic cuspidalisation (Theorem 3.7)
(resp. the Belyi cuspidalisation (Theorem 3.8)). Then, by applying Belyi cuspidalisations, one obtains a mono-anabelian reconstruction algorithm for the NF-portion of
the base field and function field of a hyperbolic curve of strictly Belyi type over a subp-adic field (Theorem 3.17), as well as a mono-anabelian reconstruction algorithm forthe base field of a hyperbolic curve of strictly Belyi type over a mixed characteristic
local field (Corollary 3.19). This motivates the philosophy of mono-analyticity andarithmetic holomorphicity (Section 3.5), as well as the theory of Kummer isomorphisms
from Frobenius-like objects to étale-like objects (cf. Remark 9.6.1).
The theory of Aut-holomorphic (orbi)spaces and related reconstruction algorithms(Section 4) is an Archimedean analogue of the mono-anabelian reconstruction algorithms
discussed above and yields another application of the technique of elliptic cuspidalisation. On the other hand, the Archimedean theory does not play a very central role ininter-universal Teichmüller theory.
The theory of the étale theta function centers around the establishment of various
rigidity properties of mono-theta environments. One applies the technique of elliptic cuspidalisation to show the constant multiple rigidity of a mono-theta environment(Theorem 7.23 (3)). The cyclotomic rigidity of a mono-theta environment is obtained
as a consequence of the (“precisely”) quadratic structure of a Heisenberg group (Theorem 7.23 (1)). Finally, by applying the “at most” quadratic structure of a Heisenberg
group (and excluding the algebraic section in the definition of a mono-theta environment), one shows the discrete rigidity of a mono-theta environment (Theorem 7.23 (2)).
The notions of étale-like and Frobenius-like objects play a very important role ininter-universal Teichmüller theory (cf. Section 4.3). The significance of Frobenius-likeobjects (cf. the theory of Frobenioids, as discussed in Section 8) lies in the fact they
allow one to construct links, or “walls”, such as the Θ-link and log-link (cf. Definition 10.8; Corollary 11.24 (3); Definition 13.9 (2); Definition 12.1 (1), (2); and Definition 12.3). (The main theorems of the theory of Frobenioids concern category-theoretic
reconstruction algorithms; however, these algorithms do not play a very central role ininter-universal Teichmüller theory (cf. [IUTchI, Remark 3.2.1 (ii)]).) By contrast, thesignificance of étale-like objects lies in the fact that they allow one to penetrate thesewalls (cf. Remark 9.6.1).
The notion of multiradiality plays a central role in inter-universal Teichmüller theory(cf. Section 11.1). The significance of the multiradial algorithms that are ultimately

a proof of the abc conjecture after Mochizuki

7

established lies in the fact that they allow one to“permute” (up to mild indeterminacies) the theta values in the source of theΘ-link and the theta values in the target of the Θ-link.
In other words, multiradiality makes it possible to “see” (up to mild indeterminacies) the“alien” ring structure on one side of Θ-link from the point of view of the ring structureon the other side (cf. the discussion at the beginning of Section 13). This multiradiality,
together with the compatibility of the algorithms under consideration with the Θ-link,
will, ultimately, lead to the desired height estimate (cf. Remark 11.1.1).
The multiradial algorithm that we ultimately wish to establish consists, roughly
speaking, of three main objects (cf. the column labelled “(1)” of the table before Corollary 13.13): (mono-analytic étale-like) log-shells (which are related to the local unitsof the number fields under consideration) equipped with log-volume functions (cf. Section 5), theta values (which are related to the local value groups of the number fields
under consideration) acting on these log-shells, and (global) number fields acting on theselog-shells. In this context, the theta function (resp. κ-coric functions (Definition 9.2))
serve(s) as a geometric container for the theta values (resp. the number fields) justmentioned and allow(s) one to establish the multiradiality of the reconstruction algorithms under consideration. Here, suitable versions of Kummer theory for the theta
function and κ-coric functions allow one to relate the respective étale-like objects andFrobenius-like objects under consideration. These versions of Kummer theory depend
on suitable versions of cyclotomic rigidity.
The cyclotomic rigidity of mono-theta environments discussed above allows one
to perform Kummer theory for the theta function in a multiradial manner (Proposition 11.4, Theorem 12.7, Corollary 12.8). Similarly, a certain version of cyclotomic
b × = {1} (Definition 9.6)
rigidity that is deduced from the elementary fact Q>0 ∩ Z
allows one to perform Kummer theory for κ-coric functions in a multiradial manner.
At a more concrete level, the cyclotomic rigidity of mono-theta environments and κcoric functions plays the role of protecting the Kummer theory surrounding the thetab × -indeterminacies that act on the local units
function and κ-coric functions from the Z
and hence ensures the compatibility of the Θ-link with the portion of the final multiradial algorithm that involves the Kummer theory surrounding the theta function andκ-coric functions (cf. the column labelled “(3)” of the table before Corollary 13.13). By
contrast, the most classical version of cyclotomic rigidity, which is deduced from local
class field theory for MLF’s (cf. Section 0.2), does not yield a multiradial algorithm (cf.
Remark 11.4.1, Proposition 11.15 (2), and Remark 11.17.2 (2)).
The Kummer theory discussed above for mono-theta environments and theta functions (resp. for κ-coric functions) leads naturally to the theory of Hodge-Arakelovtheoretic evaluation (resp. the NF-counterpart (cf. Section 0.2) of the theory of Hodge-

8

Go Yamashita

Arakelov-theoretic evaluation) and the construction of Gaussian monoids, i.e., in essence,
monoids generated by theta values (Section 11.2) (resp. the construction of elements ofnumber fields (Section 9.2, Section 11.4)). In the course of performing Hodge-Arakelovtheoretic evaluation at the bad primes, one applies a certain consequence of the theoryof semi-graphs of anabelioids (“profinite conjugates vs. tempered conjugates” Theorem 6.11). The reconstruction of mono-theta environments from (suitable types of)
topological groups (Corollary 7.22 (2) “Π 7→ M”) and tempered Frobenioids (Theorem 8.14 “F 7→ M”), together with the discrete rigidity of mono-theta environments,
allows one to derive Frobenioid-theoretic versions of the group-theoretic versions ofHodge-Arakelov evaluation and the construction of Gaussian monoids just described(Corollary 11.17). In the course of performing Hodge-Arakelov-theoretic evaluation, one
applies the Fo±l -symmetry in the Hodge theatres under consideration (Section 10.5) tosynchronise the conjugacy indeterminacies that occur (Corollary 11.16 (1)). The theoryof synchronisation of conjugacy indeterminacies makes it possible to construct “gooddiagonals”, which give rise, in the context of the log-theta-lattice, to horizontally coricobjects.
By combining the construction of Gaussian monoids just discussed with the theoryof log-links, one obtains LGP-monoids (Proposition 13.6). Here, it is of interest to
observe that this construction of LGP-monoids makes use of the compatibility of thecyclotomic rigidity of mono-theta environments with the profinite topology, which is
closely related to the isomorphism class compatibility of mono-theta environments (cf.
Remark 9.6.2 (5)). LGP-monoids are equipped with natural canonical splittings, whicharise, via canonical splittings of theta monoids (i.e., in essence, monoids generated
by theta functions), from the constant multiple rigidity of mono-theta environments(Proposition 11.7, Proposition 13.6).
The theory of log-links and log-shells, both of which are closely related to the local units of number fields under consideration (Section 5, Section 12), together with
the Kummer theory that relates corresponding Frobenius-like and étale-like versions ofobjects, gives rise to the log-Kummer correspondences for the theta values (which are
related to the local value groups of the number fields under consideration) and (global)
number fields under consideration (Proposition 13.7 and Proposition 13.11). The canonical splittings of LGP-monoids discussed above may be interpreted, in the context ofthe log-Kummer correspondence, as a non-interference property (Proposition 13.7 (2c))∏××) may be interof the LGP-monoids, while the classical fact Fmod
∩ v≤∞ Ov = µ(Fmod
preted, in the context of the log-Kummer correspondence, as a non-interference property(Proposition 13.11 (2)) of the number fields involved (cf. the column labelled “(2)” of
the table before Corollary 13.13).
By passing from arithmetically holomorphic structures to the underlying mono-a proof of the abc conjecture after Mochizuki

9analytic structures, admitting three kinds of mild indeterminacies, and applying the noninterefence properties of log-Kummer correspondences, one obtains the multiradiality of
the final multiradial algorithm (Theorem 13.12). In the final multiradial algorithm, we
use the theta values (which are related to the local value groups) to construct the Θ-pilot
objects (Definition 13.9 (1)), (global) number fields to relate (global Frobenioids arisingfrom) -line bundles to (global Frobenioids arising from) -line bundles (cf. Section 9.3),
and log-shells (which arise from the local units) as mono-analytic containers for thetavalues and (global) number fields.
Since the labels attached to the theta values depend on the arithmetically holomorphic structure, one cannot, a priori, transport these labels from one side of a Θ-link to
the other side of the Θ-link. On the other hand, by using processions, one can reducethe indeterminacy that arises from forgetting these labels (cf. Remark 13.1.1).
Finally, by combining the multiradiality of the final multiradial algorithm with thecompatibility of this algorithm with the Θ-link, the compatibility of the log-volumes
with the log-links (Section 5), and various properties concerning global Frobenioids, we
obtain an upper bound for the height of the given elliptic curve (Corollary 13.13, cf.
Remark 13.13.2). The fact that the leading term of the upper bound is of the expected
form may be regarded as a consequence of a certain calculation in Hodge-Arakelov
theory (Remark 1.15.3 (the “miracle identity”)).

Leitfaden§2. Prel. Anab.

/ §6. Prel. Temp.

,#/ §5. Log-vol./-sh.
/ §7. Ét. Theta§4.
Aut-hol.
§3. Mono-anab.
FF RRReeeeeeeeFF RRRlll
xxlll
FF RRRR xxxeeeeeeeeeeell
FF eeeRxRexRee
lll


x
vlll
reFeFeFexxx R)/ §12. Log-link/ §13. Mlt. Alg’m§10. Hodge
Th. xxFxFF / §11. H-A.
O
O Eval.
FF
x
x
FF
xx
FF
xx
FF
x
{xx
#

/ §9. Prel. NF-Eval.
/ Thm. 0.1§8. Fr’ds§1. Gen. Arith.
The above dependences are rough (or conceptual) relations. For example, we use some
portions of §7 and §9 in the constructions in §10; however, conceptually, §7 and §9 are
mainly used in §11, and so on.

10

Go Yamashita

Acknowledgments
The author feels deeply indebted to Shinichi Mochizuki for helpful and exciting
discussions on inter-universal Teichmüller theory1 , related theories, and further developments related to inter-universal Teichmüller theory2 . The author also thanks Akio
Tamagawa, Yuichiro Hoshi, and Makoto Matsumoto for attending the intensive IU seminars given by the author from May 2013 to November 2013 and for many helpful
discussions. He thanks Tomoki Mihara for some comments on topological groups. He
also thanks Koji Nuida and Takuya Sakasai for pointing out typos. He also sincerely
thanks the executives at Toyota CRDL, Inc. for oﬀering him a special position thatenabled him to concentrate on his research in pure mathematics. He sincerely thanks
Sakichi Toyoda for the generous philanthropic culture that he established when he laid
the foundations for the Toyota Group, as well as the (ex-)executives at Toyota CRDL,
Inc. (especially Noboru Kikuchi, Yasuo Ohtani, Takashi Saito and Satoshi Yamazaki)
for their continued supoort of this culture (even over 80 years after the death of Sakichi
Toyoda). He also thanks Shigefumi Mori for intermediating between Toyota CRDL,
Inc. and the author. Finally, we remark that this work was supported by the ResearchInstitute for Mathematical Sciences, a Joint Usage/Research Centre located in Kyoto
University.

§ 0.2.

Notation.

General Notation:
For a finite set A, we write #A for the cardinality of A. For a group G and asubgroup H ⊂ G of finite index, we write [G : H] for #(G/H). (For a finite extensionof fields K ⊃ F , we also write [K : F ] for dimF K. This will not result in any confusion
between the notations “[G : H]” and “[K : F ]”.) For a function f on a set X and asubset Y ⊂ X, we write f |Y for the restriction of f to Y .
1 Ivan Fesenko wrote, in the published version of his survey “Arithmetic deformation theory via

arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi
Mochizuki”, that he encouraged the author to learn and scrutinise arithmetic deformation theory
subsequent to his meeting with Mochizuki in mid-September 2012. In fact, the author had alreadysent an email to Mochizuki on the 1st of September 2012, in which the author expressed his interestin studying inter-universal Teichmüller theory.
2 In particular, the author began his study of inter-universal Teichmüller theory of his own will. In
the latest version of Fesenko’s survey (posted on Fesenko’s web site subsequent to the publicationof the published version of the survey), Fesenko replaced the expression “encouraged Yamashita”
by the expression “supported his interest to study the theory”.

11

a proof of the abc conjecture after Mochizukio±×For a prime number l > 2, we write F>:= Fl o {±1}, wherel := Fl /{±1}, Fl>`{±1} acts on Fl by multiplication, and |Fl | := Fl /{±1} = Fl{0}. We also write
>
l−1
l+1
>
±
>
l := 2 = #Fl and l := l + 1 = 2 = #|Fl |.

Categories:
For an object A in a category, we shall call an object isomorphic to A an isomorphof A.
For a category C and a filtered ordered set I 6= ∅, we write pro-CI (= pro-C)
for the category of pro-objects of C indexed by I, i.e., whose objects are of the form((Ai )i∈I , (fi,j )i<j∈I ) (= (Ai )i∈I ), where Ai is an object in C, and fi,j is a morphism
Aj → Ai satisfying fi,j fj,k = fi,k for any i < j < k ∈ I, and whose morphisms are
given by Hompro-C ((Ai )i∈I , (Bj )j∈I ) := limj limi HomC (Ai , Bj ). We also regard objects←− −→of C as objects of pro-C (by setting every transition morphism to be identity). Thus,
relative to this convention, we have Hompro-C ((Ai )i∈I , B) = limi HomC (Ai , B).
−→
For a category C, we write C 0 for the full subcategory of connected objects, i.e., thenon-initial objects which are not isomorphic to a coproduct of two non-initial objects of
C. We write C > (resp. C ⊥ ) for the category whose objects are formal (possibly empty)
countable (resp. finite) coproducts of objects in C, and whose morphisms are given by∏ ```HomC > (resp. C ⊥ ) ( i Ai , j Bj ) := i j HomC (Ai , Bj ) (cf. [SemiAnbd, §0]).
Let C1 , C2 be categories. We say that two isomorphism classes of functors f : C1 →∼C2 , f 0 : C10 → C20 are abstractly equivalent if there exist isomorphisms α1 : C1 → C10 ,∼α2 : C2 → C20 such that f 0 ◦ α1 = α2 ◦ f .
poly

Let C be a category. We define a poly-morphism A → B for A, B ∈ Ob(C) to
be a (possibly empty) set of morphisms A → B in C. A poly-morphism for which eachconstituent morphism is an isomorphism will be called a poly-isomorphism. If A = B,
poly∼then a poly-isomorphism A → B will be called a poly-automorphism. We define
full poly∼∼

the full poly-isomorphism A → B to be the set of all isomorphisms A → B.
We define the composite of poly-morphisms {fi : A → B}i∈I and {gj : B → C}j∈J
to be {gj ◦ fi : A → C}(i,j)∈I×J . We define a poly-action to be an action via polyautomorphisms.
Let C be a category. We define a capsule of objects of C to be a finite collection{Aj }j∈J of objects of C. We shall also refer to {Aj }j∈J as a #J -capsule. We define amorphism {Aj }j∈J → {A0j 0 }j 0 ∈J 0 between capsules of objects of C to be a collectionof data (ι, (fj )j∈J ) consisting of an injection ι : J ,→ J 0 and a morphism fj : Aj → A0ι(j)in C for each j ∈ J. (Thus, the capsules of objects of C and the morphisms between

12

Go Yamashita

capsules of objects of C form a category.) We define a capsule-full poly-morphism
to be a poly-morphism{}∏∼{fj : Aj → A0ι(j) }j∈J(=
IsomC (Aj , A0ι(j) ))∏0(fj )j∈J ∈j∈J IsomC (Aj ,Aι(j) )j∈Jin the category of the capsules of objects of C, associated with a fixed injection ι : J ,→J 0 . If the fixed ι is a bijection, then we shall refer to the capsule-full poly-morphism asa capsule-full poly-isomorphism.
Number Fields and Local Fields:
In this survey, we define a number field to be a finite extension of Q (i.e., we
exclude infinite extensions). We define a mixed characteristic (or non-Archimedean)
local field to be a finite extension of Qp for some p. We use the abbreviations NF
for “number field”, MLF for “mixed characteristic local field”, and CAF for “complex
Archimedean field” (i.e., a topological field isomorphic to C). For a topological fieldk which is isomorphic to R or C, we write | − |k : k → R≥0 for the absolute value
associated to k, i.e., the unique continuous map such that the restriction of | − |k to k ×determines a homomorphism k × → R>0 with respect to the multiplicative structures ofk × and R>0 , and |n|k = n for n ∈ Z≥0 . We write π ∈ R for the mathematical constantpi (i.e., π = 3.14159 · · · ).
For a number field F , we write V(F ) for the set of equivalence classes of valuationsof F and V(F )arc ⊂ V(F ) (resp. V(F )non ⊂ V(F )) for the subset of Archimedean(resp. non-Archimedean) equivalence classes of valuations. For number fields F ⊂ L
and v ∈ V(F ), we write V(L)v := V(L) ×V(F ) {v} (⊂ V(L)), where V(L)  V(F ) is
the natural surjection. For v ∈ V(F ), we write Fv for the completion of F with respect
to v. For v ∈ V(F )non (resp. v ∈ V(F )arc ), we write pv for the characteristic of theresidue field (resp. pv := e = 2.71828 · · · ) and fv for the extension degree of the residue
field at v over Fpv (resp. fv := 1). We write mv for the maximal ideal and ordv forthe valuation normalised by ordv (pv ) = 1 for v ∈ V(F )non . We normalise v ∈ V(F )nonby v(uniformiser of Fv ) = 1. (Thus, v(−) = ev · ordv (−), where we write ev for theramification index of Fv over Qpv .) We shall write ord for ordv when there is no fear ofconfusion. For v ∈ Varc , we write | − |v := | − |Fv .
For a non-Archimedean (resp. complex Archimedean) local field k, we write Ok forthe valuation ring (resp. the subset of elements of absolute value ≤ 1) of k, Ok× ⊂ Ok
for the subgroup of units (resp. the subgroup of units, i.e., elements of absolute value= 1), and Ok := Ok \ {0} ⊂ Ok for the multiplicative topological monoid of non-zeroelements of Ok . We shall also refer to Ok as the subset of integral elements of k.
When k is a non-Archimedean local field, we shall write mk for the maximal ideal of

a proof of the abc conjecture after Mochizuki

13

Ok .
For a non-Archimedean local field K with residue field k, and an algebraic closure
k of k, we write FrobK ∈ Gal(k/k) or Frobk ∈ Gal(k/k) for the (arithmetic) Frobenius
element, i.e., the map k 3 x 7→ x#k ∈ k. (Note that in this survey, neither the term
“Frobenius element”, the notation FrobK , nor the notation Frobk will be used to referto the geometric Frobenius morphism, i.e., the map k 3 x 7→ x1/#k ∈ k.)
Topological Groups and Topological Monoids:
For a Hausdorﬀ topological group G, we write (G ) Gab for the abelianisationof G as a Hausdorﬀ topological group, i.e., the quotient of G by the closure of thecommutator subgroup of G, and Gtors (⊂ G) for the subset of torsion elements in G.
For a commutative topological monoid M , we write (M →)M gp for the groupification of M (i.e., the coequaliser of the diagonal homomorphism M → M ×M and the zerohomomorphism), Mtors (⊂ M ) for the subgroup of torsion elements of M , M × (⊂ M ) for
the subgroup of invertible elements of M , and (M →)M pf for the perfection of M (i.e.,
the inductive limit limn∈N M , where the index set N≥1 is equipped with the order−→≥1
structure determined by divisibility, and the transition map from the copy of M at n
to the copy of M at m is given by multiplication by m/n).
For a Hausdorﬀ topological group G, and a closed subgroup H ⊂ G, we write

ZG (H) := {g ∈ G | gh = hg, ∀h ∈ H} ,{}⊂ NG (H) := g ∈ G | gHg −1 = H , and{}⊂ CG (H) := g ∈ G | gHg −1 ∩ H has finite index in H, gHg −1

for the centraliser, normaliser, and commensurator of H in G, respectively. (Note that
ZG (H) and NG (H) are always closed in G; however, CG (H) is not necessarily closed in
G (cf. [AbsAnab, Section 0], [Anbd, Section 0]).) If H = NG (H) (resp. H = CG (H)),
then we shall say that H is normally terminal (resp. commensurably terminal)
in G. (Thus, if H is commensurably terminal in G, then H is normally terminal in G.)
For a group G, we write Inn(G) (⊂ Aut(G)) for the group of inner automorphisms of
G and Out(G) := Aut(G)/Inn(G). We call Out(G) the group of outer automorphisms of
G. Let G be a group with ZG (G) = {1}. Then G → Inn(G) (⊂ Aut(G)) is injective, andwe have an exact sequence 1 → G → Aut(G) → Out(G) → 1. If f : H → Out(G) is aouthomomorphism of groups, we write G o H  H for the pull-back of Aut(G)  Out(G)14

Go Yamashita

with respect to f :1/GO/ Aut(G)
O

=

1

/G

/ Out(G)
O

/1

f
out

/G o H

/H

/ 1.

out

We shall call G o H the outer semi-direct product of H with G with respect tooutf . (Note that G o H is not necessarily naturally isomorphic to a semi-direct product.)
When G is a compact Hausdorﬀ topological group, then we equip Aut(G) with thecompact open topology and Inn(G), Out(G) with the induced topology. If, moreover,
H is a topological group, and f is a continuous homomorphism, then we equip without

G o H the induced topology.
Curves:For a field K, we write UP1 = UP1K := P1K \ {0, 1, ∞}. We shall call an algebraic
curve over K that is isomorphic to UP1K over K a tripod over K. We write Mell ⊂ Mellfor the fine moduli stack of elliptic curves and its canonical compactification.
If X is a generically scheme-like algebraic stack over a field k which has a finiteétale Galois covering Y → X, where Y is a hyperbolic curve over a finite extension ofk, then we call X a hyperbolic orbicurve over k ([AbsTopI, §0]).
Cyclotomes:
For a field K of characteristic 0 and a separable closure K of K, we write µZb (K) :=×

Hom(Q/Z, K ) and µQ/Z (K) := µZb (K) ⊗Zb Q/Z. Note that Gal(K/K) acts naturally
on both. We shall use the term cyclotome (associated to K) to refer to any of thefollowing objects: µZb (K), µQ/Z (K), µZl (K) := µZb (K) ⊗Zb Zl (for some prime numberl), µZ/nZ (K) := µZb (K) ⊗Zb Z/nZ (for some positive integer n). We shall refer to an
isomorph (in the category of topological abelian groups equipped with a continuous
Gal(K/K)-action) of any of the above cyclotomes associated to K (we mainly use thecase of µZb (K)) as a cyclotome. We write χcyc = χcyc,K (resp. χcyc,l = χcyc,l,K ) for the(full) cyclotomic character (resp. the l-adic cyclotomic character) of Gal(K/K) (i.e.,the character determined by the action of Gal(K/K) on µZb (K) (resp. µZl (K))).
§ 1.

Reduction Steps via General Arithmetic Geometry.

In this section, we apply arguments in elementary arithmetic geometry to reduce
Theorem 0.1 to a certain inequality −| log(q)| ≤ −| log(Θ)|, which will ultimately bea proof of the abc conjecture after Mochizuki

15

proved by applying the main theorem concerning the final multiradial algorithm (Section 13).
§ 1.1.

Height Functions.

Let Q be an algebraic closure of Q, X a normal, Z-proper, and Z-flat scheme. Ford ∈ Z≥1 , we write∪
X(Q) ⊃ X(Q)≤d :=X(F ).
[F :Q]≤dWrite X arc for the complex analytic space determined by X(C). An arithmetic line
bundle on X is defined to be a pair L = (L, || − ||L ), where L is a line bundle on
X, and || − ||L is a hermitian metric on the line bundle Larc := L|X arc (i.e., the line
bundle determined by L on X arc ) which is compatible with complex conjugation on
X arc . A morphism of arithmetic line bundles L1 → L2 is defined to be a morphism ofline bundles L1 → L2 such that, locally on X arc , sections of L1 that satisfy || − ||L1 ≤ 1
map to sections of L2 that satisfy || − ||L2 ≤ 1. We define the set of global sections
Γ(L) to be Hom(OX , L), where OX is the arithmetic line bundle on X determined by
the trivial line bundle equipped with the trivial hermitian metric. We write APic(X)for the set of isomorphism classes of arithmetic line bundles on X; thus, APic(X) isequipped with the group structure determined by forming tensor products of arithmetic
line bundles. If f : X → Y is a morphism of normal, Z-proper, Z-flat schemes, then we
have a natural pull-back map f ∗ : APic(Y ) → APic(X).
Let F be a number field. An arithmetic divisor (resp. R-arithmetic divisor) on∑
F is defined to be a finite formal sum a = v∈V(F ) cv v, where cv ∈ Z (resp. cv ∈ R) forv ∈ V(F )non and cv ∈ R for v ∈ V(F )arc . We shall call Supp(a) := {v ∈ V(F ) | cv 6= 0}
the support of a and say that a is eﬀective if cv ≥ 0 for all v ∈ V(F ). We write
ADiv(F ) (resp. ADivR (F )) for the group of arithmetic divisors (resp. R-arithmetic
divisors) on F . A principal arithmetic divisor is defined to be an arithmetic divisor∑∑of the form v∈V(F )non v(f )v− v∈V(F )arc [Fv : R] log(|f |v )v for some f ∈ F × . We have anatural isomorphism of groups ADiv(F )/(principal elements) ∼= APic(Spec OF ) sending∑
cv v to the line bundle determined by the rank one projective OF -module M =∏v∈V(F )∏( v∈V(F )non mvcv )−1 OF equipped with the hermitian metric on M ⊗Z C = v∈V(F )arc Fv
cv∏
determined by v∈V(F )arc e− [Fv :R] | − |Fv , where we write mv for the maximal ideal of OF
determined by v, and Fv denotes an algebraic closure of Fv . We have a (non-normalised)degree map
degF : APic(Spec OF ) ∼= ADiv(F )/(principal divisors) → R
that sends v ∈ V(F ) to fv log(pv ). We also define (non-normalised) degree maps degF :ADivR (F ) → R in the same way. For any finite extension K ⊃ F and any arithmetic

16

Go Yamashita1line bundle L on Spec OF , we have [F1:Q] degF (L) = [K:Q]
degK (L|Spec OK ); that is to1
say, the normalised degree [F :Q] degF is unaﬀected by passage to finite extensions of
F . Any non-zero element 0 6= s ∈ L of an arithmetic line bundle L = (L, || − ||L )on Spec OF determines a non-zero morphism OF → L and hence an isomorphism of
L−1 with some fractional ideal as of F . Thus, degF (L) can be computed as the degree∑∑
degF of the arithmetic divisor v∈V(F )non v(as )v − v∈V(F )arc ([Fv : R] log ||s||v )v for
any 0 6= s ∈ L, where v(as ) := mina∈as v(a), and || − ||v is the v-component of || − ||L``in the decomposition Larc ∼= v∈V(F )arc Fv ⊗R C.
= v∈V(F )arc Lv over (Spec OF )arc ∼For any arithmetic line bundle L on X, we define the (logarithmic) height function∪
htL : X(Q) =
X(F ) → R[F :Q]<∞

associated to L by setting htL (x) := [F1:Q] degF x∗F (L) for x ∈ X(F ), where xF ∈ X(OF )is the element ∈ X(F ) = X(OF ) corresponding to x (recall that X is proper over Z!),
and x∗F : APic(X) → APic(Spec OF ) is the pull-back map. By definition, we havehtL1 ⊗L2 = htL1 + htL2 for arbitrary arithmetic line bundles L1 , L2 on X ([GenEll,
Proposition 1.4 (i)]). For an arithmetic line bundle (L, || − ||L ) with ample generic fiber
LQ on XQ , it is well-known that #{x ∈ X(Q)≤d | htL (x) ≤ C} < ∞ for any d ∈ Z≥1
and C ∈ R (cf. [GenEll, Proposition 1.4 (iv)], Proposition C.1).
For functions α, β : X(Q) → R, we write α & β (resp. α . β, α ≈ β) if there
exists a constant C ∈ R such that α(x) > β(x) + C (resp. α(x) < β(x) + C, |α(x) −β(x)| < C) for all x ∈ X(Q). We call an equivalence class of functions relative to ≈ abounded discrepancy class. Note that htL & 0 ([GenEll, Proposition 1.4 (ii)]) for
any arithmetic line bunde L = (L, || − ||L ) such that the n-th tensor product L⊗n
Q of thegeneric fiber LQ on XQ is generated by global sections for some integer n > 0 (a condition
that holds if, for instance, LQ is ample). (Indeed, suppose that s1 , . . . , sm ∈ Γ(XQ , L⊗nQ )⊗ngenerate LQ . Write Ai := {si 6= 0}(⊂ X(Q)) for i = 1, . . . , m (so A1 ∪ · · · ∪ Am =⊗n

X(Q)). After tensoring Lwith the pull-back to X of an arithmetic line bundle on
Spec Z (cf. the property htL1 ⊗L2 = htL1 +htL2 mentioned above), we may assume (since
X arc is compact) that the section si extends to a section of L⊗n such that ||si ||L⊗n ≤ 1
on X arc . Then, for each i = 1, . . . , m, the non-negativity of the height htL of points∈ Ai ⊂ X(Q) may be verified by computing the height of such points by means ofsi and observing that both the Archimedean and non-Archimedean contributions to
the height are ≥ 0.) We also note that the bounded discrepancy class of the heighthtL of an arithmetic line bundle L = (L, || − ||L ) depends only on the isomorphismclass of the line bundle LQ on XQ ([GenEll, Proposition 1.4 (iii)]). (Indeed, for L1 and⊗(−1) ∼
L2 with (L1 )Q ∼= (L2 )Q , since both the line bundle (L1 )Q ⊗ (L2 )Q= OXQ and its
inverse are generated by global sections, we have htL1 − htL2 = htL ⊗L ⊗(−1) & 0 and
12

a proof of the abc conjecture after Mochizuki

17

htL2 − htL1 & 0.) When we are only interested in bounded discrepancy classes (andthere is no fear of confusion), we shall write htLQ for htL .
For x ∈ X(F ) ⊂ X(Q), where F denotes the minimal field of definition of x, thediﬀerent ideal of F determines an eﬀective arithmetic divisor dx ∈ ADiv(F ) supportedin V(F )non . We define the log-diﬀerent function log-diﬀ X on X(Q) as follows:
X(Q) 3 x 7→ log-diﬀ X (x) := [F1:Q] degF (dx ) ∈ R.
Let D ⊂ X be an eﬀective Cartier divisor. Write UX := X \ D. For x ∈ UX (F ) ⊂UX (Q), where F denotes the minimal field of definition of x, write xF ∈ X(OF ) forthe element in X(OF ) corresponding to x ∈ UX (F ) ⊂ X(F ) via the equality X(F ) =X(OF ). (Recall that X is proper over Z.) Write Dx for the pull-back of the Cartierdivisor D on X to Spec OF via xF : Spec OF → X. Thus, Dx may be regarded as
an eﬀective arithmetic divisor on F supported in V(F )non . We shall refer to fDx :=(Dx )red ∈ ADiv(F ) as the conductor of x. We define the log-conductor function
log-condD on UX (Q) as follows:
UX (Q) 3 x 7→ log-condD (x) := [F1:Q] degF (fDx ) ∈ R.
Note that the function log-diﬀ X on X(Q) depends only on the scheme XQ ([GenEll,
Remark 1.5.1]). By contrast, the function log-condD on UX (Q) depends on the pair ofZ-schemes (X, D). Nevertheless, the bounded discrepancy class of log-condD on UX (Q)depends only on the pair of Q-schemes (XQ , DQ ). (Indeed, this may be verified easily∼
by applying the fact that any isomorphism XQ → XQ0 that induces an isomorphism∼
0
DQ → DQ
extends to an isomorphism between the respective restrictions of X, X 0 to asuitable open dense subset of Spec Z ([GenEll, Remark 1.5.1]).)§ 1.2.

First Reduction.

In this subsection, we show that, to prove Theorem 0.1, it suﬃces to prove it in asituation subject to certain restrictions.
Let Q be an algebraic closure of Q. We shall say that a non-empty compact subset
of a topological space is a compact domain if it is the closure of its interior. Let
X be a normal, Z-proper, and Z-flat scheme and UX an open dense subscheme of X.
arc
Let V ⊂ VQ := V(Q) be a finite subset which contains Varc
Q . For each v ∈ V ∩ VQ
(resp. v ∈ V ∩ Vnon
Q ), let Qv be an algebraic closure of Qv , ∅ 6= Kv $ UX (Qv ) (resp.
∅ 6= Kv $ UX (Qv )) a Gal(Qv /Qv )-stable compact domain (resp. a Gal(Qv /Qv )-stable
subset whose intersection with each UX (K) ⊂ UX (Qv ), where K ranges over the finite
subextensions of Qv /Qv , is a compact domain in UX (K)). (Thus, there is a natural∼
Gal(Qv /Qv )-orbit of bijections X arc → X(Qv ).) Then we write KV ⊂ UX (Q) for thesubset of points x ∈ UX (F ) ⊂ UX (Q) where [F : Q] < ∞ such that for each v ∈ V ∩Varc
Q

18

Go Yamashita(resp. v ∈ V ∩ Vnon
Q ) the set of [F : Q] points of X(Qv ) (resp. X(Qv )) determined by
x is contained in Kv . We shall refer to a subset KV ⊂ UX (Q) obtained in this way
as a compactly bounded subset and to V as its support. Note that it follows
from the approximation theorem in elementary number theory that the Kv ’s and V arecompletely determined by KV .
Lemma 1.1.
([GenEll, Proposition 1.7 (i)]) Let f : Y → X be a generically finite
morphism of normal, Z-proper, Z-flat schemes of dimension two. Let e be a positive
integer, D ⊂ X, E ⊂ Y eﬀective, Z-flat Cartier divisors such that the generic fibers
DQ , EQ satisfy the following conditions: (a) DQ , EQ are reduced, (b) EQ = fQ−1 (DQ )red ,and (c) fQ restricts a finite étale morphism (UY )Q → (UX )Q , where UX := X \ D and
UY := Y \ E.
(1) We have log-diﬀ X |Y + log-condD |Y . log-diﬀ Y + log-condE on UY (Q).
(2) Consider the following condition: (d) the ramification index of fQ at each point of
EQ divides e. If, moreover, this condition (d) is satisfied, then we have)(log-diﬀ Y . log-diﬀ X |Y + 1 − 1e log-condD |Y
on UY (Q).
Proof. First, let us observe that there exists an open dense subscheme Spec Z[1/S] ⊂Spec Z such that the restriction of Y → X to Spec Z[1/S] is a finite tamely ramified morphism of proper smooth families of curves. Then it follows from an elementary propertyof diﬀerents that the prime-to-S portion of the equality log-diﬀ X |Y + log-condD |Y =log-diﬀ Y + log-condE and, if one assumes condition (d) to be in force, the prime-to-S()
portion of the inequality log-diﬀ Y ≤ log-diﬀ X |Y + 1 − 1e log-condD |Y hold. (If theramification index of fQ at each point of EQ is equal to e, then this inequality is anequality.) On the other hand, the S-portion of log-condE and log-condD |Y is ≈ 0, while
the S-portion of log-diﬀ Y − log-diﬀ X |Y is ≥ 0. Thus, it suﬃces to show that the Sportion of log-diﬀ Y − log-diﬀ X |Y is bounded above on UY (Q). Such a bound may beobtained as a consequence of the following claim:Fix a prime number p and a positive integer d. Then there exists a positive
integer n such that for any Galois extension L/K of finite extensions of Qp with
[L : K] ≤ d, the diﬀerent ideal of L/K contains pn OL .
This claim may be verified as follows. First, we observe that when the extension L/K
is tamely ramified, we may take n = 1. Thus, by considering the maximal tamely
ramified subextension of L(µp )/K, we reduce immediately to the case where L/K is
totally ramified p-power extension, and K contains µp . Since p-groups are solvable, wea proof of the abc conjecture after Mochizuki

19

reduce further to the case where [L : K] = p. Since K ⊃ µp , it follows from Kummer
theory that we have L = K(a1/p ) for some a ∈ K. Here, a1/p denotes a p-th root of ain L. By multiplying by a suitable element of (K × )p , we may assume that a ∈ OK anda∈/ mpK (⊃ pp OK ). Then we have OL ⊃ a1/p OL ⊃ pOL . We also have an inclusion of
OK -algebras OK [X]/(X p − a) ,→ OL given by sending X 7→ a1/p . Thus, the diﬀerent
ideal of L/K contains p(a1/p )p−1 OL ⊃ p1+(p−1) OL = pp OL . This completes the proofof the claim and hence of Lemma 1.1.
Proposition 1.2.
([GenEll, Theorem 2.1]) Fix a finite subset V non of Vnon
Q . To1
prove Theorem 0.1, it suﬃces to show the following: Write UP1 := PQ \ {0, 1, ∞}. Let
KV ⊂ UP1 (Q) be a compactly bounded subset with support V = V non ∪ Varc
Q . Then forany d ∈ Z>0 and  ∈ R>0 , we havehtωP1 ({0,1,∞}) . (1 + )(log-diﬀ P1 + log-cond{0,1,∞} )on KV ∩ UP1 (Q)≤d .
Proof. Let X, D, d,  be as in Theorem 0.1. For any e ∈ Z>0 , there exists (by thewell-known structure of étale fundamental groups of hyperbolic curves over algebraicallyclosed fields of characteristic zero) a connected étale Galois covering UY → UX suchthat the normalisation Y of X in UY is hyperbolic and the ramification index of Y → X
at each point in E := (D ×X Y )red is equal to e. (Later, we will take e suﬃciently large.)
First, we claim thatit suﬃces to show that, for any 0 ∈ R>0 , we have htωY . (1 + 0 )log-diﬀ Y on
UY (Q)≤d·deg(Y /X) .
This claim may be shown as follows: Let 0 ∈ R>0 be such that (1 + 0 )2 < 1 + . Then
we have
htωX (D) |Y . (1 + 0 )htωY . (1 + 0 )2 log-diﬀ Y. (1 + 0 )2 (log-diﬀ X + log-condD )|Y < (1 + )(log-diﬀ X + log-condD )|Y()−1
deg(D)
1
1−
on UY (Q)d·deg(Y /X) . Here, the first “.” follows fromfor e > deg(ω1+0
X (D))
the computation()
deg(E)deg(ωY ) = deg(ωY (E)) − deg(E) = deg(ωY (E)) 1 − deg(Y /X)deg(ωX (D))()
deg(D)
1
1= deg(ωY (E)) 1 − e·deg(ω> 1+
0 deg(ωY (E)) = 1+0 deg(ωX (D)|Y ),X (D))
together with the basic properties of height functions reviewed in Section 1.1; the second“.” is the hypothesis of the claim; the third “.” follows from Lemma 1.1 (2); the final

20

Go Yamashita

inequality “<” follows from the choice of 0 ∈ R>0 . Thus, the claim follows from the factthat the fiber at any point in UX (Q)≤d of the natural map UY (Q)≤d·deg(Y /X) → UX (Q)is non-empty. This completes the proof of the claim.
Thus, it suﬃces to show Theorem 0.1 in the case where D = ∅. We assume thathtωX . (1 + )log-diﬀ X is false on X(Q)=d . It follows from the compactness of X(K),
where K/Qv (v ∈ V ) is a finite extension, that there exists a subset Ξ ⊂ X(Q)=d and anunordered d-tuple of points Ξv ⊂ X(Qv ) for each v ∈ V such that htωX . (1+)log-diﬀ X
is false on Ξ, and the unordered d-tuples of Q-conjugates of points in Ξ converge toΞv in X(Qv ) for each v ∈ V . By Theorem C.2 (the existence of non-critical Belyimaps), there exists a morphism f : X → P1 such that f is unramified over UP1 , andf (Ξv ) ⊂ UP1 (Qv ) for each v ∈ V . In particular, it follows that, after possibly eliminating
finitely many elements from Ξ, there exists a compactly bounded subset KV ⊂ UP1 (Q)such that f (Ξ) ⊂ KV . Write X ⊃ E := f −1 ({0, 1, ∞})red . Let 00 ∈ R>0 be such that
1 + 00 ≤ (1 + )(1 − 200 deg(E)/deg(ωX )). Then we havehtωX ≈ htωX (E) − htOX (E) ≈ htωP1 ({0,1,∞}) |X − htOX (E). (1 + 00 )(log-diﬀ P1 |X + log-cond{0,1,∞} |X ) − htOX (E). (1 + 00 )(log-diﬀ X + log-condE ) − htOX (E). (1 + 00 )(log-diﬀ X + htOX (E) ) − htOX (E) = (1 + 00 )log-diﬀ X + 00 htOX (E). (1 + 00 )log-diﬀ X + 200 (deg(E)/deg(ωX ))htωX
on Ξ. Here, the second “≈” follows by applying the natural isomorphism ωP1 ({0, 1, ∞})|X∼→ ωX (E); the first “.” follows (since f (Ξ) ⊂ KV ) from the displayed inequality in thestatement of Proposition 1.2; the second “.” follows from Lemma 1.1 (1); the third “.”follows from the inequality log-condE . htOX (E) (discussed in more detail below); the⊗(2deg(E))
fourth “.” follows from the fact that ωX⊗ OX (−E)⊗(deg(ωX )) is ample since its
degree is equal to 2deg(E)deg(ωX ) − deg(E)deg(ωX ) = deg(E)deg(ωX ) > 0. (The inequality log-condE . htOX (E) may be proved by computing the height htOX (E) by meansof the section s ∈ Γ(X, OX (E)) corresponding to the natural inclusion OX ,→ OX (E):the Archimedean contribution to htOX (E) is bounded from below in light of the compactness of the space X arc , while the non-Archimedean contribution to htOX (E) is boundedfrom below by the corresponding non-Archimedean contribution to log-condE , as aconsequence of the “(−)red ” in the definition of log-condE .)The inequalities of the above display imply that (1 − 200 deg(E)/deg(ωX ))htωX .
(1+00 )log-diﬀ X on Ξ. Thus, the choice of 00 ∈ R>0 implies that htωX . (1+)log-diﬀ X
on Ξ. This contradicts the hypothesis on Ξ.

21

a proof of the abc conjecture after Mochizuki§ 1.3.

Second Reduction — Log-volume Computations.

In the present and following subsections, we further reduce Theorem 0.1 to a certain relation “−| log(q)| ≤ −| log(Θ)|” (which will be treated in later sections) thatarises naturally from the main results of inter-universal Teichmüller theory. It mightseem to some readers that it is unnatural and bizzare to consider such objects as
j2
2l
vj

the image “φ(q OKvj ⊗OKv (⊗0≤i≤j OKvi )∼ ) via an arbitrary automorphism φ of
j
⊗
⊗
×
×
1
Q ⊗ 0≤i≤j 2p1v logp (OK)
that
induces
anautomorphismof0≤i≤j 2pvi logp (OKvi )”vii(cf. Lemma 1.9). Moreover, since the reductions discussed in the present and following
subsections consist of just elementary calculations and contain nothing deep, it may
appear that the relation −| log(q)| ≤ −| log(Θ)| is no less diﬃcult than the inequalitythat we ultimately wish to show. However, we would like to give a complete treatmentof these elementary calculations in these subsections before proceeding to our discussionof anabelian geometry and inter-universal Teichmüller theory.
For a ∈ R, we write bac (resp. dae) for the largest integer ≤ a (resp. the smallest
integer ≥ a).
Lemma 1.3.
([IUTchIV, Proposition 1.2 (i)]) Let k be a finite extension of Qp .
We write e for the ramification index of k over Qp . For λ ∈ 1e Z, we write pλ Ok for thefractional ideal generated by any element x ∈ k with ord(x) = λ. Write
)
(
 l

m
 log p e 1 e
p > 2,p−1 
 − 1.
a := e p−2
and b := 
2
log p
e
p = 2,
Then we havepa Ok ⊂ logp (Ok× ) ⊂ p−b Ok .
If, furthermore, p > 2 and e ≤ p − 2, then pa Ok = logp (Ok× ) = p−b Ok .
1eProof. We have a > p−1, since, for p > 2 (resp. p = 2), we have a ≥ 1e p−2=111p−2 > p−1 (resp. a = 2 > 1 = p−1 ). We fix an embedding k ,→ Cp . Then we havepa Ok ⊂ p p−1 + OCp ∩ Ok ⊂ logp (Ok× ) for some  > 0, since the p-adic exponential map
1

11

converges on p p−1 + OCp and x = logp (expp (x)) for any x ∈ p p−1 + OCp for  > 0.
1

log(p

e

)

log

e

p−1
e
, since b + 1e >− 1 = logp−1
On the other hand, we have pb+ e > p−1
log p
p .
1
1
Note that b + e ∈ Z≥0 , and that b + e ≥ 1 if and only if e ≥ p − 1. Thus, we
1have (b + 1e ) + 1e > p−1, since, for e ≥ p − 1 (resp. for e < p − 1), we have (b +
1
1
1
1
1
1
1
1
e ) +{e > b + e ≥ 1 ≥ }p−1 (resp. (b + e ) + e = e > p−1 ). In summary, we have
b+ 1
1
1
1
min (b + 1e ) + 1e , 1e pb+ e > p−1. For b + 1e ∈ Z≥0 , we have (1 + p e OCp )p e $ 1 +11

p p−1 OCp , since ord((1 + p e x)p

b+ 1
e

b+ 1
e

1
− 1) ≥ min{(b + 1e ) + 1e , p e } > p−1
for x ∈ OCp .

22

Go YamashitaThus, we obtain pb+ e logp (Ok× ) ⊂ Ok ∩ logp (1 + p p−1 + OCp ) ⊂ Ok ∩ p p−1 + OCp ⊂ p e Ok
for some  > 0, which gives us the second inclusion. The last claim follows from thedefinition of a and b.
1

1

1

1

For finite extensions k ⊃ k0 of Qp , we write dk/k0 for ord(x), where x is any
generator of the diﬀerent ideal of k over k0 . For a ∈ Q, we write pa ∈ Qp for an elementof Qp with ord(pa ) = a.
Lemma 1.4.
([IUTchIV, Proposition 1.1]) Let {ki }i∈I be a finite set of finite ex∑tensions of Qp . Write di := dki /Qp . Fix an element ∗ ∈ I and write dI ∗ := i∈I\{∗} di .
Then we have
pdI ∗ (⊗i∈I Oki )∼ ⊂ ⊗i∈I Oki ⊂ (⊗i∈I Oki )∼ ,where (⊗i∈I Oki )∼ denotes the normalisation of ⊗i∈I Oki . (The tensor products aretaken over Zp .) Note that pdI ∗ (⊗i∈I Oki )∼ is well-defined for suitable pdI ∗ (for example,
products of pdi ∈ Oki for i ∈ I ∗ ).
Proof. It suﬃces to show that pdI ∗ (OQp ⊗Ok∗ ⊗i∈I Oki )∼ ⊂ OQp ⊗Ok∗ ⊗i∈I Oki ,since OQp is faithfully flat over Ok∗ . By applying induction on #I, we reduce immediately to the case where #I = 2. In this case, OQp ⊗Ok1 (Ok1 ⊗Zp Ok2 ) ∼= OQ p ⊗ Z p Ok 2 ,d2∼
and p (OQp ⊗Zp Ok2 ) ⊂ OQp ⊗Zp Ok2 holds by the definition of the diﬀerent ideal.
Lemma 1.5.
([IUTchIV, Proposition 1.3]) Let k ⊃ k0 be finite extensions of Qp .
We write e and e0 for the ramification indices of k and k0 over Qp , respectively. Let mbe the unique integer such that pm | [k : k0 ] and pm+1 - [k : k0 ]. Write dk := dk/Qp anddk0 := dk0 /Qp .
(1) We have dk0 + 1/e0 ≤ dk + 1/e. If, furthermore, k is tamely ramified over k0 , thenwe have dk0 + 1/e0 = dk + 1/e.
(2) If k is a finite Galois extension of a tamely ramified extension of k0 , then we havedk ≤ dk0 + m + 1/e0 .
Remark 1.5.1. Note that it is the quantity “log-diﬀ + log-cond”, not the quantity“log-diﬀ”, that behaves well under passage to field extensions (cf. also the proof of
Lemma 1.11 below). This is one of the reasons that the term log-cond appears inDiophantine inequalities (cf. Lemma 1.1 for the geometric case).
Proof. (1): We may replace k0 by its maximal unramified subextension in k ⊃ k0
and assume that k/k0 is totally ramified. Choose uniformisers $0 ∈ Ok0 and $ ∈ Ok
and let f (x) ∈ Ok0 [x] be the minimal monic polynomial of $ over Ok0 . Then there∼
exists an Ok0 -algebra isomorphism Ok0 [x]/(f (x)) → Ok which sends x to $. Moreover,23

a proof of the abc conjecture after Mochizuki

f (x) ≡ xe/e0 modulo mk0 = ($0 ). Thus, dk − dk0 ≥ min{ord($0 ), ord( ee0 $ e0 −1 ))} ≥
{
(
)}
(
)
min e10 , 1e ee0 − 1= 1e ee0 − 1 , where the inequalities are equalities if k/k0 is
tamely ramified.
(2): We apply induction on m. If m = 0, then (2) follows from (1). Thus, we assume
that m > 0. It suﬃces to show that dk ≤ dk0 +m+1/e0 + for all  > 0. By assumption,
k is a finite Galois extension of a tamely ramified extension k1 of k0 . By replacing k1 by
its maximal tamely ramified subextension in k ⊃ k1 and k0 by its maximal unramified
subextension in k ⊃ k0 , we may assume that [k : k1 ] = pm . Since any p-group is solvable,
there exists a subextension k ⊃ k2 ⊃ k1 such that [k : k2 ] = p and [k2 : k1 ] = pm−1 . By
the induction hypothesis, we have dk2 ≤ dk0 + (m − 1) + 1/e0 . By replacing k, k2 , and k1
by suitable tamely ramified extensions, we may assume that k1 ⊃ µp and (e2 ≥)e1 ≥ p/,
where we write e1 and e2 for the ramification indices of k1 and k2 over Qp , respectively.
By Kummer theory, there exists an inclusion of Ok2 -algebras Ok2 [x]/(xp − a) ,→ Ok
for some a ∈ Ok2 . Write a1/p ∈ Ok for the image of x by this homomorphism. By
replacing a by a suitable (k2× )p -multiple of a, we may assume that ord(a) ≤ p−1
e2 .
0 1/p
(p−1)/pThen we have dk ≤ ord(f (a )) + dk2 ≤ ord(pa) + dk0 + (m − 1) + 1/e0 ≤p−1 p−1pe2 + dk0 + m + 1/e0 < p/e2 + dk0 + m + 1/e0 ≤ dk0 + m + 1/e0 + . This completes
the proof of Lemma 1.5.
e

For a finite extension k of Qp , we write µlog
k for the (non-normalised) log-volume
function (i.e., the logarithm of the usual p-adic measure on k) defined on compact open
log
non-empty subsets of k and valued in R. Thus, we have µlog
(pOk ) =k (Ok ) = 0, µ
− log #(Ok /pOk ) = −[k : Qp ] log p. For a CAF k (cf. Section 0.2 for the definition oflog
the notion of a CAF), we write µlog
k for the radial log-volume function. Thus, µk
is defined on compact domains of k (cf. Section 1.2 for the definition of the notion oflog
a compact domain), is valued in R, and satisfies µlog
k (Ok ) = 0, µk (eOk ) = 1. The
non-normalised log-volume function µlog
k and twice the radial log-volume function, i.e.,log
2µk , may be regarded as local versions of the non-normalised degree map degF . (Indeed, note that if, for simplicity, F is a totally imaginary number field, and L is anarithmetic line bundle over Spec OF equipped with an embedding L ,→ O into the triv∑
ial arithmetic line bundle over Spec OF , then we have degF (L) = v∈V(F )non µlog
Fv (Lv ) +∑logv∈V(F )arc 2µFv (Lv ), where Lv denotes the ideal of OFv determined by L ,→ O.) Simi1
larly, the normalised log-volume function [k:Q
µlog and radial log-volume function µlog
kp] k
may be regarded as local versions of the normalised degree map [F1:Q] degF . (Indeed,
note that if, for simplicity, F , L and Lv are as above, then the normalised degree of L

24

Go Yamashita

may be written as a sum of weighted averages
∑
1
1
∑
=
deg
(L)
F
[F :Q][Fv :Qv ]V(F )3v|vQ

vQ ∈VnonQ+∑

vQ ∈Varc
Q

∑

Q

1

V(F )3v|vQ [Fv :QvQ ]∑1[Fv : QvQ ]( [Fv :Qµlog (Lv ))v ] Fv
Q

V(F )3v|vQ∑[Fv : QvQ ]µlog
Fv (Lv )V(F )3v|vQwith respect to the weights {[Fv : QvQ ]}V(F )3v|vQ .)For finite extensions {ki }i∈I of Qp , the sum of the normalised log-volume functionslog
1
1{ [ki :Qp ] µlogki }i∈I determines (since, for any i ∈ I, we have [ki :Qp ] µki (pOki ) = − log p) a
log-volume function µlog
kI on compact open non-empty subsets of kI := ⊗i∈I ki (where thetensor products are taken over Qp ) valued in R, normalised so that µlog
kI (⊗i∈I Oki ) = 0.
log
log
1
(Here, we note that µk{i} = [ki :Qp ] µki .) For CAF’s {ki }i∈I , we consider the tensor
product kI := ⊗i∈I ki as a tensor product of topological rings. Then kI decomposes
as a direct sum (or, equivalently, as a direct product) of CAF’s. Write BI ⊂ kI for
the subset determined by forming the direct product of the unit balls on the various
direct summands. Then we define the log-volume function µlog
kI on compact domains
in kI , valued in R, by forming the sum of the radial log-volumes on the various direct
summands. Thus, we have µlog
kI (BI ) = 0.
Lemma 1.6.
([IUTchIV, Proposition 1.2 (ii), (iv)], [IUTchIV, Proposition 1.4(iii), (iv)] and [IUTchIV, “together with the fact...” in Steps (v) and (vi) in the proofof Theorem 1.10]) Let {ki }i∈I be a finite set of finite extensions of Qp . We write ei
for the ramification index of ki over Qp ; ai and bi for the quantities a and b defined∑∑in Lemma 1.3 for ki , respectively; di := dki /Qp , aI :=i∈I bi , andi∈I ai , bI :=∑1
λdI := i∈I di . For λ ∈ ei Z, we write p Oki for the fractional ideal generated by any⊗∼ ⊗element x ∈ ki with ord(x) = λ. Let φ : i∈I logp (Ok×i ) → i∈I logp (Ok×i ) (where thetensor products are taken over Zp ) be an automorphism of Zp -modules. Thus, φ extends⊗
uniquely to a Qp -linear automorphism of the Qp -vector space Qp ⊗Zp i∈I logp (Ok×i ).
⊗We consider (⊗i∈I Oki )∼ as a submodule of Qp ⊗Zp i∈I logp (Ok×i ) via the natural isomorphisms Qp ⊗Zp (⊗i∈I Oki )∼ ∼= Qp ⊗Zp ⊗i∈I Oki ∼= Qp ⊗Zp ⊗i∈I logp (Ok×i ).
(1) Write I ⊃ I ∗ := {i ∈ I | ei > p − 2}. For any λ ∈ e1i Z, i0 ∈ I, we have0(
) ∪ bλc ⊗ 1
φ pλ (⊗i∈I Oki )∼plogp (Ok×i )2pi∈I⊗⊂ pbλ−dI −aI clogp (Ok×i ) ⊂ pbλ−dI −aI c−bI (⊗i∈I Oki )∼ , andi∈I
bλ−dI −aI c−bIµlog(⊗i∈I Oki )∼ ) ≤ (−λ + dI + 1) log(p) +
kI (p

∑
i∈I ∗

(3 + log(ei )).

a proof of the abc conjecture after Mochizuki

25(2) If p > 2 and ei = 1 for each i ∈ I, then we haveφ((⊗i∈I Oki )∼ ) ⊂

⊗ 1
i∈I

2p

logp (Ok×i ) = (⊗i∈I Oki )∼ ,∼and µlogkI ((⊗i∈I Oki ) ) = 0.
1
Remark 1.6.1.
If, for simplicity, ei < p − 2, then we have (Oki ⊂) 2p
logp (Ok×i ) =1
p mki (cf. Lemma 1.3), where we write mki for the maximal ideal of ki . Then observethat1
if we consider 2p
logp (Ok×i ) = p1 mki as a Zp -module (i.e., if we regard its submodule Oki as not being equipped with a ring structure), then, at least from an a priori point of view, there is no evident intrinsic way (i.e., e.g., no intrinsically defined algorithm that allows one) to distinguish the elements p1 $, p1 $2 , . . . , p1 $e ,where we write $ for a uniformiser of ki . One can consider this phenomenon as a kindof “diﬀerentiation over F1 ” (cf. also the point of view of Teichmüller dilations
discussed in Section 3.5).
⊗
Proof. (1): We have pdI +aI (⊗i∈I Oki )∼ ⊂ paI ⊗i∈I Oki ⊂ i∈I logp (Ok×i ), where
the first (resp. second) inclusion follows from Lemma 1.4 (resp. Lemma 1.3). Then by
Lemma 1.3, we havepλ (⊗i∈I Oki )∼ = pλ−dI −aI pdI +aI (⊗i∈I Oki )∼ ⊂ pbλ−dI −aI c pdI +aI (⊗i∈I Oki )∼⊂ pbλ−dI −aI c logp (⊗i∈I Ok×i ) ⊂ pbλ−dI −aI c−bI (⊗i∈I Oki )∼ .
Thus, we have
)
(
⊗
( λ
)
logp (Ok×i )φ p (⊗i∈I Oki )∼ ⊂ φ pbλ−dI −aI cbλ−dI −aI c

=p

⊗

i∈I

logp (Ok×i ) ⊂ pbλ−dI −aI c−bI (⊗i∈I Oki )∼ .

i∈I

If p = 2, then we have ddI + aI e ≥ dI + aI ≥ aI = 2 · (#I). If p > 2, then we haveai ≥ e1i , and di ≥ 1 − e1i by Lemma 1.5 (1), hence we have dI + aI ≥ #I. Thus, we⊗⊗
1
obtain the remaining inclusion pbλc i∈I 2p
logp (Ok×i ) ⊂ pbλ−dI −aI c i∈I logp (Ok×i ) forp ≥ 2.
2
Next, we show the upper bound on the log-volume. We have ai − e1i < p4 < log(p),where the first inequality for p > 2 (resp. p = 2) follows from the inequalities ai <
ei
1
1
1
1
4
1
1
4
ei ( p−2 + 1) = p−2 + ei and p−2 < p for p > 2 (resp. ai − ei = 2 − ei < 2 = p ),and the second inequality follows from the fact that x > 2 log x for x ≥ 2. We also

26

Go Yamashitapei
have (bi + e1i ) log(p) ≤ log( p−1) ≤ log(2ei ) < 1 + log(ei ), where the first inequalitypfollows from the definion of bi , the second inequality follows from the fact that p−1≤2
for p ≥ 2, and the third inequality follows from the fact that log(2) < 1. Thus, by
combining these inequalities, we obtain the inequality (ai + bi ) log(p) ≤ 3 + log(ei ). Fori ∈ I \ I ∗ , we have ai = −bi (= 1/ei ), hence we have (ai + bi ) log(p) = 0. Thus, webλ−dI −aI c−bI
obtain µlog(⊗i∈I Oki )∼ ) ≤ (−(λ − dI − aI − 1) + bI ) log(p) = (−λ + dI +kI (p∑aI + bI + 1) log(p) ≤ (−λ + dI + 1) log(p) + i∈I ∗ (3 + log(ei )).
(2) follows from (1) and Lemma 1.3.

For a non-Archimedean local field k, we write Ik := 2p1v logp (Ok× ). For a CAF k,Qwe write Ik := {a ∈ k | |a|k ≤ π}. If k is a non-Archimedean local field or a CAF, thenwe shall refer to Ik as the log-shell of k. Let F be a totally complex number field andvQ ∈ Vnon
Q . For V(F ) 3 v1 , . . . , vn | vQ , write
Iv1 ,··· ,vn := ⊗1≤i≤n IFvi ⊂ IvQ1 ,··· ,vn := ⊗1≤i≤n Fvi .
(Here, the tensor products are taken in the category of topological modules.) LetvQ ∈ Varc
Q . For V(F ) 3 v1 , . . . , vn | vQ , we write
Iv1 ,...,vn := π n B{1,...,n} ⊂ IvQ1 ,...,vn := ⊗1≤i≤n Fvi ,where we take the “ki ” in the definition of BI to be Fvi . (Here, the tensor products
are taken in the category of topological modules.) Let vQ ∈ Vnon (resp. vQ ∈ Varc ).
For a subset A ⊂ IvQ1 ,··· ,vn , we define the holomorphic hull of A to be the smallest
subset containing A of the form ⊕i∈I ai OLi with ai ∈ L×i in the natural direct sum
decomposition of the topological fields ⊗1≤i≤n Fvi ∼= ⊕i∈I Li .
For a CAF k, we define the group Aut(k)prim of primitive automorphisms of theunderlying additive topological module of k to be the subgroup generated by the complex√
conjugation and the multiplication by −1. (Thus, Aut(k)prim ∼= Z/4Z o {±1}.) We
prim
prim
also define the subgroup Aut(k)⊂ Aut(k)of -primitive automorphismsof the underlying additive topological module of k to be the subgroup {α ∈ Aut(k)prim |α(1) = ±1} ⊂ Aut(k)prim . (Thus, Aut(k)prim ∼= {±1} × {±1}.)
In the rest of this subsection, we consider a collection of data (F /F, EF , Vbad
mod , l, V)
satisfying the following conditions:√(1) F is a number field such that −1 ∈ F , and F is an algebraic closure of F .
(2) EF is an elliptic curve over F , whose origin we denote by 0E , such that AutF (EF ) ={±1}, where EF := EF ×F F , and AutF (EF ) denotes the group of F -automorphismsof the scheme EF that preserve 0E ; the 2 · 3 · 5 (= 30)-torsion points of EF arerational over F ; F is Galois over the field of moduli Fmod of EF , i.e., the subfield of F consisting of elements fixed by the image of the natural homomorphisma proof of the abc conjecture after Mochizuki

27

Aut(EF ) → Aut(F ) = Gal(F /Q) (⊃ Gal(F /F )) (so we have a short exact sequence
1 → AutF (EF ) → Aut(EF ) → Gal(F /Fmod ) → 1), where we write Aut(EF ) for
the group of automorphisms of the scheme EF that preserve 0E .
badnon(3) Vbadmod is a non-empty finite subset Vmod ⊂ Vmod (⊂ Vmod := V(Fmod )), such thatv - 2 holds for each v ∈ Vbad
mod , and EF has bad multiplicative reduction overw ∈ V(F )v .

(4) l is a prime number l ≥ 5 such that l is prime to the elements of Vbad
mod , as well as
badto w(−) of the q-parameters of EF at w ∈ V(F ):= V(F ) ×Vmod Vbad
mod .
(5) V is a subset V ⊂ V(K), where K := F (EF [l]) ⊂ F denotes the extension fieldof F generated by the fields of definition of the l-torsion points of EF , such thatthe restriction of the natural surjection V(K)  Vmod to V induces a bijection∼
V → Vmod .
(Note that this is not the definition of initial Θ-data, which includes more objects andconditions, as discussed in Definition 10.1.) Write
dmod := [Fmod : Q],
good
bad
good
(Varc
:= V(F ) ×Vmod Vgoodmod ⊂) Vmod := Vmod \ Vmod , and V(F )modAlso, we shall write v ∈ V for the element corresponding to v ∈ Vmod via the bijectionof (5).
Lemma 1.7.

([IUTchIV, Lemma 1.8 (ii), (iii), (iv), (v)])(1) Let EFmod be a model of EF over Fmod . Write Ftpd := Fmod (EFmod [2]) ⊂ F for theextension field of Fmod generated by the fields of definition of the 2-torsion points of
EFmod . Then Ftpd is independent of the choice of a model EFmod of EF over Fmod .
(Here, “tpd” stands for “tripod”, i.e., the projective line minus three points.)(2) The elliptic curve EF has semistable reduction at all w ∈ V(F )non .
(3) Any model EF† of EF over F such that the 3-torsion points of EF† are defined overF is isomorphic to EF over F . In particular, for any model EFtpd of EF over Ftpd
such that the 3-torsion points of EFtpd are defined over F , it holds that EFtpd ×Ftpd F
is isomorphic to EF over F .
(4) The extension K ⊃ Fmod is Galois.
Proof. (1): Consider the short exact sequence
1 → AutF (EF ) → Aut(EF ) → Gal(F /Fmod ) → 1.

28

Go Yamashita

One verifies immediately that there exists a natural bijective correspondence between
sections of the surjection Aut(EF )  Gal(F /Fmod ) and models EFmod of EF over Fmod ,and that the subfield Fmod (EFmod [2]) ⊂ F determined by such a model is the fieldof invariants with respect to the kernel of the composite of the corresponding section
Gal(F /Fmod ) ,→ Aut(EF ) and the natural homomorphism Aut(EF ) → Aut(EF [2]). On
the other hand, the assumption AutF (EF ) = {±1} implies that the natural homomorphism Aut(EF ) → Aut(EF [2]) factors through the quotient Aut(EF )  Gal(F /Fmod )(since the action of AutF (EF ) = {±1} on EF [2] is trivial, i.e., −P = P for P ∈EF [2]). This implies that the kernel of the composite Gal(F /Fmod ) ,→ Aut(EF ) →Aut(EF [2]) is independent of the choice of a section Gal(F /Fmod ) ,→ Aut(EF ), i.e.,that Fmod (EFmod [2]) is independent of the choice of model EFmod . This completes theproof of (1).
(2): For a prime r ≥ 3, there exists a fine moduli scheme X(r)Z[1/r] over Z[1/r] of
elliptic curves with level r structure. (Note that it is a scheme since r ≥ 3.) Moreover,since X(r)Z[1/r] is proper over Z[1/r], any Fw -valued point with w - r arises from aunique OFw -valued point. We apply this to the Fw -valued point defined by EF equippedwith some level r = 3 (resp. r = 5) structure (which is defined over F by condition (2)).
Thus, EF has semistable reduction for w - 3 (resp. w - 5). This completes the proof of(2).
(3): First, observe that since the image of −1 ∈ AutF (EF ) = {±1} in Aut(EF [3])}is nontrivial, we have AutF (EF ) ∩ ker(ρ) = {1}. (Note that if −P = P ∈ EF [3] thenP ∈ EF [2] ∩ EF [3] = {O}.) Next, recall from the proof of (1) that models of EF
over F correspond bijectively to sections of AutF (EF )  Gal(F /F ). Relative to thiscorrespondence, a model of EF over F all of whose 3-torsion points are rational over F
corresponds to a section of AutF (EF )  Gal(F /F ) whose image is contained in ker{ρ :AutF (EF ) → Aut(EF [3])}. Thus, it follows from the fact that AutF (EF )∩ker(ρ) = {1}that such a section is uniquely determined. This completes the proof of (3).
(4): Consider the twist (EF )g := EF ×F,g F of EF by any g ∈ Gal(F /Fmod ). (Notethat F is Galois over Fmod by condition (2).) Then, by transport of structure, the3-torsion points of (EF )g are defined over F , and F ((EF )g [l]) = K g := K ⊗F,g F . Thus,
it follows from (3) that (EF )g is isomorphic to EF over F . Hence K g is isomorphic to∼g∼K over F , which implies that we have an isomorphism K → K over F → F . Thus, K
is Galois over Fmod . This completes the proof of (4).
We further assume that(6) EF has good reduction at all v ∈ V(F )good ∩ V(F )non such that v - 2l, and√(7) F = Ftpd ( −1, EFtpd [3·5]). (Here, EFtpd is a model of EF over Ftpd which is definedby an equation in Legendre form, i.e., an equation of the form y 2 = x(x − 1)(x − λ)a proof of the abc conjecture after Mochizuki

29with λ ∈ Ftpd .)For an intermediate extension Fmod ⊂ L ⊂ K which is Galois over Fmod , we writed ∈ ADiv(L) for the eﬀective arithmetic divisor supported in V(L)non determined by1
the diﬀerent ideal of L over Q. We define log(dL ) := [L:Q]
degL (dL ) ∈ R≥0 . It makes
sense to consider the q-parameters of EF at bad places since EF has semistable reduction
at all w ∈ V(F )non by Lemma 1.7 (2). For an intermediate extension F ⊂ L ⊂ K which
is Galois over Fmod , we write qL ∈ ADivR (L) for the eﬀective R-arithmetic divisor
supported in V(L)non determined by the q-parameters of EL := EF ×F L at primes inbad
V(L)bad := V(L) ×Vmod Vbad
. (Note that
mod , and qv ∈ Kv for the q-parameter at v ∈ V
L
2l is prime to the elements in Supp(q ) even if EF has bad reduction over a place dividing
12l.) We define log(qL ) := [L:Q]
degL (qL ) ∈ R≥0 . Since log(qL ) does not depend on L, weshall write log(q) for log(qL ). We write fL ∈ ADiv(L) for the eﬀective arithmetic divisor
whose support coincides with Supp(qL ), and whose coeﬃcients are all equal to 1. (Note1degL (fL ) ∈ R≥0 .
that 2l is prime to the elements in Supp(fL ).) We define log(fL ) := [L:Q]
For an intermediate extension Ftpd ⊂ L ⊂ K which is Galois over Fmod , we writeLwhich is ramified over Q}.
V(L)dist := {w ∈ V(L)non | ∃v ∈ V(K)nonw
dist
and Vdist
in VQ and in Vmod , respectively,
We write Vdist
Q
mod for the images of V(Ftpd )via the natural surjections V(Ftpd )  Vmod  VQ . For L = Q, Fmod , we write∑
ew w ∈ ADiv(L),sL :=w∈V(L)dist1
where ew denotes the ramification index of Lw /Qpw . We define log(sL ) := [L:Q]
degL (sL ) ∈R≥0 . We write emod for the maximal ramification index of Fmod over Q. We writed∗mod := 2 · #(Z/4Z)× · #GL2 (F2 ) · #GL2 (F3 ) · #GL2 (F5 ) · dmod = 212 · 33 · 5 · dmod ,e∗mod := 2 · #(Z/4Z)× · #GL2 (F2 ) · #GL2 (F3 ) · #GL2 (F5 ) · emod = 212 · 33 · 5 · emod .
(Note that #GL2 (F2 ) = 2 · 3, #GL2 (F3 ) = 24 · 3, and #GL2 (F5 ) = 25 · 3 · 5.) We write
∑
ιvQ
s≤ :=
vQ ∈ ADivR (Q),
log(p
)
v
Q
dist
vQ ∈VQ

where ιvQ := 1 if pvQ ≤ e∗mod l and ιvQ := 0 if pvQ > e∗mod l. We define log(s≤ ) :=degQ (s≤ ) ∈ R≥0 .
∑For L a finite extension of F , a = w∈V(L) cw w ∈ ADivR (L), and v ∈ V(F ), we∑
define av := w∈V(L)v cw w.
Lemma 1.8.
([IUTchIV, Proposition 1.8 (vi), (vii)]) The extension F/Ftpd istamely ramified outside 2 · 3 · 5 and unramified outside 2 · 3 · 5 · l and Supp(qFtpd ). The

30

Go Yamashita

extension K/F is tamely ramified outside l and unramified outside 2 · l and Supp(qFtpd ).
In particular, the extension K/Ftpd is unramified outside 2 · 3 · 5 · l and Supp(qFtpd ).
Proof. First, we show that EFtpd ×Ftpd F 0 has semistable reduction at w - 2 forsome extension F 0 ⊃ Ftpd,w with [F 0 : Ftpd,w ] ≤ 2 as follows: EFtpd is defined by anequation in Legendre form y 2 = x(x − 1)(x − λ) for λ 6= 0, 1 ∈ Ftpd . If λ ∈ OFtpd,w ,then it has semistable reduction since 0 6≡ 1 in any characteristic. If λ 6∈ OFtpd,w , thenαλ ∈ OF×tpd,w for some 0 6= α ∈ OFtpd,w , so by putting x0 := αx and y 0 := α3/2 y,√
we obtain a defining equation (y 0 )2 = x0 (x0 − α)(x0 − αλ) over Ftpd,w ( α), which has
semistable reduction. Moreover, if the model of EFtpd,w defined by this equation has
good reduction, then α ∈ OF×tpd,w , so λ ∈ OF×tpd,w , a contradiction.
Take an algebraic closure Ftpd,w of Ftpd,w . Then the action of the inertia subgroupof Gal(Ftpd,w /F 0 ) on E[3 · 5] is unipotent (cf. [SGA7t1, Exposé IX, Proposition 3.5])√
for w - 2 · 3 · 5. Hence F = Ftpd ( −1, E[3 · 5]) is tamely ramified over Ftpd outside
2 · 3 · 5 and unramified outside 2 · 3 · 5 · l and Supp(qFtpd ) (cf. the assumption (6) and thedefinition of Supp(qFtpd )). By a similar argument, since EF has semistable reduction
at all w ∈ V(F )non by Lemma 1.7 (2), the action of the inertia subgroup of Gal(F /F )at w - l on E[l] is unipotent. Thus, K = F (E[l]) is tamely ramified over F outside land unramified outside 2 · l and Supp(qFtpd ) (cf. the assumption (6) and the definitionof Supp(qFtpd )).
∼In inter-universal Teichmüller theory, we will think of the bijection V → Vmod as akind of “analytic section” of the natural morphism Spec OK  Spec OFmod . From thislog
point of view, it is natural to think of [Kv :(F1mod )v ] µlog
Kv (resp. µ⊗v∈V Kv ) as a sort of stacklog
theoretic version of µlog
(Fmod )v (resp. µ⊗v∈Vmod (Fmod )v ). (Note that here the tensor productis taken with respect to V, not V(K). See also Definition 10.4 and Definition 10.5 (4).)
In particular, when considering various weighted sums, we shall regard [Kv :(F1mod )v ] µlog
Kv
log
1
or its normalised version [(Fmod1)v :Qv ] [Kv :(F1mod )v ] µlog
Kv = [Kv :Qv ] µKv for v ∈ V (not for
Q

V(K)) as having weight [(Fmod )v : QvQ ] (not [Kv : QvQ ]).

Q

Lemma 1.9.
([IUTchIV, some portions of Steps (v), (vi), (vii) of the proof of
Theorem 1.10, and Proposition 1.5]) For vQ ∈ VQ , 1 ≤ j ≤ l> (= l−12 ), and v0 , . . . , vj ∈(Vmod )vQ (where v0 , . . . , vj are not necessarily distinct), we write −| log(Θ)|{v0 ,...,vj } forthe log-volume (i.e., µlog⊗0≤i≤j Kv ) of the following:i• (Non-Archimedean vertical (Indet ↑), horizontal (Indet →), and permutation(Indet xy) indeterminacies) For vQ ∈ Vnon
Q , the holomorphic hull of the union of2

φ(qvj j /2l Iv0 ,··· ,vj )(resp. φ(Iv0 ,··· ,vj ) )31a proof of the abc conjecture after Mochizuki

for v j ∈ Vbad (resp. for v j ∈ Vgood ), and
( 2
)
j /2l
∼
φ qvj OKvj ⊗OKv (⊗0≤i≤j OKvi )

(

∼

)

(resp. φ (⊗0≤i≤j OKvi )

j

)

∼

for v j ∈ Vbad (resp. for v j ∈ Vgood ), where φ : QvQ ⊗ZvQ Iv0 ,...,vj → QvQ ⊗ZvQ Iv0 ,...,vj
varies over the automorphisms of finite-dimensional QvQ -vector spaces that stabilize
the submodule Iv0 ,...,vj , and the tensor products ⊗0≤i≤j are taken over ZvQ (cf. also
the discussion of the “ Teichmüller dilation” in Remark 1.6.1 and Section 3.5).
• (Archimedean vertical (Indet ↑), horizontal (Indet →), and permutation (Indetxy) indeterminacies) For vQ ∈ Varc
Q , the holomorphic hull of the union ofφ(Iv0 ,...,vj ),and(⊗0≤i≤j φi )(BI ),∏where (φi )0≤i≤j varies over the elements of 0≤i≤j Aut(Kvi )prim , and we recalljjthat BI = (unit ball)⊕2 in the natural direct sum decomposition ⊗0≤i≤j Kv ∼= C⊕2i(where the tensor products are taken as topological modules).
∑Write di := dKvi /QvQ and dI := 0≤i≤j di for vQ ∈ Vnon
Q . Then we have the following
upper bounds of −| log(Θ)|{v0 ,...,vj } :(1) For vQ ∈ Vdist
Q , we have( 2)
 − j ord(qv ) + dI + 1 log pv + 4(j + 1)ιv log(e∗ l),
Q
Q
mod
2l
j
− | log(Θ)|{v0 ,...,vj } ≤∗(d + 1) log p + 4(j + 1)ι log(e
l),
I

vQ

vQ

mod

if v j ∈ Vbad ,if v j ∈ Vgood .

dist(2) For vQ ∈ Vnon
Q \ VQ , we have −| log(Θ)|{v0 ,...,vj } ≤ 0.

(3) For vQ ∈ Varc
Q , we have −| log(Θ)|{v0 ,...,vj } ≤ (j + 1) log(π).
Remark 1.9.1. In Section 13, we shall see that the vertical (resp. horizontal)
indeterminacy arises from the vertical (resp. horizontal) arrows of the log-theta-lattice,i.e., the log-links (resp. the theta-links), while the permutation indeterminacy arises
from the permutation symmetry of the étale picture.
2

Proof. (1): We apply Lemma 1.6 (1) to λ := j2l ord(qvj ) (resp. 0) for v j ∈ Vbad
(resp. for v j ∈ Vgood ), I := {0, 1, . . . , j}, i0 := j, and ki := Kvi . (Note that λ ∈ ev1 Z
j1/2lsince qvj

∈ Kvj by the assumptions that K = F (EF [l]) and that EF [2] is rational over F , i.e., F = F (EF [2]).) Then by the first inclusion of Lemma 1.6 (1), both

32

Go Yamashita

( 2
)
(
)
j /2l
φ qvj OKvj ⊗OKv (⊗0≤i≤j OKvi )∼ (resp. φ (⊗0≤i≤j OKvi )∼ ) ((Indet →), (Indetjj 2 /2lbλc−ddI e−daI exy)) and qvj Iv0 ,...,vj (resp. Iv0 ,...,vj ) ((Indet ↑)) are contained in pvQ⊗i∈I×logpv (OKv ). By the second inclusion of Lemma 1.6 (1), the holomorphic hull ofQibλc−ddI e−daI e
bλc−ddI e−daI e−dbI e
××
pvQ⊗i∈I logpv (OK) is contained in pvQ(⊗i∈I OK)∼ , and
v
vi
Q
i

∑
its log-volume is ≤ (−λ + dI + 1) log(pvQ ) + i∈I ∗ (3 + log(ei )) by Lemma 1.6 (1), whereei denotes the ramification index of Kvi over Qpvi . If ei > pvQ − 2, then pvQ ≤ e∗mod lsince, for v i - l (resp. v i | l), we have pvQ ≤ 1 + ei ≤ 1 + e∗mod l/2 ≤ e∗mod l (resp.
pvQ = l ≤ e∗mod l). For ei > pvQ − 2, we also have log(ei ) ≤ −3 + 4 log(e∗mod l) since√ei ≤ emod [F (= Ftpd ( −1, EFtpd [3 · 5])) : Ftpd ][K : F ] ≤ emod · #(Z/4Z)× · #GL2 (F3 ) ·#GL2 (F5 ) · #GL2 (Fl ) < e∗mod (l2 − 1)(l2 − l) < e∗mod l4 and e3 ≤ (e∗mod )3 . Thus, we have∑(−λ+dI +1) log(pvQ )+ i∈I ∗ (3+log(ei )) ≤ (−λ+dI +1) log(pvQ )+4(j+1)ιvQ log(e∗mod l),since if ιvQ = 0, (i.e., pvQ > e∗mod l), then ei ≤ pvQ − 2 for all i, hence I ∗ = ∅. The lastequality of the claim follows from the definitions.
dist(2): For vQ ∈ Vnon
Q √\ VQ , the place vQ is unramified in K and vQ 6= 2, since 2
ramifies in K by K 3 −1. Thus, the ramification index ei of Kvi over QvQ is 1 foreach 0 ≤ i ≤ j, and pvQ > 2.
( We apply Lemma) 1.6 (2) to λ := 0, I := {0, 1, . . . , j},and ki := Kvi . Both of φ (⊗0≤i≤j OKvi )∼((Indet →), (Indet xy)) and the log-×
shell Iv0 ,...,vj (Indet ↑) are contained in ⊗i∈I logpv (OK). By the second inclusion of
v
Q

i

×
×
Lemma 1.6 (2), the holomorphic hull of ⊗i∈I logpv (OK) is contained in (⊗i∈I OK
)∼ ,
vi
vi
Q
and its log-volume is = 0.
j(3): The natural direct sum decomposition ⊗0≤i≤j Kvi ∼= C⊕2 (where the tensor
products are taken as topological modules), where Kvi ∼= C, the hermitian metric on
j
⊕2j
⊕2j
C , and the integral structure BI = (unit ball)⊂ C⊕2 are preserved by the au∏
tomorphisms of ⊗0≤i≤j Kvi induced by any (φi )0≤i≤j ∈ 0≤i≤j Aut(Kvi )prim ((Indet→), (Indet xy)). Note that, via the natural direct sum decomposition ⊗0≤i≤j Kvi ∼=⊕(j+1)⊕(j+1)C, the direct sum metric on C
induced by the standard metric on C isj
2 times the tensor product metric on ⊗0≤i≤j Kvi induced by the standard metric on
√
√
√
Kvi ∼
= C (Note that |1⊗ −1|2C⊗R C = 1 and |( −1, − −1)|2C⊕C = 2) (cf. also [IUTchIV,
Proposition 1.5 (iii), (iv)]). The log-shell Iv0 ,...,vj is contained in π j+1 BI (Indet ↑).
Thus, an upper bound of the log-volume is given by (j + 1) log(π).

Lemma 1.10.
([IUTchIV, Proposition 1.7, and some portions of Steps (v), (vi),(vii) in the proof of Theorem 1.10]) Fix vQ ∈ VQ . For 1 ≤ j ≤ l> (= l−12 ), we takethe weighted average −| log(Θ)|vQ ,j of −| log(Θ)|{v0 ,...,vj } with respect to all (j + 1)∏
tuples of elements {vi }0≤i≤j in (Vmod )vQ with weight wv0 ,...,vj := 0≤i≤j wvi , wherewv := [(Fmod )v : QvQ ] (not [Kv : QvQ ]), i.e.,∑1−| log(Θ)|vQ ,j := W
wv0 ,...,vj (−| log(Θ)|{v0 ,...,vj } ),v0 ,...,vj ∈(Vmod )vQ

33

a proof of the abc conjecture after Mochizuki∑= ( v∈(Vmod )v wv )j+1 = [Fmod : Q]j+1 , andQ
v0 ,...,vj ∈(Vmod )vQ is the summation of all (j + 1)-tuples of (not necessarily distinct)∑
elements v0 , . . . , vj ∈ (Vmod )vQ . (We write
for it from now on to lighten the nowhere W :=∑∑v0 ,...,vj ∈(Vmod )vQ wv0 ,...,vj

v0 ,...,vj

tation.) We write −| log(Θ)|vQ for the average of −| log(Θ)|vQ ,j with respect to 1 ≤j ≤ l> , (which is called procession normalised average), i.e., −| log(Θ)|vQ :=∑11≤j≤l> (−| log(Θ)|vQ ,j ).
l>(1) For vQ ∈ Vdist
Q , we have
K
Q
≤
∗
l+5
−| log(Θ)|vQ ≤ − l+1
24 log(qvQ ) + 4 log(dvQ ) + log(svQ ) + (l + 5) log(svQ ) log(emod l).

dist(2) For vQ ∈ Vnon
Q \ VQ , we have −| log(Θ)|vQ ≤ 0.

(3) For vQ ∈ Varc
Q , we have −| log(Θ)|vQ ≤ l + 1.
Remark 1.10.1.

When we think of [Kv :(F1mod )v ] µlog
Kv as a stack theoretic version ofµlog(Fmod )v and identify V with Vmod as explained before, the weighted average
∑

1
W

∑

wv0 ,...,vj

v0 ,...,vj

µlog
Kv

0≤i≤j

i

[Kvi :QvQ ]corresponds to

1
W

∑

∑

1
wv0 ,...,vj [(Fmod )v :Qiv ] = Wi

0≤i≤j v0 ,...,vj

Q



∑

µlog
(Fmod )v

∑



0≤i≤j

j

v∈(Vmod )vQ∑

j+1= [Fmod
:Q]

∑

wv 

µlog

(Fmod )vwv [(Fmod)v :Qv ]v∈(Vmod )vQQµlog(Fmod )v ,v∈(Vmod )vQ
1
which calculates (j +1) times the vQ -part of the normalised degree map [Fmod:Q] degFmod .

Proof. (1): The weighted average of the upper bound of Lemma 1.9 (1) gives us
2

1 j−| log(Θ)|vQ ,j ≤ − W
2l

∑

µlog
Kv (qv j )wv0 ,...,vj [Kv j :Qv ]jv0 ,...,vj
1+W∑
v0 ,...,vj

wv0 ,...,vj

∑
0≤i≤j

(

Q

K
µlog
Kv (dv )i[Kvi :QvQ ] +
i

Q
µlog
Qv (sv )
Q

j+1

Q

)
≤
∗
+ 4µlog
Qv (svQ ) log(emod l) .
Q

34

Go Yamashita

∑

2

1 j
Now, − W2lµlog
Kv (qv j )wv0 ,...,vj [Kv j :Qv ] is equal to



j

∑

1 j 
−W
2l
2

Q

j

v0 ,...,vj

∑

wv 

v∈(Vmod )vQQv∈(Vmod )vQ∑2= − [Fmod :Q] j2l

1µlog
K (qv )

wv [Kvv:Qv ]µlog
Kv (qv )∑2

j1[Kv :(Fmod )v ] = − [Fmod :Q] 2lv∈(Vmod )vQ

∑

2

1 j
= − [K:Q]
2llog[Kv :(Fmod )v ] µKw (qw )[K:Fmod ] [Kv :(Fmod )v ]w∈V(K)vQ2

jµlog
Kw (qw ) = − 2l log(qvQ ),w∈V(K)vQ
log
where the second equality follows from that µlog
Kw (qw ) = µKv (qv ), [Kw : (Fmod )v ] = [Kv :mod ](Fmod )v ], and #V(K)v = [K[K:F
for any w ∈ V(K)v with a fixed v ∈ Vmod , since K
v :(Fmod )v ]∑∑ µlog(dKv )K
1
is Galois over Fmod (Lemma 1.7 (4)). On the other hand, W
wv0 ,...,vj( [Kvvi:Qv i ] +v0 ,...,vjQµlog
Qv (sv )
Q

Q

j+1

1
W

∑

0≤i≤j

i

Q

≤
∗
+ 4µlog
Qv (svQ ) log(emod l)) is equal to




0≤i≤j
j+1
= [Fmod
:Q]

Q

j

∑

wv 

v∈(Vmod )vQ

(

∑

wv

v∈(Vmod )vQ
j+1= [Fmod
:Q]

∑

∑

(
wv

[Kv :QvQ ] +Qi[Kvi :QvQ ] +iv∈(Vmod )vQKµlog
Kv (dv )Qµlog
Qv (sv )Kµlog
Kv (dv )Qµlog
Qv (sv )
Q

j+1

Q

j+1

Q

)
∗
≤
+ 4µlog
Qv (svQ ) log(emod l)
Q)∗≤+ 4µlog
Qv (svQ ) log(emod l)
Q

K
µlog
Kv (dv )

log
log
∗
≤
Q
[Kv :(Fmod )v ] + µQvQ (svQ ) + 4(j + 1)µQvQ (svQ ) log(emod l)v∈(Vmod )vQ
j+1= [Fmod
:Q]

∑

log
K
[Kv :(Fmod )v ] µKw (dw )
log
log
≤
∗
Q
[K:Fmod ] [Kv :(Fmod )v ] + µQvQ (svQ ) + 4(j + 1)µQvQ (svQ ) log(emod l)w∈V(K)vQ∗Q≤= (j + 1) log(dKvQ ) + log(svQ ) + 4(j + 1) log(svQ ) log(emod l),∑where the second equality follows fromv∈(Vmod )v wv = [Fmod : Q] and the thirdQ

logequality follows from that µlog
Kw (dw ) = µKv (dv ), [Kw : (Fmod )v ] = [Kv : (Fmod )v ], andmod ]#V(K)v = [K[K:F
for any w ∈ V(K)v with a fixed v ∈ Vmod as before. Thus, by
v :(Fmod )v ]combining these, we have2Q≤∗−| log(Θ)|vQ ,j ≤ − j2l log(qvQ ) + (j + 1) log(dKvQ ) + log svQ + 4(j + 1) log(svQ ) log(emod l).
∑
∑
>
1
2
Then (1) holds since we have l1> 1≤j≤l> (j +1) = l 2+1 +1 = l+51≤j≤l> j =4 , and l>(l> +1)(2l> +1)= (l+1)l6
12 . Next, (2) trivally holds by Lemma 1.9 (2). Finally, (3) holds byl+5
Lemma 1.9 (3) with l+5
4 log(π) < 4 2 ≤ l + 1 since l ≥ 3.

35

a proof of the abc conjecture after Mochizuki

Lemma 1.11.
([IUTchIV, Steps (ii), (iii), (viii) in the proof of Theorem 1.10,
and Proposition 1.6])(1) We have the following bound of log(dK ) in terms of log(dFtpd ) and log(fFtpd ):log(dK ) ≤ log(dFtpd ) + log(fFtpd ) + 2 log l + 21.
(2) We have the following bound of log(sQ ) in terms of log(dFtpd ) and log(fFtpd ):log(sQ ) ≤ 2dmod (log(dFtpd ) + log(fFtpd )) + log l + 5.
(3) We have the following bound of log(s≤ ) log(e∗mod l): there is ηprm ∈ R>0 (which is aconstant determined by using the prime number theorem) such thatlog(s≤ ) log(e∗mod l) ≤ 34 (e∗mod l + ηprm ).
∑∑1
1
Proof. Note that log(dL )+log(fL ) = [L:Q]w∈V(L)non ew dw log(qw )+ [L:Q]w∈Supp(fL )∑1log(qw ) = [L:Q] w∈V(L)non (dw + ιfL ,w /ew )ew log(qw ) for L = K, F, Ftpd , Fmod , where
qw is the cardinality of the residue field of Lw , ew is the ramification index of Lw overQpw and ιfL ,w := 1 if w ∈ Supp(fL ), and ιfL ,w := 0 if w ∈/ Supp(fL ).
(1): The extension F/Ftpd is tamely ramified outside 2 · 3 · 5 (Lemma 1.8). Thenby using Lemma 1.5 (1) (dL0 + 1/e0 = dL + 1/e) for the primes outside 2 · 3 · 5 andLemma 1.5 (2) (dL + 1/e ≤ dL0 + 1/e0 + m + 1/e ≤ dL0 + 1/e0 + (m + 1)) for the primes
dividing 2 · 3 · 5, we have log(dF ) + log(fF ) ≤ log(dFtpd ) + log(fFtpd ) + log(211 · 33 · 52 ) ≤√√log(dFtpd ) + log(fFtpd ) + 21 since [F : Ftpd ] = [Ftpd ( −1) : Ftpd ][F : Ftpd ( −1)] ≤#(Z/4Z)× ·#GL2 (F3 )·#GL2 (F5 ) = 2·(24 ·3)·(25 ·3·5) = 210 ·32 ·5, and log 2 < 1, log 3 < 2,
log 5 < 2. In a similar way, we have log(dK ) + log(fK ) ≤ log(dF ) + log(fF ) + 2 log l, since
K/F is tamely ramified outside l (Lemma 1.8). Then we have log(dK ) ≤ log(dK ) +log(fK ) ≤ log(dF ) + log(fF ) + 2 log l ≤ log(dFtpd ) + log(fFtpd ) + 2 log l + 21.
Fmod) for vQ ∈ Vnon
(2): We have log(sQvQ ) ≤ dmod log(svQ
Q . By using Lemma 1.5 (1),
F

F

mod
) ≤ 2(log(dvQtpd ) + log(fvQtpd )) for Vnon
3 vQ - 2 · 3 · 5 · l, since 1 =we have log(sF
vQ
Q
dQvQ + 1/eQvQ ≤ dFmod,v + 1/eFmod,v ≤ 2(dFmod,v + ιfFmod ,v /eFmod,v ), where ιfFmod ,v := 1
for v ∈ Supp(fFmod ) and ιfFmod ,v := 0 for v ∈/ Supp(fFmod ). Thus, we have log(sQ ) ≤2dmod (log(dFtpd ) + log(fFtpd )) + log(2 · 3 · 5 · l) ≤ 2dmod (log(dFtpd ) + log(fFtpd )) + log l + 5,
since log 2 < 1, log 3 < 2, and log 5 < 2.
∑(3): We have log(s≤ ) log(e∗mod l) ≤ log(e∗mod l) p≤e∗ l 1. By the prime number themod
orem limn→∞ n log(pn )/pn = 1 (where pn is the n-th prime number), there exists ηprm ∈∑∑4η
∗
R>0 such that
1
≤
for
η
≥
η
.
Then
log(e
l)
1 ≤
prm
mod
prime p≤η
p≤e∗
3 log(η)
mod l
∗
∑
e
l
4
4 ∗
∗
∗
∗
mod
p≤e∗ l 1 ≤ log(ηprm )3 log(emod l) log(e∗ l) = 3 emod l if emod l ≥ ηprm , and log(emod l)ηprm
4
3 log(ηprm )ηprm ).

modmod= 43 ηprm if e∗mod l < ηprm . Thus, we have log(s≤ ) log(e∗mod l) ≤ 43 (e∗mod l +36

Go Yamashita

1
Proposition 1.12.
([IUTchIV, Theorem 1.10]) We set −| log(q)| := − 2llog(q).
∑We have the following an upper bound of −| log(Θ)| := − vQ ∈VQ | log(Θ)|vQ :1− | log(Θ)| ≤ − 2llog(q)+
(
(
)
(
)(
)
)
12dmodl+1− 16 1 − 12log(dFtpd ) + log(fFtpd ) + 10(e∗mod l + ηprm ) .
4
l2 log(q) + 1 +l

In particular, we have −| log(Θ)| < ∞. If −| log(q)| ≤ −| log(Θ)| , then we have1
6 log(q) ≤(1 + 20dlmod)()
log(dFtpd ) + log(fFtpd ) + 20(e∗mod l + ηprm ) ,where ηprm is the constant in Lemma 1.11.
Proof. By Lemma 1.10 (1), (2), (3) and Lemma 1.11 (1), (2), (3), we have
(
)
Ftpd
Ftpd
l+5
−| log(Θ)| ≤ − l+1log(q)
+
log(d
)
+
log(f
)
+
2
log
l
+
21
24
4
(
)
Ftpd
+ 2dmod (log(d) + log(fFtpd )) + log l + 5 + (l + 5) 43 (e∗mod l + ηprm ) + l + 1.
2

2

l +5l
4
l+1
l+1 1620 l+1Since l+5
< l +5l+4
= l+1
4 =
4l
4l
4 (1 + l ), 4 < 4 l = 4 l , and l + 5 ≤ 3 4 (forl ≥ 5), this is bounded above by

( 1
(
)(
)
Ftpd
Ftpd
4
−
log(q)
+
1
+
log(d
)
+
log(f
)
+
2
log
l
+
21
< l+1
4
6
l
)
(
)
Ftpd
4 ∗
4
l
+
η
)
+
4
) + log(fFtpd )) + log l + 5 + 20(e
+ l 2dmod (log(dprm
3 3 mod
( 1
(
)(
)
8dmod
Ftpd
l+1
4
= 4 − 6 log(q) + 1 + l + llog(d) + log(fFtpd )
(
)
)
80 ∗
+ 1 + 4l (2 log l + 21) + 16
l (log l + 5) + 9 (emod l + ηprm ) + 4 .
Since 4 + 8dmod ≤ 12dmod , (1 + 4l )(2 log l + 21) = 2 log l + 8 logl l + (1 + 4l )21 < 2 log l +8 12 + (1 + 1)21 = 2 log l + 46 (for l ≥ 5), 16 logl l < 16 12 = 8, and 16
l 5 ≤ 16 (for l ≥ 5),
this is bounded above by
)
( 1
(
)(
)
12dmod
Ftpd
Ftpd
80 ∗
< l+1
−
log(q)
+
1
+
log(d
)
+
log(f
)
+
2
log
l
+
(e
l
+
η
)
+
74
.
prm
mod
4
6
l
9
Since 2 log l + 74 < 2l + 74 < 2 · 74l + 2 · 74l = 22 · 74l < 22 · 212 · 3 · 5l < 49 e∗mod l <4 ∗80
49 (emod l + ηprm ), and 9 + 9 < 10, this is bounded above by
( 1
(
)(
)
)
12dmod
Ftpd
Ftpd
∗
< l+1
−
log(q)
+
1
+
log(d
)
+
log(f
)
+
10(e
l
+
η
)
.
prm
mod
4
6
l
1 12
1
1
1
Since l+1
4 6 l2 = 2 (1 + l ) > 2l , this is bounded above by

)
)
( 1(
)
(
)(
12dmod< l+1− 6 1 − 12log(dFtpd ) + log(fFtpd ) + 10(e∗mod l + ηprm )4l2 log(q) + 1 +l
1− 2llog(q).

a proof of the abc conjecture after Mochizuki

37

If −| log(q)| ≤ −| log(Θ)|, then for any −| log(Θ)| ≤ CΘ log(q), we have −| log(q)| ≤1−| log(Θ)| ≤ CΘ log(q), hence CΘ ≥ −1 since | log(q)| = 2llog(q) > 0. By taking CΘ
to be

)
(
)(
)
)
2l(l+1) ( 1 (12dmod12log(dFtpd ) + log(fFtpd ) + 10(e∗mod l + ηprm ) − 1,
4 log(q) − 6 1 − l2 log(q) + 1 +l
we have1
6 log(q) ≤(1 − 12l2))−1 (()()
1 + 12dlmod log(dFtpd ) + log(fFtpd ) + 10(e∗mod l + ηprm ) .

20dmod−1
Since (1 − 12
≤ 2 and (1 − 12
) ≥ 1 + 12dlmod ⇔ 12 ≤ dmod (8l − 240l2 )l2 )(1 +ll ) which
240
240
240
holds for l ≥ 7 (by dmod (8l − l ) ≥ 8l − l ≥ 56 − 7 > 12), we have1
6 log(q) ≤§ 1.4.

()()
1 + 20dlmod log(dFtpd ) + log(fFtpd ) + 20(e∗mod l + ηprm ).

Third Reduction — Choice of Initial Θ-Data.

In this subsection, we regard UP1 as the λ-line, i.e., the fine moduli scheme whose Svalued points (where S is an arbitrary scheme) are the isomorphism classes of the triples[E, φ2 , ω], where E is an elliptic curve f : E → S equipped with an isomorphism φ2 :∼(Z/2Z)⊕2 → E[2] of S-group schemes, and an S-basis ω of f∗ Ω1E/S to which an adaptedx ∈ f∗ OE (−2(origin)) satisfies x(φ2 (1, 0)) = 0, x(φ2 (0, 1)) = 1. Here, a section x ∈f∗ OE (−2(origin)), for which {1, x} forms Zariski locally a basis of f∗ OE (−2(origin)), iscalled adapted to an S-basis ω of f∗ Ω1E/S , if Zariski locally, there is a formal parameter
T at the origin such that ω = (1 + higher terms)dT and x = T12 (1 + higher terms)(cf. [KM, (2.2), (4.6.2)]). Then λ ∈ UP1 (S) corresponds to E : y 2 = x(x − 1)(x − λ),φ2 ((1, 0)) = (x = 0, y = 0), φ2 ((0, 1)) = (x = 1, y = 0), and ω = − dx
2y . For a cyclic
subgroup scheme H ⊂ E[l] of order l > 2, a level 2 structure φ2 gives us a level 2
structure Im(φ2 ) of E/H. An S-basis ω also gives us an S-basis Im(ω) of f∗ Ω1(E/H)/S .
For α = (φ2 , ω), write Im(α) := (Im(φ2 ), Im(ω)).
Let F be a number field. For a semi-abelian variety E of relative dimension 1 over anumber Spec OF whose generic fiber EF is an elliptic curve, we define Faltings height of
E as follows: Let ωE be the module of invariant diﬀerentials on E (i.e., the pull-back ofΩ1E/OF via the zero section), which is finite flat of rank 1 over OF . We equip an hermi√
∫
−1
arc
2
by (||a||Falt
tian metric ||−||Falta∧a,Ev ) := 2
Ev on ωEv := ωE ⊗OF Fv for v ∈ V(F )Ev
where Ev := E ×F Fv and a is the complex conjugate of a. We also equip an hermitian
metric || − ||Falt
on ωE ⊗Z C ∼= ⊕real:v∈V(F )arc ωEv ⊕ ⊕complex:v∈V(F )arc (ωEv ⊕ ωEv ), by
E
Falt
Falt
|| − ||Ev (resp. || − ||Ev and its complex conjugate) for real v ∈ V(F )arc (resp. forcomplex v ∈ V(F )arc ), where ωEv is the complex conjugate of ωEv . Then we obtain

38

Go Yamashita

an arithmetic line bundle ω E := (ωE , || − ||Falt
E ). We define Faltings height of E by
Falt
1
ht (E) := [F :Q] degF (ω E ) ∈ R. Note that for any 0 6= a ∈ ωE , the non-Archimedean(resp. Archimedean) portion htFalt (E, a)non (resp. htFalt (E, a)arc ) of htFalt (E) is given∑∑
by [F1:Q] v∈V(F )non v(a) log pfvv = [F1:Q] log #(ωE /aωE ) (resp. − [F1:Q] v∈V(F )arc [Fv :
(√ ∫
(√ ∫
)1/2
)
∑
−1
−1
1
a
∧
[F
:
R]
log
a
∧
R] loga=−a), where htFalt (E)arcvv∈V(F )
2
2[F :Q]2
Ev
Ev= htFalt (E, a)non + htFalt (E, a)arc is independent of the choice of 0 6= a ∈ ωE (cf. Section 1.1).
Let Q be an algebraic closure of Q. For any point [E, α] ∈ UP1 (Q) of the λ-line, we
define htFalt ([E, α]) := htFalt (E). When [E, α] ∈ UP1 (C) varies, the hermitian metricon ωE continuously varies, and gives a hermitian metric on the line bundle ωE|| − ||Falt
E
on UP1 (C), where E is the universal elliptic curve of the λ-line. Note that this metriccannot be extended to the compactification P1 of the λ-line, and the Faltings height has
logarithmic singularity at {0, 1, ∞}. (See also Lemma 1.13 (1) and its proof below.)
We also introduce some notation. We write htnon
ωP1 ({0,1,∞}) for the non-Archimedeannonportion of htωP1 ({0,1,∞}) ([E, α]), i.e., htωP1 ({0,1,∞}) ([E, α]) := [F1:Q] degF (x−1F ({0, 1, ∞}))
1
1
1∼
for xF : Spec OF → P representing [E, α] ∈ P (F ) = P (OF ) (Note that x−1
F ({0, 1, ∞})non
is supported in V(F )and degF is the degree map on ADiv(F ), not on APic(Spec OF )).
Note that we havehtnon
ωP1 ({0,1,∞}) . htωP1 ({0,1,∞})
on P1 (Q), since the Archimedean portion is bounded below in light of the compactnessof (P1 )arc (cf. the proof of Proposition 1.2).
We also note that ht∞ in [GenEll, Section 3] is a function on Mell (Q), on the other1
hand, our htnon
ωP1 ({0,1,∞}) is a function on λ-line P (Q), and that the pull-back of ht∞ tothe λ-line is equal to 6 times our htnon
ωP1 ({0,1,∞}) ([IUTchIV, Corollary 2.2 (i)], cf. also
the proof of Lemma 1.13 (1) below).
Lemma 1.13.
([GenEll, Proposition 3.4, Lemma 3.5], [Silv2, Proposition 2.1,
Corollary 2.3]) Let l > 2 be a prime, E an elliptic curve over a number field F suchthat E has everywhere semistable reduction, and H ⊂ E[l] a cyclic subgroup scheme oforder l. Then we have(1) (relation between htωP1 ({0,1,∞}) and htFalt )2htFalt . htωP1 ({0,1,∞}) . 2htFalt + log(htωP1 ({0,1,∞}) ) . 2htFalt + htωP1 ({0,1,∞})for any  ∈ R>0 on UP1 (Q),(2) (relation betwen htFalt ([E, α]) and htFalt ([E/H, Im(α)]))htFalt ([E, α]) −1
1log l ≤ htFalt ([E/H, Im(α)]) ≤ htFalt ([E, α]) + log l.
22

a proof of the abc conjecture after Mochizuki

39non(3) (relation between htnon
ωP1 ({0,1,∞}) ([E, α]) and htωP1 ({0,1,∞}) ([E/H, Im(α)]))Furthermore, we assume that l is prime to v(qE,v ) ∈ Z>0 for any v ∈ V(F ), where
E has bad reduction with q-parameter qE,v (e.g., l > v(qE,v ) for any such v’s).
Then we havenonl · htnon
ωP1 ({0,1,∞}) ([E, α]) = htωP1 ({0,1,∞}) ([E/H, Im(α)]).

Proof. (1): We have the Kodaira-Spencer isomorphism ωE⊗2 ∼= ωP1 ({0, 1, ∞}),
where E is the universal generalised elliptic curve over the compactification P1 of the
λ-line, which extends E over the λ-line UP1 . Thus we have htωP1 ({0,1,∞}) ≈ 2htωE on
P1 (Q). Thus, it is reduced to compare htωE and htFalt . Here, htωE is defined by equipping a hermitian metric on the line bundle ωE . On the other hand, htFalt is defined byequipping a hermitian metric on the line bundle ωE , which is the restriction of ωE ot UP1 .
Thus, it is reduced to compare the Archimedean contributions of htωE and htFalt . The
former metric is bounded on the compact space (P1 )arc . On the other hand, we show
the latter metric defined on the non-compact space (UP1 )arc has logarithmic singularity
along {0, 1, ∞}. Let 0 6= dz ∈ ωE be an invariant diﬀerential over OF . Then dz de`composes as ((dzv )real:v∈V(F )arc , (dzv , dzv )complex:v∈V(F )arc ) on E arc ∼= real:v∈V(F )arc Ev```
Ev ), where dzv , Ev are the complex conjugates of dzv , Ev
complex:v∈V(F )arc (Ev×respectively. For v ∈ V(F )arc , we have Ev ∼= Fv /q Z ∼= Fv /(Z ⊕ τv Z) and dzv
E,v

is the descent of the usual Haar measure on√ Fv , where qE,v = e2πiτv and τv is in∫−1
dzv ∧ dzv )1/2 = (Im(τv ))1/2 =the upper half plane. Then ||dzv ||Falt
=
(
Ev
2
Ev
∑
1
log(|qE,v |2v ))1/2 and htFalt (E, dz)arc ≈ − 2[F1:Q] v∈V(F )arc [Fv : R] log(− log |qE,v |v )(− 4π
has a logarithmic singularity at |qE,v |v = 0. Thus, it is reduced to calculate the logarithmic singularity of htFalt (E, dz)arc in terms of htωP1 ({0,1,∞}) . We have |jE |v = |jEv |v ≈|qE,v |−1near |qE,v |v = 0, where jE is the j-invariant of E. Then by the arithmeticv∏
geometric inequality, we have htFalt (E, dz)arc ≈ − 2[F1:Q] log v∈V(F )arc (log |jE |v )[Fv :R])(∏∑≥ − 12 log [F1:Q] v∈V(F )arc log |jE |v near v∈V(F )arc |jE |v = ∞. On the other hand,
arc
2
2
2
, since
we have |j|−1v ≈ |λ|v , |λ−1|v , 1/|λ|v near |λ|v = 0, 1, ∞ respectively for v ∈ V(F )
∑
non
1
8 2
3
2
2
j = 2 (λ −λ+1) /λ (λ−1) . Thus, we have htωP1 ({0,1,∞}) ([E, α]) = [F :Q] v∈V(F )non (v(λE )+
∑
∑
−1
−1
v(λE −1)+v(1/λE )) log qv = 2[F1:Q] v∈V(F )non v(jE
) log qv = 2[F1:Q] v∈V(F )non log |jE|v .
∑1
By the product formula, this is equal to 2[F :Q] v∈V(F )arc log |jE |v . By combining these,non1
we obtain htFalt (E, dz)arc & − 12 log(2htnon
ωP1 ({0,1,∞}) ([E, α])) ≈ − 2 log(htωP1 ({0,1,∞}) ([E, α]))∏∏near v∈V(F )arc |jE |v = ∞, or equivalently, near v∈V(F )non |jE |v = 0. We also have1htnon
ωP1 ({0,1,∞}) . htωP1 ({0,1,∞}) on P (Q) since the Archimedean contribution is bounded

below in light of the compactness of (P1 )arc . Therefore, we have htFalt . htωE . htFalt +Falt
1. htωP1 ({0,1,∞}) . 2htFalt +log(htωP1 ({0,1,∞}) ).
2 log(htωP1 ({0,1,∞}) ). This implies 2ht
The remaining portion comes from log(1 + x) . x for any  ∈ R>0 .

40

Go Yamashita(2): We have htFalt ([E, α])non − log l ≤ htFalt ([E/H, Im(α)])non ≤ htFalt ([E, α])non
since #coker{ωE/H ,→ ωE } is killed by l. We also have htFalt ([E/H, Im(α)])arc =2
Falt 2Falt
htFalt ([E, α])arc + 12 log l, since (|| − ||Falt
E/H ) = l(|| − ||E ) by the definition of || − ||by the integrations on E(C) and (E/H)(C). By combining the non-Archimedean portionand the Archimedean portion, we have the second claim.
(3): Let v ∈ V(F )non where E has bad reduction. Then the l-cyclic subgroup
H ×F Fv is the canonical multiplicative subgroup Fl (1) in the Tate curve E ×F Fv , by
the assumption l - v(qE,v ). Then the claim follows from that the Tate parameter of
E/H is equal to l-th power of the one of E.
Corollary 1.14.
have([GenEll, Lemma 3.5]) In the situation of Lemma 1.13 (3), welhtnon([E, α]) ≤ htωP1 ({0,1,∞}) ([E, α]) + log l + C
1 +  ωP1 ({0,1,∞})for some constant C ∈ R which (may depend on , however) is independent of E, F ,H and l.
Remark 1.14.1. The above corollary says that if E[l] has a global multiplicative
subgroup, then the height of E is bounded. Therefore, a global multiplicative subspace
M ⊂ E[l] does not exist for general E in the moduli of elliptic curves. A “global multiplicative subgroup” is one of the main themes of inter-universal Teichmüller theory.
In inter-universal Teichmüller theory, we construct a kind of “global multiplicative subgroup” for suﬃciently general E in the moduli of elliptic curves, by going out the scheme
theory. cf. also Appendix A.
10
Proof. For  > 0, take 0 > 0 such that 1−
0 < 1 + . There is a constant A ∈ Rsuch that htωP1 ({0,1,∞}) ≤ 2htFalt + 0 htωP1 ({0,1,∞}) + A0 on UP1 (Q) by the second andthe third inequalities of Lemma 1.13 (1). We have htωP1 ({0,1,∞}) ≤ 2(1 + )htFalt + A10
on UP1 (Q) by the choice of 0 > 0, where A := 1−
By the first inequality of
0 A .
Falt
Lemma 1.13 (1), we have 2ht≤ htωP1 ({0,1,∞}) + B for some constant B ∈ R. Writenonl1C := A +B. Then we have 1+ htnon
ωP1 ({0,1,∞}) ([E, α]) = 1+ htωP1 ({0,1,∞}) ([E/H, Im(α)]) ≤2htFalt ([E/H, Im(α)]) + A ≤ 2htFalt ([E, α]) + log l + A ≤ htωP1 ({0,1,∞}) ([E, α]) + log l +C , where the equality follows from Lemma 1.13 (3), and the first inequality follows
from Lemma 1.13 (2).

From now on, we use the assumptions and the notation in the previous subsection. We also write log(q∀ ) (resp. log(q-2 )) for the R-valued function on the λ-line
1
UP1 obtained by the normaised degree [L:Q]
degL of the eﬃctive (Q-)arithmetic divisor
determined by the q-parameters of an elliptic curve over a number field L at arbitrary
non-Archimedean primes. (resp. non-arcihmedean primes which do not divide 2). Notea proof of the abc conjecture after Mochizuki

41

that log(q) in the previous subsection avoids the primes dividing 2l, and that for acompactly bounded subset K ⊂ UP1 (Q) whose support contains the prime 2, we havelog(q∀ ) ≈ log(q-2 ) on K (cf. [IUTchIV, Corolarry 2.2 (i)]). We also note that we have1log(q∀ ) ≈ htnon
ωP1 ({0,1,∞}) ≈ htωP1 ({0,1,∞})
6
on K (⊂ UP1 (Q)). (For the first equivalence, see the argument just before Lemma 1.13,
and the proof of Lemma 1.13 (1); For the second equivalence, note that K includes theunique Archimedean place of Q.)
Proposition 1.15.
([IUTchIV, Corollary 2.2]) Let K ⊂ UP1 (Q) be a compactly
bounded subset with support containing Varc
and 2 ∈ Vnon
Q }Q , and A ⊂ UP1 (Q) a finite{
set containing [(E, α)] | #AutQ (E) 6= {±1} . Then there exists CK ∈ R>0 , which
depends only on K, satisfying the following property: Let d ∈ Z>0 ,  ∈ R>0 , and set
d∗ := 212 · 33 · 5 · d. Then there exists a finite subset ExcK,d, ⊂ UP1 (Q)≤d such that
ExcK,d, ⊃ A and satisfies the following property: Let x = [(EF , α)] ∈ (UP1 (F ) ∩ K) \ExcK,d, with [F : Q] ≤ d. Write Fmod for the field of moduli of EF := EF ×F F ,and Ftpd := Fmod (EFmod [2]) ⊂ F where EFmod is a model of EF over Fmod (Notethat Fmod (EFmod [2]) is independent of the choice of the model EFmod by the assumptionof AutF (EF ) 6= {±1}, and that Fmod (EFmod [2]) ⊂ F since [(EF , α)] ∈ UP1 (F ). cf.
Lemma 1.7 (1)). We assume that all the points of EF [3 · 5] are rational over F and that√F = Ftpd ( −1, EFtpd [3 · 5]), where EFtpd is a model of EF over Ftpd which is definedby the Legendre form (Note that EF ∼= EFtpd ×Ftpd F and EF has semistable reduction
nonfor all w ∈ V(F )by Lemma 1.7 (2), (3)). Then EF and Fmod arise from an initialΘ-data (cf. Definition 10.1)(F /F, XF , l, C K , V, Vbad
mod , ).
(Note that it is included in the definition of initial Θ-data that the image of the outerhomomorphism Gal(Q/F ) → GL2 (Fl ) determined by EF [l] contains SL2 (Fl ).) Furthermore, we assume that −| log(q)| ≤ −| log(Θ)| for EF and Fmod , which arise from an
initial Θ-data. Then we havehtωP1 ({0,1,∞}) (x) ≤ (1 + )(log-diﬀ P1 (x) + log-cond{0,1,∞} (x)) + CK .
Remark 1.15.1.
We take A = {[(E, α)] ∈ UP1 (Q) | E does not admit Q-core}. cf.
Definition
3.3 and Lemma C.3 for the{} definition of k-core, the finiteness of A, and thatA ⊃ [(E, α)] | #AutQ (E) 6= {±1} .
Remark 1.15.2.
By Proposition 1.15, Theorem 0.1 is reduced to show −| log(q)| ≤−| log(Θ)| for EF and Fmod , which arise from an initial Θ-data. The inequality −| log(q)| ≤42

Go Yamashita−| log(Θ)| is almost a tautological translation of the inequality which we want to show
(cf. also Appendix A). In this sense, these reduction steps are just calculations to reducethe main theorem to the situation where we can take an initial Θ-data, i.e., the situation
where the inter-universal Teichmüller theory works, and no deep things happen in these
reduction steps.

Proof. First we write ExcK,d := A, and we enlarge the finite set ExcK,d several
times in the rest of the proof in the manner that depends only on K and d, but not on
x. When it will depend on  > 0, then we will change the notation ExcK,d by ExcK,d, .
Let x = [(EF , α)] ∈ (UP1 (F ) ∩ K) \ ExcK,d .
Let ηprm ∈ R>0 be the constant in Lemma 1.11. We take another constant ξprm ∈R>0 determined by using the prime number theorem as follows (cf. [GenEll, Lemma∑4.1]): We define ϑ(x) :=prime: p≤x log p (Chebychev’s ϑ-function). By the prime
number theorem (and Lemma C.4), we have ϑ(x) ∼ x (x → ∞), where ∼ means thatthe ration of the both side goes to 1. Hence there exists a constant R 3 ξprim ≥ 5 suchthat
42
x < ϑ(x) ≤ x
3
3

(s0)

for any x ≥ ξprm .
∑Let h := h(EF ) = log(q∀ ) = [F1:Q] v∈V(F )non hv fv log(pv ) be the summation of thecontributions from qv for v ∈ V(F )non . and the degree of extension of the residue fieldover Fpv respectively. Note also that hv ∈ Z≥0 and that hv = 0 if and only if EF has
good reduction at v. By 61 log(q∀ ) ≈ htωP1 ({0,1,∞}) and Proposition C.1, we there are
only finiely many isomorphism classes of EF (hence finiely many x = [EF , α]) satisfying
1h 2 < ξprm + ηprm . Therefore, by enlarging the finite set ExcK,d , we may assume that1h 2 ≥ ξprm + ηprm .

(s1)
1Note that h 2 ≥ 5 since ξprm ≥ 5 and ηprm > 0. We have(s2)
2d∗ h 2 log(2d∗ h) ≥ 2[F : Q]h 2 log(2[F : Q]h) ≥
1

1

≥

∑
hv 6=0− 21h

log(hv )hv ≥

∑
hv

≥h1/2

∑
hv 6=0− 12h

log(hv )hv ≥2h− 2 log(2hv fv log(pv ))hv fv log(pv )
1

∑

hv

log(hv ),≥h1/2

where the third inequality follows from 2 log(pv ) ≥ 2 log 2 = log 4 > 1. By [F : Q] ≤ d∗ ,

43a proof of the abc conjecture after Mochizuki

we also have∑d∗ h 2 ≥ [F : Q]h 2 =
1

(s3)

≥

1

1

v∈V(F )non∑hv

h− 2 hv fv log(pv ) ≥h− 2 hv log(pv ) ≥
1

≥h1/2

∑

hv

∑

h− 2 hv log(pv )1v∈V(F )nonlog(pv ).

≥h1/2

Let A be the set of prime numbers satisfying either
1(S1) p ≤ h 2 ,(S2) p | hv 6= 0 for some v ∈ V(F )non , or1(S3) p = pv for some v ∈ V(F )non and hv ≥ h 2 .
Then we have
∑
1
1
4 12
2
2
(S’1)
p:(S1) log p = ϑ(h ) ≤ 3 h by the second inequality of (s0), and h ≥ ξprm , which
follows from (s1),
(S’2)
(S’3)

∑
∑

p:(S2), not (S3) log p ≤∗ 21
p:(S3) log p ≤ d h

∑

∗ 12
∗
hv >h1/2 log(hv ) ≤ 2d h log(2d h) by (s2), andby (s3).

Then we obtain(S’123)ϑA :=∑log(p) ≤ 2h 2 + d∗ h 2 + 2d∗ h 2 log(2d∗ h)
1

1

1

p∈A

≤ 4d∗ h 2 log(2d∗ h) ≤ −ξprm + 5d∗ h 2 log(2d∗ h),11

where the first inequality follows from (S’1), (S’2), and (S’3), the second inequality
1
1
1
follows from 2h 2 ≤ d∗ h 2 and log(2d∗ h 2 ) ≥ log 4 > 1, and the last inequality follows
from (s1). Then there exists a prime number l 6∈ A such that l ≤ 2(ϑA + ξprm ), because
otherwise we have ϑA ≥ ϑ(2(ϑA + ξprm )) ≥ 23 (2(ϑA + ξprm )) ≥ 43 ϑA , by the second
inequality of (s0), which is a contradiction. Since l 6∈ A, we have(P1) (upper bound of l)11(5 ≤ ) h 2 < l ≤ 10d∗ h 2 log(2d∗ h) ( ≤ 20(d∗ )2 h2 ),where the second inequality follows from that l does not satisfy (S1), the third
inequality follows from l ≤ 2(ϑA + ξprm ) and (S’123), and the last inequality follows
3
from log(2d∗ h) ≤ 2d∗ h ≤ 2d∗ h 2 (since log x ≤ x for x ≥ 1),(P2) (monodromy non-vanishing modulo l)
l - hv for any v ∈ V(F )non such that hv 6= 0, since l does not satisfy (S2), and44

Go Yamashita(P3) (upper bound of monodromy at l)
1
if l = pv for some v ∈ V(F )non , then hv < h 2 , since l does not satisfy (S3).

Claim 1: We claim that, by enlarging the finite set ExcK,d , we may assume that(P4) there does not exist l-cyclic subgroup scheme in EF [l].
Proof of Claim 1: If there exists an l-cyclic subgroup scheme in EF [l], then by applying
nonCorollary 1.14 for  = 1, we have l−2
2 htωP1 ({0,1,∞}) (x) ≤ log l+TK ≤ l+TK (since log x ≤x for x ≥ 1) for some TK ∈ R>0 , where TK depends only on K. Thus, htnon
ωP1 ({0,1,∞}) (x)
non
2l
2
14
2
is bounded because we have htωP1 ({0,1,∞}) (x) ≤ l−2 + l−2 TK < 7−2 + 7−2
TK . Notenonalso that htωP1 ({0,1,∞}) ≈ htωP1 ({0,1,∞}) on K, since K includes the unique Archimedean
place of Q. Therefore, there exist only finitely many such x = [EF , α]’s by Proposition
C.1. The claim is proved.
Claim 2: Next, we claim that, by enlarging the finite set ExcK,d , we may assume thatnon(P5) ∅ =6 Vbadmod := {v ∈ Vmod | v - 2l, and EF has bad multiplicative reduction at v}

Proof of Claim 2: First, we note that we have
(p5a)
(p5b)

h 2 log l ≤ h 2 log(20(d∗ )2 h2 ) ≤ 2h 2 log(5d∗ h)
1

1

1≤ 8h 2 log(2(d∗ ) 4 h 4 ) ≤ 8h 2 2(d∗ ) 4 h 4 = 16(d∗ ) 4 h 4 .
1

1

1

1

1

1

1

3

-2
where the first inequality follows from (P1). If Vbad
mod = ∅, then we have h ≈ log(q ) ≤1
3
1
h 2 log l ≤ 16(d∗ ) 4 h 4 on K, where the first inequality follows from (P3), and the last1inequality is (p5b). Thus, h 4 , hence h as well, is bounded. Therefore, there exist only
finitely many such x = [EF , α]’s by Proposition C.1. The claim is proved.

Claim 3: We also claim that, by enlarging the finite set ExcK,d , we may assume that(P6) The image of the outer homomorphism Gal(Q/F ) → GL2 (Fl ) determined by EF [l]contains SL2 (Fl ).
Proof of Claim 3 (cf. [GenEll, Lemma 3.1 (i), (iii)]): By (P2) l - hv 6= 0 and( (P5))
1
1
Vbad.
mod 6= ∅, the image H of the outer homomorphism contains the matrix N+ :=01
Here, N+ generates an l-Sylow subgroup S of GL2 (Fl ), and the number of l-Sylow
subgroups of GL2 (Fl ) is precisely l + 1. Note that the normaliser of S in GL2 (Fl ) is
the subgroup of the upper triangular matrices. By (P4) E[l] 6⊃ (l-cyclic subgroup),
the image contains a matrix which is not upper triangluar. Thus, the number nH ofa proof of the abc conjecture after Mochizuki

45l-Sylow subgroups of H is greater than 1. On the other hand, nH ≡ 1 (mod l) by thegeneral theory of Sylow subgroups.
Then(we )have nH = l + 1 since 1 < nH ≤ l + 1. In
( )
11
10
particular, we have N+ =, N− :=∈ H. Let G ⊂ SL2 (Fl ) be the subgroup
01
11
generated by N+ and N− . Then it suﬃces to show that G = SL2 (Fl ). We note thatball
for a,(b ∈)Fl , the( matrix) N− N+ (this makes sense since N+ = N− = 1) takes the vector
(
)
0
a
v :=
to
. This implies that we have F×l × Fl ⊂ G. This also implies
1
ab + 1( )cthat for c ∈ F×(= cA1 v). Then we havel , there exists Ac ∈ G such that Ac v =
0
{( )}
0
cv = A−1⊂ Gv. Let M ∈ SL2 (Fl )1 Ac v ∈ Gv. Thus, we proved that (Fl × Fl ) \0
be any matrix. By
M by an element in G,{(
we may)}assume that M v = v,{(multiplying)}
100. Thus, M is a power
since (Fl × Fl ) \⊂ Gv. This means that M ⊂0∗1
of N− . The claim is proved.

Then we take, as parts of initial Θ-data, F to be Q so far, F , XF , l to be thenumber field F , once-punctured elliptic curve associated to EF , and the prime number,badrespectively, in the above discussion, and Vbad
mod to be the set Vmod of (P 5). By using(P 1), (P 2), (P 5), and (P 6), there exist data C K , V, and , which satisfy the conditionsof initial Θ-data (cf. Definition 10.1. The existence of V and  is a consequence of (P 6)),and moreover,(P7) the resulting initial Θ-data (F /F, XF , l, C K , V, Vbad
mod , ) satisfies the conditions in
Section 1.3.
Now, we have −| log(q)| ≤ −| log(Θ)| by assumption, and apply Proposition 1.12.
(Note that we are in the situation where we can apply it.)
Then we obtain())
1
20dmod (log(q) ≤ 1 +log(dFtpd ) + log(fFtpd ) + 20(e∗mod l + ηprm )
6
l
(
)(
)
1
∗ − 12(A)≤ 1+d hlog(dFtpd ) + log(fFtpd ) + 200(d∗ )2 h 2 log(2d∗ h) + 20ηprm ,where the second inequality follows from the second and third inequalities in (P1) and20dmod < d∗mod (:= 212 · 33 · 5 · dmod ) ≤ d∗ (:= 212 · 33 · 5 · dmod ). We also have
(B)

1
1
1
1 1
1 1
log(q-2 ) − log(q) ≤ h 2 log l ≤ h 2 log(5d∗ h) ≤ h 2 log(2d∗ h),
6
6
6
3

46

Go Yamashita

where the first inequality follows from (P3) and (P5), the second inequality follows from(p5a), and the last inequality follows from 5 < 23 . We also note that11log(q∀ ) − log(q-2 ) ≤ BK
6
6

(C)

for some constant BK ∈ R>0 , which depends only on K, since log(q∀ ) ≈ log(q-2 ) on K
as remarked when we introduced log(q∀ ) and log(q-2 ) just before this proposition. By
combining (A), (B), and (C), we obtain
(
)(
)
1
1
1
1
1
h = log(q∀ ) ≤ 1 + d∗ h− 2 log(dFtpd ) + log(fFtpd ) + (15d∗ )2 h 2 log(2d∗ h) + CK
6
6
2
(
)(
)
1
1
2
1
1
(ABC)
≤ 1 + d∗ h− 2 log(dFtpd ) + log(fFtpd ) + h (60d∗ )2 h− 2 log(2d∗ h) + CK ,6 52
where we write CK := 40ηprm +2BK , the first inequality follows from 200 < 152 , the sec12 2∗ 2 − 12
ond inequality follows from 1 < 32
log(2d∗ h) (≥30 = 6 5 4 . Here, we put E := (60d ) h
15d∗ h− 2 ). We have(Epsilon)E ≤ 4(60d∗ )2 h− 2 log(2(d∗ ) 4 h 4 ) ≤ 4(60d∗ )3 h− 2 h 4 = 4(60d∗ )3 h− 4 .
1

1

1

1

1

1

1

Let  > 0. If E > min{1, }, then h 4 , hence h as well, is bounded by (Epsilon).
Therefore, by Proposition C.1, by replacing the finite set ExcK,d by a finite set ExcK,d, ,we may assume that E ≤ min{1, }. Then finally we obtain
(
)−1 (
)
(
)−1
(
)
2
1
2
1
Ftpd
Ftpd
1 − E
1 + E log(d) + log(f) + 1 − E
CK
5
5
5
2
(
)
≤ (1 + E ) log(dFtpd ) + log(fFtpd ) + CK()
≤ (1 + ) log-diﬀ P1 (xE ) + log-cond{0,1,∞} (xE ) + CK ,1h≤6where the first inequality follows from the definition of E and  ≥ 5d∗ h− 2 , the sec1+ 1 
ond inequality follows from 1− 52 E ≤ 1 + E (i.e., E (1 − E ) ≥ 0, which holds since1

5 E

E ≤ 1), and 1 − 25 E ≥ 21 (i.e., E ≤ 54 , which holds since E ≤ 1), and thethird inequality follows from E ≤ , log-diﬀ P1 (xE ) = log(dFtpd ) by definition, andlog(fFtpd ) ≤ log-cond{0,1,∞} (xE ). (Note that Supp(f) excludes the places dividing 2l inthe definition.) Now the proposition follows from 16 log(q∀ ) ≈ htωP1 ({0,1,∞}) on P1 (Q) as
remarked just before this proposition (by the eﬀect of this ≈, the CK in the statementof the proposition may diﬀer from the CK in the proof).
Remark 1.15.3. (Miracle Identity) As shown in the proof, the reason that themain term of the inequality is 1 (i.e., ht ≤ ( 1 +)(log-diﬀ +log-cond) + bounded term)
is as follows (cf. the calculations in the proof of Lemma 1.10): On one hand (ht-side), we( )∑l/2 21
1 1
1 1 1 l 3
= 4l . Note that we multiply 2lsince thehave an average 6 2l
j=1 j ≈ 6 2l l/2 3 2l/2

a proof of the abc conjecture after Mochizuki

47

theta function under consideration lives in a covering of degree 2l, and that we multiply
6 since the degree of λ-line over j-line is 6. On the other hand ((log-diﬀ + log-cond)( )∑l/2
1
1 1 l 2side), we have an average l/2= 4l . These two values miraculously
j=1 j ≈ l/2 2 2coincide! In other words, the reason that the main term of the inequality is 1 comesfrom the equality
1(theta function involves a double covering)
2
n
∑
1
1
× 2 (the exponent of theta series is quadratic) × (the main term ofj 2 ≈ n3 /3)2
3
j=1
6 (the degree of λ-line over j-line) ×n∑
11= 1 (the terms of diﬀerents are linear) × (the main term ofj ≈ n2 /2).
22
j=1This equality was already observed in Hodge-Arakelove theory, and motivates the definition of the Θ-link (cf. also Appendix A). Mochizuki firstly observed this equality, andnext he established the framework (i.e. going out of the scheme theory and studying
inter-universal geometry) in which these calculations work (cf. also [IUTchIV, Remark
1.10.1]).
Note also that it is already known that this main term 1 cannot be improved byMasser’s calculations in analytic number theory (cf. [Mass2]).
Remark 1.15.4.
(-term) In the proof of Proposition 1.15, we also obtained an
upper bound of the second main term (i.e., the main behaviour of the term involved to) of the Diophantine inequality (when restricted to K):1ht ≤ δ + ∗δ 2 log(δ)
on K, where ∗ is a positive real constant, ht := htωP1 ({0,1,∞}) and δ := log-diﬀ P1 +log-cond{0,1,∞} (cf. (ABC) in the proof of Proposition 1.15) It seems that the exponent
1
2 suggests a possible relation to Riemann hypothesis. For more informations, see[IUTchIV, Remark 2.2.1] for remarks on a possible relation to inter-universal Mellinetransformation, and [vFr], [Mass2] for lower bounds of the -term from analytic number theory.
Remark 1.15.5. (Uniform ABC) So-called the uniform abc Conjecture (uniformity with respect to d of the bounded discrepancy in the Diophantine inequality)
is not proved yet; however, we have an estimate of the dependence on d of our upperbound as follows (cf. [IUTchIV, Corollary 2.2 (ii), (iii)]): For any 0 < d ≤ 1, write

48

Go Yamashita1∗d := 16
d (< 12 ). Then we havemin{1, }−1 E = min{1, }−1 (60d∗ )2 h− 2 log(2d∗ h)
1

∗

∗

∗

= (min{1, }∗d )−1 (60d∗ )2 h− 2 log(2d (d∗ )d hd )() 1 −∗∗∗1≤ (min{1, }∗d )−1 (60d∗ )2+d h−( 2 −d ) ≤ (min{1, }∗d )−3 (60d∗ )4+d h−1 2 d ,11

where the first inequality follows from h 2 ≥ 5, and x ≤ log x for x ≥ 1, and the second
31∗inequality follows from −3( 21 − ∗d ) = − 32 + 16
d ≤ − 21
16 < −1 and ( 2 − d )(4 + d ) =1 2− 16
d + 14 d + 2 ≥ 14 d + 2 ≥ ∗d + 2. We recall that, at the final stage of the proof of
Proposition 1.15, we enlarged ExcK,d to ExcK,d, so that it includes the points satisfying
E > min{1, }. Now, we enlarge ExcK,d to ExcK,d,,d , which depends only on K, d, ,and d , so that it includes the points satisfying E > min{1, }. Therefore, we obtainan inequality
14+dht := h ≤ Hunif min{1, }−3 −3
+ HK
d d
6
on ExcK,d,,d , where Hunif ∈ R>0 is independent of K, d, , and d , and HK ∈ R>0
depends only on K. The above inequality shows an explicit dependence on d of ourupper bound.
§ 2.

Preliminaries on Anabelian Geometry.

In this section, we give some reviews on the preliminaries on anabelian geometry
which will be used in the subsequent sections.
§ 2.1.

Some Basics on Galois Groups of Local Fields.

Proposition 2.1.
([AbsAnab, Proposition 1.2.1]) For i = 1, 2, let Ki be a finite
extension of Qpi with residue field ki , and Ki be an algebraic closure of Ki with residue
field ki (which is an algebraic closure of ki ). We write e(Ki ) for the ramification indexof Ki over Qpi and write f (Ki ) := [ki : Fpi ]. Write GKi := Gal(Ki /Ki ), and wewrite PKi ⊂ IKi (⊂ GKi ) for the wild inertia subgroup and the inertia subgroup of GKi∼
respectively. Let α : GK1 → GK2 be an isomorphism of profinite groups. Then we havethe following:(1) p1 = p2 (=: p).
∼

×
×
×
ab
ab
(2) The abelianisation αab : Gab
K1 → GK2 , and the inclusions ki ⊂ OKi ⊂ Ki ⊂ GKi ,where the last inclusion is defined by the local class field theory, induce isomorphisms∼(a) αab : k1× → k2× ,∼××(b) αab : OK
→ OK
,
1
2

a proof of the abc conjecture after Mochizuki

49

∼




(c) αab : OK
→ OK
(cf. Section 0.2 for the notation OK), and12i∼(d) αab : K1× → K2× .
(3) (a) [K1 : Qp ] = [K2 : Qp ],(b) f (K1 ) = f (K2 ), and(c) e(K1 ) = e(K2 ).
(4) The restrictions of α induce∼(a) α|IK1 : IK1 → IK2 , and∼(b) α|PK1 : PK1 → PK2 .
∼ab(5) The induced map Gab
K1 /IK1 → GK2 /IK2 preserves the Frobenius element FrobKi(i.e., the automorphism given by ki 3 x 7→ x#ki ).
}{∼α(6) The collection of the isomorphisms (α|U1 )ab : U1ab → U2ab
inopen∼GK1 ⊃ U1 −→U2 ⊂GK2∼

duces an isomorphism µQ/Z (K1 ) → µQ/Z (K2 ), which is compatible with the actions∼of GKi for i = 1, 2, via α : GK1 → GK2 . In particular, α preserves the cyclotomic
characters χcyc,i for i = 1, 2.
∼(7) The isomorphism α∗ : H 2 (Gal(K2 /K2 ), µQ/Z (K2 )) → H 2 (Gal(K1 /K1 ), µQ/Z (K1 ))∼induced by α is compatible with the isomorphisms H 2 (Gal(Ki /Ki ), µQ/Z (Ki )) →Q/Z in the local class field theory for i = 1, 2.
Remark 2.1.1.
In the proof, we can see that the objects in the above (1)–(7) are
functorially reconstructed by using only K1 (or K2 ), and we have no need of both K1
and K2 , nor the isomorphism α (i.e., no need of referred models). In this sense, thereconstruction algorithms in the proof are in the “mono-anabelian philosophy” of
Mochizuki (cf. also Remark 3.4.4 (2), (3)).
Proof. We can group-theoretically reconstruct the objects in (1)-(7) from GKi as
follows:{}(1): pi is the unique prime number which attains the maximum of rankZl Gab
Ki l: prime ,× ∧∼by the local class field theory Gab
Ki = (Ki ) .
prime-to-p(2a): k × ∼the prime-to-p part of the torsion subgroup of Gab ,= (Gab )i

Ki tors

Ki

where p is group-theoretically reconstructed in (1).
(3a): [Ki : Qp ] = rankZp Gab
Ki − 1, where p is group-theoretically reconstructed in(1).
(3b): pf (Ki ) = #(ki× ) + 1, where ki and p are group-theoretically reconstructed in(2a) and (1) respectively.

50

Go Yamashita(3c): e(Ki ) = [Ki : Qp ]/f (Ki ), where the numerator and the denominator aregroup-theoretically reconstructed in (3a) and (3b) respectively.
∩(4a): IKi = GK ⊃U : open, e(U )=e(GK ) U , where we write e(U ) for the number groupiitheoretically constructed from U in (3c) (i.e., e(U ) := (rankZp U ab −1)/ logp (#(U ab )prime-to-p
+
tors
}
{
ab
ab
1), where {p} := p | rankZp GKi = maxl rankZl GKi and logp is the (real) logarithmwith base p).
(4b): PKi = (IKi )pro-p the pro-p part of IKi , where IKi is group-theoreticallyreconstructed in (4a).
}{× ∼by the local class field theory,(2b): OK
= Im (IKi ) := Im IKi ,→ GKi  Gab
Ki
i
where IKi is group-theoretically reconstructed in (4a).
(5): The Fronbenius element FrobKi is characterised by the element in GKi /IKi (∼=f (Ki )abGKi /Im (IKi )) such that the conjugate action on IKi /PKi is a multiplication by p
(Here we regard the topological group IKi /PKi additively), where IKi and PKi aregroup-theoretically reconstructed in (4a) and (4b) respectively.
(2c): We reconstruc OK
by the following pull-back diagram:i0/ Im (IKi )
O

/ Gab
Ki
O

/ Gab /Im (IKi )
Ki
O

/0

?
/ O

?
/ Z≥0 Frob
Ki

/ 0,

=

0

/ Im (IKi )Ki

where IKi and FrobKi are group-theoretically reconstructed in (4a) and (5) respectively.
(2d): In the same way as in (2c), we reconstruc Ki× by the following pull-backdiagram:/ Gab/ Gab /Im (IK )/0/ Im (IK )
0
i
Ki
Ki
O i
O
O
=

0

/ Im (IKi )

?
/ K×
i

?
/ ZFrobKi/ 0,

where IKi and FrobKi are group-theoretically reconstructed in (4a) and (5) respectively.
(6): Let L be a finite extension of Ki . Then we have the Verlangerung (or transfer)
ab
ab ∼
∼ ab
GKi → Gab
L of GL ⊂ GKi by the norm map GKi = H1 (GKi , Z) → H1 (GL , Z) = GL ingroup homology, which is a group-theoretic construction (Or, we can explicitly constructabthe Verlangerung Gab
Ki ,→ GL without group homology as follows: For x ∈ GKi , take a`lift xe ∈ GKi of x. We write GKi = i gi GL for the coset decomposition, and we writexegi = gj(i) xi for each i, where xi ∈ GL . Then the Verlangerung is given by Gab
Ki 3
∏
ab
x 7→ ( i xi mod [GL , GL ]) ∈ GL , where we write [GL , GL ] for the topological closureof the commutator subgroup [GL , GL ] of GL ). Then this reconstructs the inclusion
Ki× ,→ L× , by the local class field theory and the reconstruction in (2d). The conjugate×abaction of GKi on GL  Gab
L preserves L ⊂ GL by the reconstruction of (2d). This

51

a proof of the abc conjecture after Mochizuki×reconstructs the action of GKi on L× . By taking the limit, we reconstruct Ki , hence
×µQ/Z (Ki ) = Q/Z ⊗Zb Hom(Q/Z, Ki ) equipped with the action of GKi .
∼(7): The isomorphism H 2 (Gal(Ki /Ki ), µQ/Z (Ki )) → Q/Z is defined by the composition
×∼∼

H 2 (Gal(Ki /Ki ), µQ/Z (Ki )) → H 2 (Gal(Ki /Ki ), Ki ) ←− H 2 (Gal(Kiur /Ki ), (Kiur )× )∼∼∼→ H 2 (Gal(Kiur /Ki ), Z) ←− H 1 (Gal(Kiur /Ki ), Q/Z) = Hom(Gal(Kiur /Ki ), Q/Z) → Q/Z,
×

where the first isomorphism is induced by the canonical inclusion µQ/Z (Ki ) ,→ Ki , themultiplicative group (Kiur )× (not the field Kiur ) of the maximal unramified extension Kiur
of Ki and the Galois group Gal(Kiur /K) are group-theoretically reconstructed in (2d)and (4a) respectively, the third isomorphism is induced by the valuation (Kiur )×  Z,
which is group-theoretically reconstructed in (2b) and (2d), the fourth isomorphism
is induced by the long exact sequence associated to the short exact sequence 0 →Z → Q → Q/Z → 0, and the last isomorphism is induced by the evaluation at FrobKi ,which is group-theoretically reconstructed in (5). Thus, the above composition is grouptheoretically reconstructed.
§ 2.2.

Arithmetic Quotients.

Proposition 2.2.
([AbsAnab, Lemma 1.1.4]) Let F be a field, and write G :=Gal(F /F ) for a separable closure F of F . Let
1→∆→Π→G→1
be an exact sequence of profinite groups. We assume that ∆ is topologically finitely
generated.
(1) Assume that F is a number field. Then ∆ is group-theoretically characterised in Πby the maximal closed normal subgroup of Π which is topologically finitely generated.
(2) (Tamagawa) Assume that F is a finite extension of Qp . For an open subgroupΠ0 ⊂ Π, we write ∆0 := Π0 ∩ ∆ and G0 := Π0 /∆0 , and let G0 act on (∆0 )ab by the
conjugate. We also assume that(Tam1)() /abb∀Π0 ⊂ Π : open, Q := (∆0 )(tors) is a finitely generated free Z-module,
0G

where we write (−)G0 for the G0 -coinvariant quotient, and (tors) for the torsion
part of the numerator. Then ∆ is group-theoretically characterised in Π as theintersection of those open subgroups Π0 ⊂ Π such that, for any prime number l 6= p,

52

Go Yamashita

we have(Tam2)

dimQp (Π0 ) ⊗Zb Qp − dimQl (Π0 ) ⊗Zb Ql
)
(
ab
ab
= [Π : Π0 ] dimQp (Π) ⊗Zb Qp − dimQl (Π) ⊗Zb Ql ,ababwhere p is also group-theoretically characterised as the unique prime number suchababthat dimQp (Π) ⊗Zb Qp − dimQl (Π) ⊗Zb Ql 6= 0 for infinitely many prime numbersl.
Proof. (1): This follows from the fact that every topologically finitely generated
closed normal subgroup of Gal(F /F ) is trivial (cf. [FJ, Theorem 15.10]).
(2): We have the inflation-restriction sequence associated to 1 → ∆ → Π → G → 1:
1 → H 1 (G, Q/Z) → H 1 (Π, Q/Z) → H 1 (∆, Q/Z)G → H 2 (G, Q/Z),
where we write (−)G for the G-invariant submodule. For the last term H 2 (G, Q/Z), we
also have H 2 (G, Q/Z) = limn H 2 (G, n1 Z/Z) ∼Hom(H 0 (G, µn ), Q/Z) ∼H 0 (G, µn ),= lim= Hom(lim
−→
−→n
←−n
Q/Z) = 0 by the local class field theory. Thus, by taking Hom(−, Q/Z) of the above
exact sequence, we obtain an exact sequence()
0 → ∆ab G → Πab → Gab → 0.
Let F 0 denote the finite extension corresponding to an open subgroup G0 ⊂ G. Thenby the assumption of (Tam1), we obtain
dimQp (Π0 )ab⊗Zb Qp − dimQl (Π0 )= dimQp (G0 )abab⊗Zb Ql⊗Zb Qp − dimQl (G0 )ab⊗Zb Ql = [F 0 : Qp ],where the last equality follows from the local class field theory. The group-theoretic
characterisation of p follows from the above equalies. The above equalites also imply
that (Tam2) is equivalent to [F 0 : Qp ] = [Π : Π0 ][F : Qp ], which is equivalent to[Π : Π0 ] = [G : G0 ], i.e., ∆ = ∆0 . This proves the second claim of the proposition.
Lemma 2.3.
([AbsAnab, Lemma 1.1.5]) Let F be a non-Archimedean local field,
and A a semi-abelian variety over F . Let F be an algebraic closure of F , and write
G := Gal(F /F ). We write T (A) := Hom(Q/Z, A(F )) for the Tate module of A. Thenb
Q := T (A)G /(tors) is a finitely generated free Z-module.
Proof. We have an extension 0 → S → A → A0 → 0 of group schemes over F ,b ⊕n for somewhere S is a torus and A0 is an abelian variety over F . Then T (S) ∼= Z(1)
n after restristing on an open subgroup of G, where T (S) is the Tate module of T .

a proof of the abc conjecture after Mochizuki

53

Thus, the image of T (S) in Q is trivial. Therefore, we may assume that A is an abelianvariety. By [SGA7t1, Exposé IX §2], we have extensions
0 → T (A)≤−1 → T (A) → T (A)0 → 0,
0 → T (A)−2 → T (A)≤−1 → T (A)−1 → 0
of G-modules, where T (A)≤−1 and T (A)≤−2 are the “fixed part” and the “toric part”
of T (A) respectively in the terminology of [SGA7t1, Exposé IX §2], and we have isomorphisms T (A)−1 ∼= T (B) for an abelian variety B over F which has potentially good rebb T (A)−2 ∼where M 0 and M −2 are finitely
duction, and T (A)0 ∼= M −2 ⊗Z Z(1),= M 0 ⊗Z Z,
generated free Z-modules and G acts both on M 0 and M −2 via finite quotients. Thus,
the images of T (A)−2 and T (A)−1 in Q are trivial (by the Weil conjecture proved by
Weil for abelian varieties in the latter case). Therefore, we obtain Q ∼= (T (A)0 )G /(tors),b since Z
b is flat over Z. Now the lemma follows
which is isomorphic to (M 0 )G /(tors)⊗Z Z,
since (M 0 )G /(tors) is free over Z.
Corollary 2.4.
We have a group-theoretic characterisation of ∆ = π1 (XF , x)in Π = π1 (X, x) as Proposition 2.2 (2) (Tam2), where X is a geometrically connectedsmooth hyperbolic curve over a finite extension F of Qp , and s : Spec F → X a geometricpoint lying over Spec F (which gives a geometric point s on XF := X ×F F via XF →X).
Remark 2.4.1.
Let Σ be a set of prime numbers such that p ∈ Σ and #Σ ≥ 2.
In the situation of Corollary 2.4, let ∆Σ be the maximal pro-Σ quotient, and writeΠΣ := Π/ker(∆  ∆Σ ). Then the algorithm of Proposition 2.2 (2) works for ΠΣ aswell, hence Corollary 2.4.1 holds for ΠΣ as well.
Proof. The corollary immediately follows from Proposition 2.2 (2) and Lemma 2.3.

§ 2.3.

Slimness and Commensurable Terminality.

Definition 2.5.
(1) Let G be a profinite group. We say that G is slim if we have ZG (H) = {1} for any
open subgroup H ⊂ G.
(2) Let f : G1 → G2 be a continuous homomorhism of profinite groups. We say that
G1 relatively slim over G2 (via f ) if we have ZG2 (Im{H → G2 }) = {1} for any
open subgroup H ⊂ G1 .
Lemma 2.6.
([AbsAnab, Remark 0.1.1, Remark 0.1.2]) Let G be a profinite
gruop, and H ⊂ G a closed subgroup of G.

54

Go Yamashita(1) If H ⊂ G is relatively slim, then both H and G are slim.
(2) If H ⊂ G is commensurably terminal and H is slim, then H ⊂ G is relatively slim.
Proof. (1): For any open subgroup H 0 ⊂ H, we have ZH (H 0 ) ⊂ ZG (H 0 ) = {1}.
For any open subgroup G0 ⊂ G, we have ZG (G0 ) ⊂ ZG (H ∩ G0 ) = {1} since H ∩ G0 is
open in H.
(2): Let H 0 ⊂ H be an open subgroup. The natural inclusion CG (H) ⊂ CG (H 0 ) isan equality since H 0 is open in H. Then we have ZG (H 0 ) ⊂ CG (H 0 ) = CG (H) = H.
This combined with ZH (H 0 ) = {1} implies ZG (H 0 ) = {1}.
Proposition 2.7.
([AbsAnab, Theorem 1.1.1, Corollary 1.3.3, Lemma 1.3.1,
Lemma 1.3.7]) Let F be a number field, and v a non-Archimedean place. Let Fv be
an algebraic closure of Fv , F the algebraic closure of F in Fv .
(1) Write G := Gal(F /F ) ⊃ Gv := Gal(Fv /Fv ).
(a) Gv ⊂ G is commensurably terminal,(b) Gv ⊂ G is relatively slim,(c) Gv is slim, and(d) G is slim.
(2) Let X be a hyperbolic curve over F . Let s : Spec Fv → XFv := X ×F Fv bea geometric point lying over Spec Fv (which gives geometric points s on XF :=X ×F F , XFv := X ×F Fv , and X via XFv → XF → X, and XFv → XFv → X).
Write ∆ := π1 (XF , s) ∼= π1 (XFv , s), Π := π1 (X, s), and Πv := π1 (XFv , s). Let x
be any cusp of XF (i.e., a point of the unique smooth compactification of XF overF which does not lie in XF ), and we write Ix ⊂ ∆ (well-defined up to conjugates)b
for the inertia subgroup at x (Note that Ix is isomorphic to Z(1)).
For any prime(l)(l)
number l, we write Ix → ∆ for the maximal pro-l quotient of Ix ⊂ ∆ (Note that(l)(l)
Ix is isomorphic to Zl (1) and that it is easy to see that Ix → ∆(l) is injective).
(a) ∆ is slim,(b) Π and Πv are slim, and(l)(c) Ix ⊂ ∆(l) and Ix ⊂ ∆ are commensurably terminal.
Remark 2.7.1. Furthermore, we can show that Gal(F /F ) is slim for any Kummerfaithful field F (Remark 3.17.3).
Proof. (1a)(cf. also [NSW, Corollary 12.1.3, Corollary 12.1.4]): First, we claimthat any subfield K ⊂ F with K 6= F has at most one prime ideal which is indecomposable in F . Proof of the claim: Let p1 6= p2 be prime ideals in K which do not split

a proof of the abc conjecture after Mochizuki

55

in F . Let f1 ∈ K[X] be any irreducible polynomial of degree d > 0, and f2 ∈ K[X]
a completely split separable polynomial of the same degree d. By the approximation
theorem, for any  > 0 there exists f ∈ K[X] a polynomial of degree d, such that|f − f1 |p1 <  and |f − f2 |p2 < . Then for suﬃciently small  > 0 the splitting fields
of f and fi over Kpi coincide for i = 1, 2 by Krasner’s lemma. By assumption that
p1 6= p2 do not split in F , the splitting fields of f1 and f2 over K coincide. Then we
have K = F since splitting field of f2 is K, and f1 is any irreducible polynomial. The
claim is proved. We show (1a). We specify a base point of Gv to kill the conjugacy
indeterminacy, that is, we take a place ve in Kv over v, and we use Gve instead of Gv .
Let g ∈ CG (Gve). Then Gve ∩ Ggev 6= {1}, since Gve ∩ gGveg −1 = Gve ∩ Ggev has finite index
in Gve. Then the above claim implies that gev = ve. Thus, we have g ∈ Gve.
(c): Let GK ⊂ Gv be an open subgroup, and g ∈ ZGv (GK ). Then for any finite
Galois extension L over K, the action of g on GL , hence on Gab
L , is trivial. By the local
×class field theory, the action of g on L is also trivial. Thus, we have g = 1 since L isany extension over K.
(b) follows from (a), (c), and Lemma 2.6 (2).
(d) follows from (b) and Lemma 2.6 (1).
(2a): This is similar to the proof of (1c). Let H ⊂ ∆ be an open subgroup. We write
XH → XF for the finite étale covering corresponding to H. We take any suﬃciently
small open normal subgroup H 0 ⊂ ∆ such that H 0 ⊂ H and the corresponding finiteétale covering XH 0 → XH has the canonical compactification XH 0 of genus > 1. We
have an identification H 0 = π1 (XH 0 , y) for a basepoint y. We write JH 0 := Jac(XH 0 )with the origin O for the Jacobian variety of XH 0 . Let g ∈ ∆. Then we have thefollowing commutative diagram of pointed schemes:(XH 0 , y) 
gX





(XH 0 , g(y)) / (XH 0 , y)fygX

/ (JH 0 , O)
gJ



f
/ (XH 0 , g(y)) g(y) / (JH 0 , g(O)),

which induces
π1 (XH 0 , y)g∗Xπ1 (XH 0 , g(y))/ / π1 (JH 0 , O)

∼

g∗J


/ / π1 (JH 0 , g(O))/ T (JH 0 , O)
g∗J

∼


/ T (JH 0 , g(O)),

where we write T (JH 0 , O) and T (JH 0 , g(O)) for the Tate modules of JH 0 with origin O
and g(O) respectively (Note that we have the isomorphisms from π1 to the Tate modules,
since F is of characteristic 0). Here, the morphism g J : (JH 0 , O) → (JH 0 , g(O)) is the56

Go Yamashita

composite of an automorphism (g J )0 : (JH 0 , O) → (JH 0 , O) of abelian varieties andan addition by g(O). We also have a conjugate action conj(g) : H 0 = π1 (XH 0 , y) →π1 (XH 0 , g ∗ (y)) = gH 0 g −1 = H 0 , which induces an action conj(g)ab : (H 0 )ab → (H 0 )ab .
This is also compatible with the homomorphism induced by (g J )0 :(H 0 )abconj(g)ab(H 0 )ab/ / T (JH 0 , O)
(g J )0∗/ / T (JH 0 , O).

Assume that g ∈ Z∆ (H). Then the conjugate action of g on H 0 , hence on (H 0 )ab ,is trivial. By the surjection (H 0 )ab  T (JH 0 , O), the action (g J )0∗ : T (JH 0 , O) →T (JH 0 , O) is trivial. Thus, the action (g J )0 : (JH 0 , O) → (JH 0 , O) is also trivial, since
the torsion points of JH 0 are dense in JH 0 . Therefore, the morphism g J : (JH 0 , O) →(JH 0 , g ∗ (O)) of pointed schemes is the addition by g(O). Then the compatibility ofg X : (XH 0 , y) → (XH 0 , g(y)) and g J : (JH 0 , O) → (JH 0 , g(O)) with respect to fy andfg(y) (i.e., the first commutative diagram) implies that g X : (XH 0 , y) → (XH 0 , g(y)),
hence g X : (XH 0 , y) → (XH 0 , g(y)), is an identity morphism by (the uniqueness assertionof) Torelli’s theorem (cf. [Mil, Theorem 12.1 (b)]). Then we have g = 1 since H 0 is any
suﬃciently small open subgroup in H.
(b) follows from (a), (1c), and (1d).
(c): This is similar to the proof of (1a). We assume that C∆ (Ix ) 6= Ix (resp.
(l)
(l)
(l)
C∆(l) (Ix ) 6= Ix ). Let g ∈ C∆ (Ix ) (resp. C∆(l) (Ix )) be such that g is not in Ix (resp.
(l)(l)
Ix ). Since g 6∈ Ix (resp. g 6∈ Ix ), we have a finite Galois covering (resp. a finite
Galois covering of degree a power of l) Y → XF (which is unramified over x) and acusp y of Y over x such that y 6= g(y). By taking suﬃciently small ∆Y ⊂ ∆ (resp.
∆Y ⊂ ∆(l) ), we may assume that Y has a cusp y 0 6= y, g(y). We have Ig(y) = gIy g −1
(l)
(l)
(l)
(l)
(resp. Ig(y) = gIy g −1 ). Since Iy ∩ Ig(y) (resp. Iy ∩ Ig(y) ) has a finite index in Iy (resp.
(l)Iy ), we have a finite Galois covering (resp. a finite Galois covering of degree a powerof l) Z → Y such that Z has cusps z, g(z), and z 0 lying over y, g(y), and y 0 respectively,(l)(l)
and Iz = Ig(z) (resp. Iz = Ig(z) ), i.e., z and g(z) have conjugate inertia subgroups in(l)∆Z (resp. ∆Z ) (Note that inertia subgroups are well-defined up to inner conjugate).
On the other hand, we have abelian coverings of Z which are totally ramified over z
and not ramified over g(z), since we have a cusp z 0 other than z and g(z) (Note that∏gthe abelianisation of a surface relation γ1 · · · γn i=1 [αi , βi ] = 1 is γ1 · · · γn = 1, andthat if n ≥ 3, then we can choose the ramifications at γ1 and γ2 independently). This(l)contradicts that z and g(z) have conjugate inertia subgroups in ∆Z (resp. ∆Z ).

57

a proof of the abc conjecture after Mochizuki§ 2.4.

Characterisation of Cuspidal Decomposition Groups.

Let k a finite extension of Qp . For a hyperbolic curve X of type (g, r) over k, wewrite ∆X and ΠX for the geometric fundamental group (i.e., π1 of Xk := X ×k k) andthe arithmetic fundamental group (i.e., π1 of X) of X for some basepoint, respectively.
Note that we have a group-theoretic characterisation of the subgroup ∆X ⊂ ΠX (hence
the quotient ΠX  Gk ) by Corollary 2.4. For a cusp x, we write Ix and Dx for theinertia subgroup and the decomposition subgroup at x in ∆X and in ΠX respectively(they are well-defined up to inner automorphism). For a prime number l, we also write(l)(l)
Ix and ∆X for the maximal pro-l quotient of Ix and ∆X , respectively. Write also
(l)
(l)
(l)
(l)
ΠX := ΠX /ker(∆X  ∆X ). Then we have a short exact sequence 1 → ∆X → ΠX →Gk → 1.
Lemma 2.8.
([AbsAnab, Lemma 1.3.9], [AbsTopI, Lemma 4.5]) Let X be a
hyperbolic curve of type (g, r) over k.
(1) X is not proper (i.e., r > 0) if and only if ∆X is a free profinite group (Note that
this criterion is group-theoretic ).
(2) We can group-theoretically reconstruct (g, r) from ΠX as follows:
(
)wt=2
(
)wt=0
r = dimQl ∆ab− dimQl ∆ab
+1
b Ql
b Ql
X ⊗Z
X ⊗Z
 (
 1 dim ∆ab ⊗ Q − r + 1) if r > 0,
Ql
l
b
X
Z
g= 2
1
ab
 dimQ ∆ ⊗b Ql
if r = 0 for any l,2lX

if r > 0, for l 6= p,

Z

where (−)wt=w with w ∈ Z is the subspace on which the Frobenius at p acts withw
eigenvalues of weight w, i.e., algebraic numbers with absolute values q 2 (Note thatthe weight is independent of the choice of a lifting of the Frobenius element Frobkb
to Gk in the extension 1 → Ik → Gk → ZFrob
k → 1, since the action of theabinertia subgroup on ∆X is quasi-unipotent). Here, note also that Gk and ∆X aregroup-theoretically reconstructed from ΠX by Corollary 2.4, the prime number p,
the cardinality q of the residue field, and the Frobenius element Frobk are grouptheoretically reconstructed from Gk by Proposition 2.1 (1), (1) and (3b), and (5)respectively (cf. also Remark 2.1.1).
Remark 2.8.1. By the same group-theoretic algorithm as in Lemma 2.8, we can(l)(l)
also group-theoretically reconstruct (g, r) from the extension datum 1 → ∆X → ΠX →(l)
Gk → 1 for any l 6= p (i.e., in the case where the quotient ΠX  Gk is given).
Proof. (1): Trivial (Note that, in the proper case, the non-vanishing of H 2 implies
the non-freeness of ∆X ). (2): Let X ,→ X be the canonical smooth compactification.

58

Go Yamashita

Then we have
{
}
{ ab
}wt=2
ab
ab
r − 1 = dimQl ker ∆abb Ql  ∆ X ⊗Z
b Ql = dimQl ker ∆X ⊗Z
b Ql
b Ql  ∆X ⊗Z
X ⊗Z
wt=2
= dimQl (∆ab− dimQl (∆ab
⊗Zb Ql )wt=2
b Ql )
X ⊗Z
X
wt=2
ab
= dimQl (∆ab− dimQl (∆X⊗Zb Ql )wt=0
b Ql )
X ⊗Z
wt=0
wt=2
− dimQl (∆ab,= dimQl (∆abb Ql )b Ql )X ⊗Z
X ⊗Z

where the forth equality follows from the self-duality of ∆X . The rest of the lemma (theformula for g) is trivial.
Corollary 2.9.
([NodNon, Lemma 1.6 (ii)⇒(i)]) Let X be an aﬃne hyperbolic
curves over k, and X the canonical smooth compactification. We have the following
group-theoretic characterisations or reconstruction algorithms from ΠX :
(l)

(l)

(1) The natural surjection ∆X  ∆X (resp. ∆X  ∆X for any l 6= p) is group(l)theoretically characterised as follows: An open subgroup H ⊂ ∆X (resp. H ⊂ ∆X )(l)(l)
is contained in ker(∆X  ∆X ) (resp. ker(∆X  ∆X )) if and only if r(XH ) =(l)[∆X : H]r(X) (resp. r(XH ) = [∆X : H]r(X)), where XH is the coverings corresponding to H ⊂ ∆X , and r(−)’s are their number of cusps (Note that r(−)’s aregroup-theoretically computed by Lemma 2.8 (2) and Remark 2.8.1.
(l)(2) The inertia subgroups of cusps in ∆X for any l 6= p are characterised as follows:(l)A closed subgroup A ⊂ ∆X which is isomorphic to Zl is contained in the inertia(l)(l)
subgroup of a cusp if and only if, for any open subgroup ∆Y ⊂ ∆X , the composite
(l)

(l)

(l)

(l)

A ∩ ∆Y ⊂ ∆Y  ∆Y  (∆Y )abvanishes. Here, we write Y for the canonical smooth compactification of Y (Note(l)(l)
that the natural surjection ∆Y  ∆Y has a group-theoretic characterisation in
(1)).
(l)

(3) We can reconstruct the set of cusps of X as the set of ∆X -orbits of the inertia(l)subgroups in ∆X via conjugate actions by Proposition 2.7 (2c) (Note that inertia(l)subgroups in ∆X have a group-theoretic characterisation in (2)).
(4) By functorially reconstructing the cusps of any covering Y → X from ∆Y ⊂ ∆X ⊂e → X (NoteΠX , we can reconstruct the set of cusps of the universal pro-covering X
that the set of cusps of Y is reconstructed in (3)).
(5) We can reconstruct inertia subgroups in ∆X as the subgroups that fix some cusp
e → X of X determined by the basepoint underof the universal pro-covering X
e is reconstructed in (4)).
consideration (Note that the set of cusps of X

a proof of the abc conjecture after Mochizuki

59(6) We have a characterisation of decomposition groups D of cusps in ΠX (resp. in(l)ΠX for any l 6= p) as D = NΠX (I) (resp. D = NΠ(l) (I)) for some inertia subgroupX(l)in ∆X (resp. in ∆X ) by Proposition 2.7 (2c) (Note that inertia subgroups in ∆X(l)and ∆X are reconstructed in (5) and in (2) respectively).
Remark 2.9.1.
(cf. also [IUTchI, Remark 1.2.2, Remark 1.2.3]) The arguments in[AbsAnab, Lemma 1.3.9], [AbsTopI, Lemma 4.5 (iv)], and [CombGC, Theorem 1.6 (i)]are wrong, because there is no covering of degree l of proper curves, which is ramified
at one point and unramified elsewhere (Note that the abelianisations of the geometricfundamental group of a proper curve is equal to the one of the curve obtained byremoving one point from the curve).
Proof. The claims (1) is trivial. (2): The “only if” part is trivial since an inertia
(l)
(l)
(l)
subgroup is killed in ∆Y . We show the “if” part. Write ∆Z := A∆Y ⊂ ∆X . The
(l)
(l)
(l)
(l)
(l)
(l)
natural surjection ∆Z  ∆Z /∆Y ∼= A/(A ∩ ∆Y ) factors as ∆Z  (∆Z )ab (l)(l)
A/(A∩∆Y ), since A/(A∩∆Y ) is isomorphic to an abelian group Z/lN Z for some N . By
(l)
(l)
(l)
the assumption of the vanishing of A∩∆Y in (∆Y )ab , the image Im{A∩∆Y → (∆Y )ab }(l)is contained in the subgroup generated by the image of the inertia subgroups in ∆Y .
(l)
(l)
(l)
(l)
Hence the image Im{A ∩ ∆Y → (∆Y )ab → (∆Z )ab  A/(A ∩ ∆Y )(∼= Z/lN Z)} is(l) ∼contained in the image of the subgroup in A/(A ∩ ∆Y )(= Z/lN Z) generated by the
(l)
(l)
(l)
(l)
image of the inertia subgroups in ∆Y . Since the composite A ⊂ ∆Z  ∆Z /∆Y ∼
=
(l) ∼
N
N
A/(A ∩ ∆Y )(= Z/l Z) is a surjection, and since Z/l Z is cyclic, there exists the(l)image I z ⊂ (∆Z )ab of the inertia subgroup of a cusp z in Z, such that the composite(l)(l)
I z ⊂ (∆Z )ab  A/(A∩∆Y )(∼= Z/lN Z) is surjective (Note that if we are working in the
profinite geometric fundamental groups, instead of pro-l geometric fundamental groups,
then the cyclicity does not hold, and we cannot use the same argument). This meansthat the corresponding subcovering Y → Z(→ X) is totally ramified at z. The claims(3), (4), (5), and (6) are trivial.
Remark 2.9.2.
(Generalisation to l-cyclotomically full fields, cf. also [AbsTopI,
Lemma 4.5 (iii)], [CombGC, Proposition 2.4 (iv), (vii), proof of Corollary 2.7 (i)]) We cangeneralise the results in this subsection for an l-cyclotomically full field k for some l (cf.
Definition 3.1 (3) below), under the assumption that the quotient ΠX  Gk is given, as
follows: For the purpose of a characterisation of inertia subgroups of cusps, it is enoughto consider the case where X is aﬃne. First, we obtain a group-theoretic reconstructionof a positive power χ+
cyc,l,up to fin of the l-adic cyclotomic character up to a character of∧dimQ (H ab ⊗Zb Ql ) ablfinite order by the actions of Gk on(H ⊗Zb Ql ) for characteristic open
torsion-free subgroups H ⊂ ∆X . Next, we group-theoretically reconstruct the l-adic
cyclotomic character χcyc,l,up to fin up to a character of finite order as χcyc,l,up to fin =60

Go Yamashitaχmax , where χmax is the maximal power of χ+
cyc,up to fin by which Gk acts in someabsubquotient of H ⊗Zb Ql for suﬃciently small characteristic open torsion-free subgroupsH ⊂ ∆X . Once we reconstruct the l-adic cyclotomic character χcyc,l,up to fin up to acharacter of finite order, then, for a finite-dimensional Ql -vector space V with continuous0
Gk -action, we take any filtration V = V 0 ⊃ V 1 ⊃ . . . (resp. V (χ−1cyc,l,up to fin ) = V ⊃V 1 ⊃ . . .) of Ql [Gk ]-modules (Here we write V (χ−1 ) for the twist of V by χ−1 ) such thateach graded quotient either has the action of Gk factoring through a finite quotient or
has no nontrivial subquotients, and we use, instead of dimQl V wt=0 (resp. dimQl V wt=2 )in Lemma 2.8, the summation of dimQl V j /V j+1 , where the Gk -action on V j /V j+1
factors through a finite quotient of Gk , and the rest is the same.
§ 3.

Mono-anabelian Reconstruction Algorithms.

In this section, we show mono-anabelian reconstruction algorithms, which are crucial ingredients of inter-universal Teichmüller theory.
§ 3.1.

Some Definitions.

Definition 3.1.
([pGC, Definition 1.5.4 (i)], [AbsTopIII, Definition 1.5], [CombGC,
Definition 2.3 (ii)]) Let k be a field.
(1) We say that k is sub-p-adic, if there is a finitely generated field L over Qp for somep such that we have an injective homomorphism k ,→ L of fields.
(2) We say that k is Kummer-faithful, if k is of characteristic 0, and if for any
finite extension k 0 of k and any semi-abelian variety A over k 0 , the Kummer map∩
A(k 0 ) → H 1 (k 0 , T (A)) is injective (which is equivalent to N ≥1 N A(k 0 ) = {0}),where we write T (A) for the Tate module of A.
(3) We say that k is l-cyclotomically full, if the l-adic cyclotomic character χcyc,l :Gk → Z×l has an open image.
Remark 3.1.1. ([pGC, remark after Definition 15.4]) For example, the following
fields are sub-p-adic:(1) finitely generated extensions of Qp , in particular, finite extensions of Qp ,(2) finite extensions of Q, and(3) the subfield of an algebraic closure Q of Q which is the composite of all number
fields of degree ≤ n over Q for some fixed integer n (Note that such a field can beembedded into a finite extension of Qp by Krasner’s lemma).

a proof of the abc conjecture after Mochizuki

Lemma 3.2.

61([AbsTopIII, Remark 1.5.1, Remark 1.5.4 (i), (ii)])(1) If k is sub-p-adic, then k is Kummer-faithful.
(2) If k is Kummer-faithfull, then k is l-cyclotomically full for any l.
(3) If k is Kummer-faithfull, then any finitely generated field over k is also Kummerfaithful.
Proof. (3): Let L be a finitely generated extension of k. By Weil restriction, theinjectivity of the Kummer map for a finite extension L0 of L is reduced to the one for
L, i.e., we may assume that L0 = L. Let A be a semi-abelian variety over L. Let U be
an integral smooth scheme over k such that A extends to a semi-abelian scheme A over
U and the function field of U is L. By a commutative diagram/ H 1 (L, T (A))

A(L)
_
∏


A
x∈|U | x (Lx )


1
H
(L
x , T (Ax )),x∈|U |/∏

where we write |U | for the set of closed points, Lx is the residue field at x, and Ax is thefiber at x (Note that a ∈ A(L) is zero on any fiber of x ∈ |U |, then a is zero since |U |is dense in U ), we may assume that L is a finite extension of k. In this case, again by
Weil restriction, the injectivity of the Kummer map for a finite extension L is reduced
to the one for k, which holds by assumption.
(1): By the same way as in (3), by Weil restriction, the injectivity of the Kummer
map for a finite extension k 0 of k is reduced to the one for k, i.e., we may assume thatk 0 = k. Let k embed into a finitely generated field L over Qp . By the base change fromk to L and the following commutative diagram
A(k)
_

A(L)

/ H 1 (k, T (A))

/ H 1 (L, T (A)),

the injectivity of the Kummer map for k is reduced to the one for L, i.e., we may
assume that k is a finitely generated extension over Qp . Then by (3), we may assume∩
that k = Qp . If A is a torus, then N ≥1 N A(Qp ) = {0} is trivial. Hence the claim is
reduced to the case where A is an abelian variety. Then A(Qp ) is a compact abelianp-adic Lie group, which contains Z⊕nfor some n as an open subgroup. Hence we havep∩N ≥1 N A(Qp ) = 0. Thus, the Kummer map is injective. We are done.

62

Go Yamashita(2): For any finite extensin k 0 over k, the Kummer map for Gm over k 0 is injectiveby the assumption. This implies that the image of l-adic cyclotomic character Gk → Z×l
has an open image.
Definition 3.3.
([CanLift, Section 2]) Let k be a field. Let X be a geometricallynormal, geometrically connected algebraic stack of finite type over k.
(1) We write Lock (X) for the category whose objects are generically scheme-like algebraic stacks over k which are finite étale quotients (in the sense of stacks) of(necessarily generically scheme-like) algebraic stacks over k that admit a finite étalemorphism to X over k, and whose morphisms are finite étale morphisms of stacks
over k.
(2) We say X admits k-core if there exists a terminal object in Lock (X). We shallrefer to a terminal object in Lock (X) as a k-core.
For an elliptic curve E over k with the origin O, we shall refer to the hyperbolic
orbicurve (cf. Section 0.2) obtained as the quotient (E \ {O})//±1 in the sense of stacks
as a semi-elliptic orbicurve over k (cf. [AbsTopII, §0]. It is also called “punctured
hemi-elliptic orbicurve” in [CanLift, Definition 2.6 (ii)]).
Definition 3.4.
([AbsTopII, Definition 3.5, Definition 3.1]) Let X be a hyperbolic orbicurve (cf. Section 0.2) over a field k of characteristic 0.
(1) We say that X is of strictly Belyi type if (a) X is defined over a number field,
and if (b) there exist a hyperbolic orbicurve X 0 over a finite extension k 0 of k, a
hyperbolic curve X 00 of genus 0 over a finite extension k 00 of k, and finite étale
coverings X  X 0  X 00 .
(2) We say that X is elliptically admissible if X admits k-core X  C, where C isa semi-elliptic orbicurve.
Remark 3.4.1.
In the moduli space Mg,r of curves of genus g with r cusps, theset of points corresponding to the curves of strictly Belyi type is not Zariski open for2g − 2 + r ≥ 3, g ≥ 1. cf. [Cusp, Remark 2.13.2] and [Corr, Theorem B].
Remark 3.4.2.
If X is elliptically admissible and defined over a number field, then
X is of strictly Belyi type (cf. also [AbsTopIII, Remark 2.8.3]), since, for an elliptic curve[2]

doubleE, we have a diagram of finite étale coverings E \ {0}  E \ E[2]  P1 \ {4 pts}.
cover

For a hyperbolic curve X over a field k of characteristic zero with the canonicalsmooth compactification X. A closed point x in X is called algebraic, if there area proof of the abc conjecture after Mochizuki

63

a finite extension K of k, a hyperbolic curve Y over a number field F ⊂ K with thecanonical smooth compactification Y , and an isomorphism X ×k K ∼= Y ×F K over K
such that x maps to a closed point under the composition X ×k K ∼= Y ×F K → Y .
§ 3.2.

Belyi and Elliptic Cuspidalisations — Hidden Endomorphisms.

Let k be a field of characteristic 0, and k an algebraic closure of k. Write Gk :=Gal(k/k). Let X be a hyperbolic orbicurve over k (cf. Section 0.2). We write ∆X andΠX for the geometric fundamental group (i.e., π1 of Xk := X ×k k) and the arithmetic
fundamental group (i.e., π1 of X) of X for some basepoint, respectively. Note that
we have an exact sequence 1 → ∆X → ΠX → Gk → 1. We consider the following
conditions on k and X:(Delta)X : We have a “group-theoretic characterisation” (for example, like Proposition 2.2 (1),(2)) of the subgroup ∆X ⊂ ΠX (or equivalently, the quotient ΠX  Gk ).
(GC): Isom-version of the relative Grothendieck conjecture (cf. also Theorem B.1) for the
profinite fundamental groups of any hyperbolic (orbi)curves over k holds, i.e., thenatural map Isomk (X, Y ) → IsomOut
Gk (∆X , ∆Y ) := IsomGk (∆X , ∆Y )/Inn(∆Y ) isbijective for any hyperbolic (orbi)curve X, Y over k.
(slim): Gk is slim (Definition 2.5 (1)).
(Cusp)X : We have a “group-theoretic characterisation” (for example, like Proposition 2.9 (3))
of decomposition groups in ΠX of cusps.
We also consider the following condition (of diﬀerent nature):(Delta)’X : Either• ΠX is given and (Delta)X holds, or• ∆X ⊂ ΠX are given.
Note that (Delta)X , (GC), and (slim) are conditions on k and X; however, as for(Delta)’X , “the content of a theorem” depends on which case of (Delta)’X is satisfied,i.e., in the former case, the algorithm in a theorem requires only ΠX as (a part of) an
input datum, on the other hand, in the latter case, the algorithm in a theorem requires
both ∆X ⊂ ΠX as (a part of) input data.
Remark 3.4.3.
(1) (Delta)X holds for any X in the case where k is an NF by Proposition 2.2 (1) or k
is an MLF by Corollary 2.4.

64

Go Yamashita(2) (GC) holds in the case where k is sub-p-adic by Theorem B.1.
(3) (slim) holds in the case where k is an NF by Proposition 2.7 (1) (d) or k is an MLF
by Proposition 2.7 (1) (c). More generally, it holds for Kummmer-faithful field k
by Remark 3.17.3, which is shown without using the results in this subsection.
(4) (Cusp)X holds for any X in the case where k is an MLF by Corollary 2.9. Moregenerally, (Cusp)X holds for l-cyclotomically full field k for some l under the assumption (Delta)’X by Remark 2.9.2.
In short, we have the following table (cf. also Lemma 3.2):
NF, MLF(Delta)X holds
for any X⇒

sub-p-adic(GC) holds⇒

Kummer-faithful(slim) holds⇒

l-cyclotomically full
(Cusp)X holds
under (Delta)’X .

Remark 3.4.4.
(1) It seems diﬃcult to rigorously formulate the meaning of “group-theoretic characterisation”. Note that the formulation for (Delta)X like “any isomorphism ΠX1 ∼= ΠX2of topological groups induces an isomorphism ∆X1 ∼= ∆X2 of topological groups”(it is called bi-anabelian approach) is a priori weaker than the notion of “grouptheoretic characterisation” of ∆X in ΠX (this is called mono-anabelian approach),which allows us to reconstruct the object itself (not the morphism between two objects).
(2) (Important Convention) In the same way, it also seems diﬃcult to rigorously formulate “there is a group-theoretic algorithm to reconstruct” something in the senseof mono-anabelian approach (Note that it is easy to rigorously formulate it in thesense of bi-anabelian approach). To rigorously settle the meaning of it, it seems
that we have to state the algorithm itself, i.e., the algorithm itself have to be a partof the statement. However, in this case, the statement must be often rather lengthy
and complicated. In this survey, we use the phrase “group-theoretic algorithm”
loosely in some sense, for the purpose of making the input data and the output
data of the algorithms in the statement clear. However, the rigorous meaning will
be clear in the proof, since the proof shows concrete constructions, which, properly
speaking, should be included in the statement itself. We sometimes employ this convention of stating propositions and theorems in this survey (If we use the languageof species and mutations (cf. [IUTchIV, §3]), then we can rigorously formulatemono-anabelian statements without mentioning the contents of algorithms).
(3) Mono-anabelian reconstruction algorithms have an advantage, as contrasted withbi-anabelian approach, of avoiding “a referred model” of a mathematical object like

a proof of the abc conjecture after Mochizuki

65“the C”, i.e., it is a “model-free” (or “model-implicit”) approach. For more informations on Mochizuki’s philosophy of mono-anabelian reconstruction algorithms
versus bi-anabelian reconstruction algorithms, see [AbsTopIII, §I.3, Remark 3.7.3,
Remark 3.7.5].
In this subsection, to avoid settling the meaning of “group-theoretic characterisation” in (Delta)X and (Cusp)X (cf. Remark 3.4.4 (1)), we assume that k is sub-p-adic,
and we include the subgroup ∆X (⊂ ΠX ) as an input datum. More generally, the results in this section hold in the case where k and X satisfy (Delta)’X , (GC), (slim),and (Cusp)X . Note that if we assume that k is an NF or an MLF, then (Delta)X ,(GC), (slim), and (Cusp)X hold for any X, and we do not need include the subgroup∆X (⊂ ΠX ) as an input datum.
Lemma 3.5.
Let ψ : H → Π be an open homomorphism of profinite groups,
and φ1 , φ2 : Π → G two open homomorphisms of profinite groups. We assume that G
is slim. If φ1 ◦ ψ = φ2 ◦ ψ, then we have φ1 = φ2 .
Proof. By replacing H by the image of ψ, we may assume that H is an open
subgroup of Π. By replacing H by ∩g∈Π/H gHg −1 , we may assume that H is an open
normal subgroup of Π. For any g ∈ Π and h ∈ H, we have ghg −1 ∈ H, and φ1 (ghg −1 ) =φ2 (ghg −1 ) by assumption. This implies that φ1 (g)φ1 (h)φ1 (g)−1 = φ2 (g)φ2 (h)φ2 (g)−1 =φ2 (g)φ1 (h)φ2 (g)−1 . Hence we have φ1 (g)φ2 (g)−1 ∈ ZIm(Π) (G). By the assumption of
the slimness of G, we have ZIm(Π) (G) = {1}, since Im(Π) is open in G. Therefore, we
obtain φ1 (g) = φ2 (g), as desired.
Remark 3.5.1.
In the algebraic geometry, a finite étale covering Y  X is an
epimorphism. The above lemma says that the inclusion map ΠY ⊂ ΠX correspoinding
to Y  X is also an epimorphism if ΠX is slim. This enables us to make a theory forprofinite groups (without using 2-categories and so on.) which is parallel to geometry,when all involved profinite groups are slim. This is a philosophy behind the geometryof anabelioids ([Anbd]).
Choose a hyperbolic orbicurve X over k, and we write ΠX for the arithmetic fundamental group of X for some basepoint. We have the surjection ΠX  Gk determinedby (Delta)’X . Note that now we are assuming that k is sub-p-adic, hence Gk is slim
by Lemma 3.2 (1) and Remark 3.17.3. Let G ⊂ Gk be an open subgroup, and writeΠ := ΠX ×Gk G, and ∆ := ∆X ∩ Π. In this survey, we do not adopt the convention that
we always write (−)0 for the commutator subgroup for a group (−).
In the elliptic and Belyi cuspidalisations, we use the following three types of operations:

66

Go Yamashita

Lemma 3.6.
Write Π0 := ΠX 0 to be the arithmetic fundamental group of a
hyperbolic orbicurve X 0 over a finite extension k 0 of k. Write ∆0 := ker(Π0  Gk0 ).
(1) Let Π00 ,→ Π0 be an open immersion of profinite groups. Then Π00 arises as a finiteétale covering X 00  X 0 of X 0 , and ∆00 := Π00 ∩ ∆0 reconstructs ∆X 00 .
(2) Let Π0 ,→ Π00 be an open immersion of profinite groups such that there exists a
surjection Π00  G00 to an open subgroup of G, whose restriction to Π0 is equal to
the given homomorphism Π0  G0 ⊂ G. Then the surjection Π00  G00 is uniquelydetermined (hence we reconstruct the quotient Π00  G00 as the unique quotient of
Π00 having this property), and Π00 arises as a finite étale quotient X 0  X 00 of X 0 .
(3) Assume that X 0 is a scheme i.e., not a (non-scheme-like) stack (We can treat orbicurves as well; however, we do not use this generalisation in this survey. cf.
[AbsTopI, Definition 4.2 (iii) (c)]). Let Π0  Π00 be a surjection of profinite groups
such that the kernel is generated by a cuspidal inertia subgroup group-theoreticallycharacterised by Corollary 2.9 and Remark 2.9.2 (We shall refer to it as a cuspidal
quotient). Then Π00 arises as an open immersion X 0 ,→ X 00 , and we reconstruct∆X 00 as ∆0 /∆0 ∩ ker(Π0  Π00 ).

Proof. (1) is trivial by the definition of ΠX 0 .
The first asserion of (2) comes from Lemma 3.5, since G is slim. Write (Π0 )Gal :=∩g∈Π00 /Π0 gΠ0 g −1 ⊂ Π0 , which is normal in Π00 by definition. Then (Π0 )Gal arises froma finite étale covering (X 0 )Gal  X 0 by (1). By the conjugation, we have an action ofΠ00 on (Π0 )Gal . By (GC), this action determines an action of Π00 /(Π0 )Gal on (X 0 )Gal .
We take the quotient X 00 := (X 0 )Gal //(Π00 /(Π0 )Gal ) in the sense of stacks. Then ΠX 00
is isomorphic to Π00 by definition, and the quotinet (X 0 )Gal  X 00 factors as (X 0 )Gal 
X 0  X 00 since the intermediate quotient (X 0 )Gal //(Π0 /(Π0 )Gal ) is isomorphic to X 0 .
This proves the second assertion of (2).
(3) is also trivial.
3.2.1. Elliptic Cuspidalisation. Let X be an elliptically admissible orbicurve overk. By definition, we have a k-core X  C = (E \ {O})//{±1} where we write E for
an elliptic curve over k with the origin O. Let N ≥ 1 be a positive integer. We write
UC,N := (E \ E[N ])//{±1} ⊂ C for the open sub-orbicurve of C determined by theimage of E \E[N ]. Write UX,N := UC,N ×C X ⊂ X, which is an open suborbicurve of X.
For a finite extension K of k, write XK := X ×k K, CK := C ×k K, and EK := E ×k K.
For a suﬃciently large finite extension K of k, all points of EK [N ] are rational over K.

67

a proof of the abc conjecture after Mochizuki

We have the following key diagram for elliptic cuspidalisation:(EllCusp)

X

//Coo

E \ {O} o oN

E \ E[N_ ]/ / UC,N o o
_

UX,N
_


E \ {O}


//Coo


X,

where ’s are finite étale coverings, ,→’s are open immersions, and two sqauares arecartesian.
We will use the technique of elliptic cuspidalisation three times:(1) Firstly, in the theory of Aut-holomorphic space in Section 4, we will use it for thereconstruction of “local linear holomorphic structure” of an Aut-holomorphic space(cf. Proposition 4.5 (Step 2)).
(2) (This is the most important usage) Secondly, in the theory of the étale theta function
in Section 7, we will use it for the constant multiple rigidity of the étale theta function(cf. Proposition 7.9).
(3) Thirdly, we will use it for the reconstruction of “pseudo-monoids” (cf. Section 9.2).
Theorem 3.7.
(Elliptic Cuspidalisation, [AbsTopII, Corollary 3.3]) Let X be
an elliptically admissible orbicurve over a sub-p-adic field k. Let N ≥ 1 be a positive
integer, and we write UX,N for the open sub-orbicurve of X defined as above. Then fromthe profinite groups ∆X ⊂ ΠX , we can group-theoretically reconstruct (cf. Remark 3.4.4(2)) the surjection
πX : ΠUX,N  ΠXof profinite groups, which is induced by the open immersion UX,N ,→ X, and the set ofthe decomposition groups in ΠX at the points in X \ UX,N .
We shall refer to πX : ΠUX,N  ΠX as an elliptic cuspidalisation.
Proof. (Step 1): By (Delta)’X , we have the quotient ΠX  Gk with kernel ∆X .
Let G ⊂ Gk be a suﬃciently small (which will depend on N later) open subgroup, andwrite Π := ΠX ×Gk G, and ∆ := ∆X ∩ Π.
(Step 2): We define a category LocG (Π) as follows: The objects are profinite groups0Π such that there exist open immersions Π ←- Π00 ,→ Π0 of profinite groups and68

Go Yamashita

surjections Π0  G0 , Π00  G00 to open subgroups of G, and that the diagramΠo? _ Π00  

/ Π0


G


G _ 00


G
_0

=


Go

=


G

=


/ G.

is commutative. Note that, by this compatibility, the surjections Π0  G0 and Π00  G00
are uniquely determined by Lemma 3.6 (1), (2) (or Lemma 3.5). The morphisms from
Π1 to Π2 are open immersions Π1 ,→ Π2 of profinite groups up to inner conjugates byker(Π2  G2 ) such that the uniquely determined homomorphisms Π1  G1 ⊂ G andΠ2  G2 ⊂ G are compatible. The definition of the category LocG (Π) depends only
on the topological group structure of Π and the surjection Π  G of profinite groups.
∼
By (GC), the functor X 0 7→ ΠX 0 gives us an equivalence LocK (XK ) → LocG (Π) of
categories, where K is the finite extension of k corresponding to G ⊂ Gk . Then wegroup-theoretically reconstruct (ΠXK ⊂)ΠCK as the terminal object (Π ⊂)Πcore of thecategory LocG (Π).
(Step 3): We group-theoretically reconstruct ∆CK (⊂ ΠCK ) as the kernel ∆core :=ker(Πcore → G). We group-theoretically reconstruct ∆EK \{O} as an open subgroup ∆ellof ∆core of index 2 such that ∆ell is torsion-free (i.e., the corresponding covering is ascheme, not a (non-scheme-like) stack), since the covering is a scheme if and only if thegeometric fundamental group is torsion-free (cf. also [AbsTopI, Lemma 4.1 (iv)]). We
take any (not necessarily unique) extension 1 → ∆ell → Πell → G → 1 such that the
push-out of it via ∆ell ⊂ ∆core is isomorphic to the extension 1 → ∆core → Πcore →00 \{O} , where E
G → 1 (Note that Πell is isomorphic to ΠEK
K \ {O} is a twist of order 1
0 \{O} as Πell (Note that if weor 2 of EK \ {O}). We group-theoretically reconstruct ΠEK
replace G by a subgroup of index 2, then we may reconstruct ΠEK \{O} ; however, we do
not detect group-theoretically which subgroup of index 2 is correct. However, the final
output does not depend on the choice of Πell ).
(Step 4): Let(a) Πell,N ,→ Πell be an open immersion of profinite groups with Πell /Πell,N ∼= (Z/N Z)⊕2
cptsuch that the composite Πell,N ,→ Πell  Πcpt
ell factors through as Πell,N  Πell,N →
cpt
cpt
Πcpt
ell , where we write Πell  Πell , Πell,N  Πell,N for the quotients by all of the
conjugacy classes of the cuspidal inertia subgroups in Πell , Πell,N respectively, and(b) Πell,N  Π0 a composite of (N 2 − 1) cuspidal quotients of profinite groups suchthat there exists an isomorphism Π0 ∼= Πell of profinite groups.

a proof of the abc conjecture after Mochizuki

69

cpt
Note that the factorisation Πell,N  Πcpt
ell,N → Πell means that the finite étale coveringcorresponding to Πell,N ,→ Πell extends to a finite étale covering of their compactifications i.e., the covering corresponding to Πell,N ,→ Πell is unramified at all cusps as well.
Note that there exists such a diagram

Πell ←- Πell,N  Π0 ∼= Πellby (EllCusp). Note that for any intermediate composite Πell,N  Π∗  Π0 of cuspidal
quotients in the composite Πell,N  Π0 of cuspidal quotients, and for the uniquelydetermined quotient Π∗  G∗ , we have G∗ = G for suﬃciently small open subgroup
G ⊂ Gk , and we take such an open subgroup G ⊂ Gk .
0 \E 0 [N ]  ΠE 0 \{O}We group-theoretically reconstruct the surjection πE 0 : ΠEK
K
K
0
0
0
\ EK
[N ] ,→ EK\ {O} as the composite πE 0 ? :induced by the open immersion EK
Πell,N  Π0 ∼= Πell , since we can identify πE 0 ? with πE 0 by (GC).
(Step 5): We write Πcore,1 for Πcore for G = Gk . If necessary, by changing Πell ,we may take Πell such that there exists a unique lift of Πcore,1 /Πell → Out(Πell ) toout

Out(Πell,N ) by (EllCusp). We form o (Πcore,1 /Πell ) (cf. Section 0.2) to the surjection
out

out

Πell,N  Πell i.e., Πell,N o (Πcore,1 /Πell )  Πell o (Πcore,1 /Πell ) = Πcore,1 , whereoutΠcore,1 /Πell → Out(Πell ) (in the definition of o (Πcore,1 /Πell )) is the natural one, andoutΠcore,1 /Πell → Out(Πell,N ) (in the definition of o (Πcore,1 /Πell )) is the unique lift ofΠcore,1 /Πell → Out(Πell ) to Out(Πell,N ). Then we obtain a surjection πC? : Πcore,N :=outΠell,N o (Πcore,1 /Πell )  Πcore,1 . We group-theretically reconstruct the surjection
πC : ΠUC,N  ΠC induced by the open immersion UC,N ,→ C as the surjection πC? :Πcore,N  Πcore,1 , since we can identify πC? with πC by (GC).
(Step 6): We form a fiber product ×Πcore,1 ΠX to the surjection Πcore,N  Πcore,1i.e., ΠX,N := Πcore,N ×Πcore,1 ΠX  Πcore,1 ×Πcore,1 ΠX = ΠX . Then we obtain asurjection πX? : ΠX,N  ΠX . We group-theretically reconstruct the surjection πX :
ΠUX,N  ΠX induced by the open immersion UX,N ,→ X as the surjection πX? :ΠX,N  ΠX , since the identification of πC? with πC induces an identification of πX?
with πX .
(Step 7): We group-theretically reconstruct the decomposition groups at the pointsof X \UX,N in ΠX as the image of the cuspidal decomposition groups in ΠX,N , which aregroup-theoretically characterised by Corollary 2.9, via the surjection ΠX,N  ΠX .
3.2.2. Belyi Cuspidalisation. Let X be a hyperbolic orbicurve of strictly Belyi
type over k. We have finite étale coverings X  Y  P1 \ (N points), where Y is a
hyperbolic curve over a finite extension k 0 of k, and N ≥ 3. We assume that Y  X
is Galois. For any open sub-orbicurve UX ⊂ X defined over a number field, write
UY := Y ×X UX . Then by the theorem of Belyi (cf. also Theorem C.2 for its refinement),

70

Go Yamashita

we have a finite étale covering UY0  UP1 from an open sub-orbicurve UY0 ⊂ UY to the
tripod UP1 (cf. Section 0.2) over k 0 . For a suﬃciently large finite extension K of k 0 , all
the points of Y \ UY0 are defined over K. We have the following key diagram for Belyi
cuspidalisation:
UY0 (BelyiCusp)

XooY/ / P1 \ (N points) 






/ UP1

/ UY  

/Y




/ X,

UX



where ’s are finite étale coverings, ,→’s are open immersions, and the square is cartesian.
Theorem 3.8.
(Belyi Cuspidalisation, [AbsTopII, Corollary 3.7]) Let X be anorbicurve over a sub-p-adic field k. We assume that X is of strictly Belyi type. Thenfrom the profinite groups ∆X ⊂ ΠX , we can group-theoretically reconstruct (cf. Remark 3.4.4 (2)) the set{ΠUX  ΠX }UXof the surjections of profinite groups, where UX runs through the open subschemes of X
defined over a number field. We can also group-theoretically reconstruct the set of thedecomposition groups in ΠX at the points in X \ UX , where UX runs through the open
subschemes of X defined over a number field.
We shall refer to ΠUX  ΠX as a Belyi cuspidalisation.
Proof. (Step 1): By (Delta)’X , we have the quotient ΠX  Gk with kernel ∆X .
For suﬃciently small (which will depend on U later) open subgroup G ⊂ Gk , writeΠ := ΠX ×Gk G.
(Step 2): Let(a) Π ←- Π∗ be an open immersion of profinite grouops,(b) Π∗ ,→ Πtpd,U an open immersion of profinite groups, such that the group-theoretic
algorithms described in Lemma 2.8 and Remark 2.9.2 tell us that the hyperbolic
curve corresponding to Πtpd,U has genus 0,(c) Πtpd,U  Πtpd a composite of cuspidal quotients of profinite groups, such that the
number of the conjugacy classes of cuspidal inertia subgroups of Πtpd is three,0(d) Πtpd ←- Π∗,U an open immersion of profinite groups,0(e) Π∗,U  Π∗,U a composite of cuspidal quotients of profinite groups, anda proof of the abc conjecture after Mochizuki

71(f) Π∗,U  Π∗∗ a composite of cuspidal quotients of profinite groups such that there
exists an isomorphism Π∗∗ ∼= Π∗ of profinite groups.
Note that there exists such a diagramΠ ←- Π∗ ,→ Πtpd,U  Πtpd ←- Π∗,U  Π∗,U  Π∗∗ ∼= Π∗
0by (BelyiCusp). Note also that any algebraic curve over an algebraically closed field of
characteristic 0, which is finite étale over a tripod, is defined over a number field (i.e.,
converse of Belyi’s theorem, essentially the descent theory) and that algebraic points ina hyperbolic curve are sent to algebraic points via any isomorphism of hyperbolic curves
over the base field (cf. [AbsSect, Remark 2.7.1]). Write πY ? : Π∗,U  Π∗∗ ∼= Π∗ to be0
the composite. Note that for any intermediate composite Π∗,U  Π#  Π∗∗ in the0composite Π∗,U  Π∗∗ of cuspidal quotients and for the uniquely determined quotient
Π#  G# , we have G# = G for suﬃciently small open subgroup G ⊂ Gk , and we takesuch an open subgroup G ⊂ Gk .
We group-theoretically reconstruct the surjection πY : ΠUY  ΠY induced by someopen immersion UY ,→ Y as πY ? : Π∗,U  Π∗ , since we can identify πY ? with πY by(GC) (Note that we do not prescribe the open immersion UY ,→ Y ).
(Step 3): We choose the data (a)-(e) such that the natural homomorphism ΠX /Π∗ →Out(Π∗ ) has a unique lift ΠX /Π∗ → Out(Π∗,U ) to Out(Π∗,U ) (Note that this corresponds to that UY ⊂ Y is stable under the action of Gal(Y /X), thus descends to
out

out

UX ⊂ X). We form o (ΠX /Π∗ ) to the surjection Π∗,U  Π∗ i.e., ΠX,U := Π∗,U oout(ΠX /Π∗ )  Π∗ o (ΠX /Π∗ ) = ΠX . Then we obtain a surjection πX? : ΠX,U  ΠX .
We group-theretically reconstruct the surjection πX : ΠUX  ΠX induced by the open
immersion UX ,→ X as the surjection πX? : ΠX,U  ΠX , since we can identify πX?
with πX by (GC) (Note again that we do not prescribe the open immersion UX ,→ X.
We just group-theoretically reconstruct a surjection ΠUX  ΠX for some UX ⊂ X suchthat all of the points in X \ UX are defined over a number field).
(Step 4): We group-theretically reconstruct the decomposition groups at the pointsof X \ UX in ΠX as the image of the cuspidal decomposition groups in ΠX,U , which aregroup-theoretically characterised by Corollary 2.9, via the surjection ΠUX  ΠX .
Corollary 3.9.
([AbsTopII, 3.7.2]) Let X be a hyperbolic orbicurve over a nonArchimedean local field k. We assume that X is of strictly Belyi type. Then from theprofinite group ΠX , we can reconstruct the set of the decomposition groups at all closed
points in X.
Proof. The corollary follows from Theorem 3.8 and the approximation of a decomposition group in (the proof of) Lemma 3.10 below.

72

Go Yamashita

Since the geometric fundamental group ∆X of X (for some basepoint) is topologically finitely generated, there exist characteristic open subgroups. . . ⊂ ∆X [j + 1] ⊂ ∆X [j] ⊂ . . . ⊂ ∆X∩
of ∆X for j ≥ 1 such that j ∆X [j] = {1}. Let k be an algebraic closure of k and write
Gk := Gal(k/k). For any section σ : Gk → ΠX , we write
ΠX[j,σ] := Im(σ)∆X [j] ⊂ ΠX ,and we obtain a corresponding finite étale coverings. . . → X[j + 1, σ] → X[j, σ] → . . . → X.
Lemma 3.10.
([AbsSect, Lemma 3.1]) Let X be a hyperbolic curve over a nonArchimedean local field k. Suppose X is defined over a number field. Let σ : Gk → ΠX
be a section such that Im(σ) is not contained in any cuspidal decomposition group of
ΠX . Then the following conditions on σ is equivalent:(1) Im(σ) is a decomposition group Dx of a point x ∈ X(k).
(2) For any j ≥ 1, the subgroup ΠX[j,σ] contains a decomposition group of an algebraic
closed point of X which surjects onto Gk .
Proof. (1)⇐(2): For j ≥ 1, take points xj ∈ X[j, σ](k). Since the topological∏space j≥1 X[j, σ](k) is compact, there exists an infinite set of positive integers J 0 suchthat for any j ≥ 1, the images of xj 0 in X[j, σ](k) for j 0 ≥ j with j 0 ∈ J 0 converges to apoint yj ∈ X[j, σ](k). By definition of yj , the point yj1 maps to yj2 in X[j2 ](k) for any
j1 > j2 . We write y ∈ X(k) for the image of yj in X(k). Then we have Im(σ) ⊂ Dy
(up to conjugates), and y is not a cusp by the assumption that Im(σ) is not containedin any cuspidal decomposition group of ΠX .
(1)⇒(2): By using Krasner’s lemma, we can approximate x ∈ X(k) by a point0x ∈ XF (F ) ⊂ X(k), where XF is a model of X ×k k over a number field F , which is
suﬃciently close to x so that x0 lifts to a point x0j ∈ X[j, σ](k), which is algebraic.
§ 3.3.

Uchida’s Lemma.

Let X be a hyperbolic curve over a field k. Let k be an algebraic closure of k.
Write Gk := Gal(k/k), and Xk := X ×k k. We write k(X) for the function field of
X. We write ∆X and ΠX for the geometric fundamental group (i.e., π1 of Xk ) andthe arithmetic fundamental group (i.e., π1 of X) of X for some basepoint, respectively.
Note that we have an exact sequence 1 → ∆X → ΠX → Gk → 1.
We recall that we have Γ(X, O(D)) = {f ∈ k(X)× | div(f ) + D ≥ 0} ∪ {0} for adivisor D on X.

a proof of the abc conjecture after Mochizuki

73

Lemma 3.11.
([AbsTopIII, Proposition 1.2]) Assume that k be an algebraicallyclosed, and X proper.
(1) There are distinct points x, y1 , y2 ∈ X(k) and a divisor D on X such that x, y1 , y2 6∈
Supp(D) and l(D) := dimk Γ(X, O(D)) = 2, and l(D − E) = 0 for any E = e1 + e2
with e1 , e2 ∈ {x, y1 , y2 }, e1 6= e2 .
(2) Let x, y1 , y2 , D be as in (1). For i = 1, 2, and λ ∈ k × , there exists a uniquefλ,i ∈ k(X)× such thatdiv(fλ,i ) + D ≥ 0, fλ,i (x) = λ, fλ,i (yi ) 6= 0, fλ,i (y3−i ) = 0.
(3) Let x, y1 , y2 , D be as in (1), and λ, µ ∈ k × be such that µλ 6= −1. Let also fλ,1 , fµ,2 ∈k(X)× be as in (2). Then fλ,1 + fµ,2 ∈ k(X)× is characterised as a unique elementg ∈ k(X)× such thatdiv(g) + D ≥ 0, g(y1 ) = fλ,1 (y1 ), g(y2 ) = fµ,2 (y2 ).
In particular, λ + µ ∈ k × is characterised as g(x) ∈ k × .
Proof. (1): For any divisor D of degree ≥ 2g − 2 + 3 on X, then we have l(D) =l(KX − D) + deg(D) + 1 − g = deg(D) + 1 − g ≥ g + 2 ≥ 2, by the theorem ofRiemann-Roch (Here, we write KX for the canonical divisor of X). For any divisor D
on X with d := l(D) ≥ 2, we write Γ(X, O(D)) = hf1 , . . . , fd ik , and take a point P
in the locus “f1 f2 · · · fd 6= 0” in X of non-vanishing of the section f1 f2 · · · fd such thatP 6∈ Supp(D) (Note that this locus is non-empty since there is a non-constant function inΓ(X, O(D)) by l(D) ≥ 2). Then we have l(D − P ) < l(D). On the other hand, we havel(D)−l(D−P ) = l(KX −D)−l(KX −D+P )+1 ≤ 1. Thus, we have l(D−P ) = l(D)−1.
Therefore, by substracting a suitable divisor from a divisor of degree ≥ 2g − 2 + 3, there
is a divisor D on X with l(D) = 2. In the same way, take x ∈ X(k) \ Supp(D) such that
there is f ∈ Γ(X, OX (D)) with f (x) 6= 0 (this implies that l(D −x) = l(D)−1 = 1). Let
y1 ∈ X(k) \ (Supp(D) ∪ {x}) be such that there is g ∈ Γ(X, OX (D − x)) with g(y1 ) 6= 0
(this implies that l(D − x − y1 ) = l(D − x) − 1 = 0), and y2 ∈ X(k) \ (Supp(D) ∪ {x, y1 })such that there are h1 ∈ Γ(X, OX (D − x)) and h2 ∈ Γ(X, OX (D − y1 )) with h1 (y2 ) 6= 0
and h2 (y2 ) 6= 0 (this implies that l(D − x − y2 ) = l(D − y1 − y2 ) = 0). The first claim(1) is proved. The claims (2) and (3) trivially follow from (1).
Proposition 3.12.
(Uchida’s Lemma, [AbsTopIII, Proposition 1.3]) Assume
that k be an algebraically closed, and X proper. There exists a functorial (with respect to isomorphisms of the following triples) algorithm for constructing the additive
structure on k(X)× ∪ {0} from the following data:

74

Go Yamashita(a) the (abstract) group k(X)× ,(b) the set of surjective homomorphisms VX := {ordx : k(X)×  Z}x∈X(k) of the valuation maps at x ∈ X(k), and{{}}(c) the set of the subgroups Uv := f ∈ k(X)× | f (x) = 1 ⊂ k(X)× v=ordx ∈V ofX
k(X)× .
Proof. From the above data (a), (b), and (c), we reconstruct the additive structureon k(X)× as follows:∩(Step 1): We reconstruct k × ⊂ k(X)× as k × := v∈VX ker(v). We also reconstruct
the set X(k) as VX .
(Step 2): For each v = ordx ∈ VX , we have inclusions k × ⊂ ker(v) and Uv ⊂ ker(v)with k × ∩ Uv = {1}, thus we obtain a direct product decomposition ker(v) = Uv × k × .
We write prv for the projection ker(v) → k × Then we reconstruct the evaluation map
ker(v) 3 f 7→ f (x) ∈ k × as f (x) := prv (f ) for f ∈ ker(v).
(Step 3): We reconstruct divisors (resp. eﬀective divisors) on X as formal finite
sums of v ∈ VX with coeﬃcient Z (resp. Z≥0 ). By using ordx ∈ VX , we reconstruct thedivisor div(f ) for an element f in an abstract group k(X)× .
(Step 4): We reconstruct a (multiplicative) k × -module Γ(X, O(D)) \ {0} for adivisor D as {f ∈ k(X)× | div(f ) + D ≥ 0}. We also reconstruct l(D) ≥ 0 for a divisor
D as the smallest non-negative integer d such that there is an eﬀective divisor E of
degree d on X such that Γ(X, O(D − E)) \ {0} = ∅ (cf. also the proof of Lemma 3.11(1)). Note that dimk of Γ(X, O(D)) is not available yet here, since we do not have theadditive structure on {f ∈ k(X)× | div(f ) + D ≥ 0} ∪ {0} yet.
(Step 5): For λ, µ ∈ k × , µλ 6= −1 (Here, −1 is the unique element of order 2 in k × ),we take ordx , ordy1 , ordy2 ∈ VX corresponding to x, y1 , y2 in Lemma 3.11 (1). Then we
obtain unique fλ,1 , fµ,2 , g ∈ k(X)× as in Lemma 3.11 (2), (3) from abstract data (a),(b), and (c). Then we reconstruct the addition λ + µ ∈ k × of λ and µ as g(x). We also
reconstruct the addition λ + µ := 0 for µλ = −1, and λ + 0 = 0 + λ := λ for λ ∈ k × ∪ {0}.
These reconstruct the additive structure on k × ∪ {0}.
(Step 6): We reconstruct the addition f + g of f, g ∈ k(X)× ∪ {0} as the uniqueelement h ∈ k(X)× ∪ {0} such that h(x) = f (x) + g(x) for any ordx ∈ VX withf, g ∈ ker(ordx ) (Here, we write f (x) := 0 for f = 0). This reconstructs the additive
structure on k(X)× ∪ {0}.
§ 3.4.

Mono-anabelian Reconstruction of the Base Field and Function
Field.

We continue the notation in Section 3.3 in this subsection. Furthermore, we assume
that k is of characteristic 0.

a proof of the abc conjecture after Mochizuki

75

Definition 3.13.
(1) We assume that X has genus ≥ 1. Let (X ⊂)X be the canonical smooth compactification of X. We defineb Z).
bµZb (ΠX ) := Hom(H 2 (∆X , Z),
We shall refer to µZb (ΠX ) as the cyclotome of ΠX as orientation.
(2) In the case where the genus of X is not necessarily greater than or equal to 2, wetake a finite étale covering Y  X such that Y has genus ≥ 2, and we define thecyclotome of ΠX as orientation to be µZb (ΠX ) := [∆X : ∆Y ]µZb (ΠY ). It does
not depend on the choice of Y in the functorial sense, i.e., For any such coverings
Y  X, Y 0  X, take Y 00  X which factors through Y 00  Y  X andb → H 2 (∆ 00 , Z),b H 2 (∆ 0 , Z)b →Y 00  Y 0  X. Then the restrictions H 2 (∆Y , Z)YYb (where Y , Y 0 , and Y 00 are the canonical compactifications of Y , Y 0 ,
H 2 (∆Y 00 , Z)b induce natural isomorphisms [∆X :and Y 00 respectively), and taking Hom(−, Z)∼∆Y ]µZb (ΠY ) ← [∆X : ∆Y ][∆Y : ∆Y 00 ]µZb (ΠY 00 ) = [∆X : ∆Y 00 ]µZb (ΠY 00 ) = [∆X :∼∆Y 0 ][∆Y 0 : ∆Y 00 ]µZb (ΠY 00 ) → [∆X : ∆Y 0 ]µZb (ΠY 0 ) (cf. [AbsTopIII, Remark 1.10.1(i), (ii)]).
(3) For an open subscheme ∅ 6= U ⊂ X, let ∆U  ∆cusp-cent( ∆X ) be the maximalUintermediate quotient ∆U  Q  ∆X such that ker (Q  ∆X ) is in the center ofQ, and ΠU  Πcusp-cent
the push-out of ∆U  ∆cusp-centwith respect to ∆U ⊂ ΠU .
U
U
We shall refer to them as the maximal cuspidally central quotient of ∆U andΠU respectively.
Remark 3.13.1. In this subsection, by the functoriality of cohomology with µZb (Π(−) )coeﬃcients for an open injective homomorphism of profinite groups ∆Z ⊂ ∆Y , we always
mean multiplying [∆Y 1:∆Z ] on the homomorphism between the cyclotomes ΠY and ΠZ(cf. also [AbsTopIII, Remark 1.10.1 (i), (ii)]).
Proposition 3.14.
(Cyclotomic Rigidity for Inertia Subgroups, [AbsTopIII, Proposition 1.4]) Assume that X has genus ≥ 2. Let (X ⊂)X be the canonical smooth compactification of X. Let U ⊂ X be a non-empty open subscheme. We have an exact
sequence 1 → ∆U → ΠU → Gk → 1. For x ∈ X(k) \ U (k), write Ux := X \ {x}. We
write Ix for the inertia subgroup of x in ∆U (it is well-defined up to inner automorphismbof ∆U ), which is naturally isomorphic to Z(1).
(1) ker (∆U  ∆Ux ) and ker (ΠU  ΠUx ) are topologically normally generated by theinertia subgroups of the points of Ux \ U .

76

Go Yamashita(2) We have an exact sequence
1 → Ix → ∆cusp-cent→ ∆X → 1,
Uxwhich induces the Leray spectral sequence E2p,q = H p (∆X , H q (Ix , Ix )) ⇒ H p+q (∆cusp-cent, Ix )
Ux
cusp-cent(Here, Ix and ∆Ux
act on Ix by the conjugates). Then the composite
0,1
b = Hom(Ix , Ix ) ∼Z= H 0 (∆X , H 1 (Ix , Ix )) = E2
→ E 2,0 = H 2 (∆ , H 0 (Ix , Ix )) ∼= Hom(µb (ΠX ), Ix )2ZXb to the natural isomorphism
sends 1 ∈ Z(Cyc. Rig. Iner.)∼µZb (ΠX ) −→ Ix .

bb(this is a natural identification between “Z(1)”arising from H 2 and “Z(1)”arisingfrom Ix .) Therefere, we obtain a group-theoretic reconstruction of the isomorphism(Cyc. Rig. Iner.) from the surjection ∆Ux  ∆X (Note that the intermediate quotient ∆Ux  ∆cusp-cent ∆X is group-theoretically characterised). We shall referUx
to the isomorphism (Cyc. Rig. Iner.) as the cyclotomic rigidity for inertia subgroup.

Proof. (1) is trivial. (2): By the definitions, for any intermediate quotient ∆Ux 
Q  ∆X such that ker (Q  ∆X ) is in the center of Q, the kernel ker (Q  ∆X ) is
generated by the image of Ix . Thus, we have the exact sequence 1 → Ix → ∆cusp-cent→
Ux∆X → 1 (cf. also [Cusp, Proposition 1.8 (iii)]). The rest is trivial.
Remark 3.14.1. In the case where the genus of X is not necessarily greater thanor equal to 2, we take a finite étale covering Y  X such that Y has genus ≥ 2, anda point y ∈ Y (k 0 ) lying over x ∈ X(k) for a finite extension k 0 of k. Then we have thecyclotomic rigidity µZb (ΠY ) ∼= Iy by Proposition 3.14. This induces isomorphisms
1[∆X :∆Y ]µZb (ΠX ) = [∆X : ∆Y ]µZb (ΠY )∼−→µZb (ΠY ) ∼= Iy = Ix .

We shall also refer to this as the cyclotomic rigidity for inertia subgroup. It does
not depend on the choice of Y and y in the functorial sense of Definition 3.13 (2), i.e., Forsuch Y  X, Y 0  X with y ∈ Y (kY ), y 0 ∈ Y 0 (kY 0 ), take Y 00  X with y 00 ∈ Y 00 (kY 00 )lying over Y, Y 0 and y, y 0 , then we have the following commutative diagram (cf. alsoa proof of the abc conjecture after Mochizuki

77

Remark 3.13.1)
b = Hom(Iy , Iy )Z

/ Hom(µb (ΠY ), Iy )
Z

=


b
Z = Hom(Iy00 , Iy00 )O

1[∆Y :∆ 00 ]
Y

∼
=

1
[∆ 0 :∆ 00 ]YY
/ Hom(µb (ΠY 00 ), Iy00 )Z O=

b = Hom(Iy0 , Iy0 )Z∼=/ Hom(µb (ΠY 0 ), Iy0 ).
Z

For a proper hyperbolic curve X over k, we write J d for the Picard scheme
parametrising line bundles of degree d on X (Note that J d is a J := J 0 -torsor). We havea natural map X → J 1 (P 7→ O(P )), which induces ΠX → ΠJ 1 (for some basepoint).
For x ∈ X(k), let tx : Gk → ΠJ 1 be the composite of the section Gk → ΠX determinedby x and the natural map ΠX → ΠJ 1 . The group structure of Picard schemes also
determines a morphism ΠJ 1 × · · · (d-times) · · · × ΠJ 1 → ΠJ d for d ≥ 1. For any divisor
D of degree d on X such that Supp(D) ⊂ X(k), by forming a Z-linear combination oftx ’s, we have a section tD : Gk → ΠJ d .
Lemma 3.15.
([AbsTopIII, Proposition 1.6]) Assume that k is Kummer-faithful,and that X is proper. Let ∅ 6= U ⊂ X be an open subscheme, and we write×κU : Γ(U, OU) → H 1 (ΠU , µZb (k(X))) = H 1 (ΠU , µZb (k)) ∼= H 1 (ΠU , µZb (ΠX ))for the composite of the Kummer map (for an algebraic closure k(X) of k(X)) and theb(which comes from the scheme theory).
natural isomorphism µZb (k) ∼= µZb (ΠX )(∼= Z(1))(1) κU is injective.
(2) (cf. also [Cusp, Proposition 2.3 (i)]) For any divisor D of degree 0 on X such that
Supp(D) ⊂ X(k), the section tD : Gk → ΠJ is equal to (up to conjugates by ∆X ) the
section determined by the origin O of J(k) if and only if the divisor D is principal.
(3) (cf. also [Cusp, Proposition 2.1 (i)]) We assume that U = X \ S, where S ⊂ X(k)is a finite set. Then the quotient ΠU  Πcusp-centinduces an isomorphismU∼

H 1 (Πcusp-cent, µZb (ΠX )) → H 1 (ΠU , µZb (ΠX )).
U(4) (cf. also [Cusp, Proposition 1.4 (ii)]) We have an isomorphism
H 1 (ΠX , µZb (ΠX )) ∼= (k × )∧ ,where we write (k × )∧ for the profinite completion of k × .

78

Go Yamashita(5) (cf. also [Cusp, Proposition 2.1 (ii)]) We have a natural exact sequence induced by
the restrictions to Ix (x ∈ S):∏⊕
0 → H 1 (ΠX , H 0 (Ix , µZb (ΠX ))) → H 1 (Πcusp-cent
,
µ
(Π
)))
→
H 0 (ΠX , H 1 (Ix , µZb (ΠX ))).
X
b
U
Z
x∈S

x∈S

The cyclotomic rigidity isomorphism (Cyc. Rig. Iner.) µZb (ΠX ) ∼= Ix in Proposition 3.14 induces an isomorphismb
H 0 (ΠX , H 1 (Ix , µZb (ΠX ))) = HomΠX (Ix , µZb (ΠX )) ∼=Z
(Hence note that we can use the above isomorphism for a group-theoretic reconstruction later). Then by the isomorphisms in (3) and (4) and the above cyclotomic
rigidity isomorphism, the above exact sequence is identified with⊕b
1 → (k × )∧ → H 1 (ΠU , µZb (ΠX )) →Z.
x∈S×(6) The image of Γ(U, OU) in H 1 (ΠU , µZb (ΠX ))/(k × )∧ via κU is equal to the inverse⊕⊕b deimage in H 1 (ΠU , µZb (ΠX ))/(k × )∧ of the submodule PU0 of x∈S Z (⊂ x∈S Z)
termined by the principal divisors with support in S.

Remark 3.15.1. (A general remark to the readers who are not familiar with theculture of anabelian geometers) In the above lemma, note that we are currently studying
in a scheme theory here, and that the natural isomorphism µZb (k) ∼= µZb (ΠX ) comes fromthe scheme theory. A kind of “general principle” of studying anabelian geometry is likethis:(1) First, we study some objects in a scheme theory to obtain group-theoretic properties
or group-theoretic characterisations.
(2) Next, by using the group-theoretic properties or group-theoretic characterisations
obtained in the first step, we formulate group-theoretic reconstruction algorithms,
and we cannot use a scheme theory in this situation.
When we consider cyclotomes as abstract abelian groups with Galois action (i.e., when
we are working in the group theory), we only know a priori that two cyclotomes are
abstractly isomorphic (this is the definition of the cyclotomes), the way to identify themb × -ways (or we have a Z
b × -torsor) for the identification (i.e.,
is not given, and there are Z
b × -indeterminacy for the choice). It is important to note that the cylotomic
we have Z
rigidity isomorphism (Cyc. Rig. Iner.) is constructed in a purely group-theoretic manner,
and we can reconstruct the identification even when we are working in the group theory.
cf. also the (Step 3) in Theorem 3.17.

a proof of the abc conjecture after Mochizuki

79

Proof. (1): By the assumption that k is Kummer-faithful, k(X) is also Kummerfaithful by Lemma 3.2 (3).
(2): The origin O ∈ J determines a section sO : Gk → ΠJ , and, by taking (inthe additive expression) the substraction ηD := tD − sO : Gk → ∆J (⊂ ΠJ ) (i.e., thequotient ηD := tD /sO in the multiplicative expression), which is a 1-cocycle, of two
sections tD , sO : Gk → ΠJ , we obtain a cohomology class [ηD ] ∈ H 1 (Gk , ∆J ). On theother hand, the Kummer map for J(k) induces an injection (J(k) ⊂)J(k)∧ ⊂ H 1 (k, ∆J ),since k is Kummer-faithful (Here, we write J(k)∧ for the profinite completion of J(k)).
Then we claim that [D] = [O(D)] ∈ J(k) is sent to ηD ∈ H 1 (Gk , ∆J ) (cf. also [NTs,Lemma 4.14] and [Naka, Claim (2.2)]). We write αD : J → J for the morphism which
sends x to x − [D], and for a positive integer N , let JD,N → J be the pull-back of
αD : J → J via the morphism [N ] : J → J of multiplication by N :
/J

JD,N

[N ]

J \ {O}


/J



αD


/ J.

[N ]

The origin O ∈ J( → J) corresponds to a k-rational point N1 [D] ∈ JD,N (k) lying over[D] ∈ J(k). By the k-rationality of N1 [D], we have tD (σ) ∈ ΠJD,N (⊂ ΠJ ) for σ ∈ Gk .
The inertia subgroup IO (⊂ ∆J\{O} ) of the origin O ∈ J( JD,N ) determines a systemof geometric points QD,N ∈ JD,N (k) corresponding to the divisor N1 (−[D]) for N ≥ 1such that IO always lies over QD,N . The conjugation conj(tD (σ)) ∈ Aut(∆J\{O} )∗by tD (σ) coincides with the automorphism induced by σN:= id ×Spec k Spec (σ −1 ) ∈Aut((J \ {O}) ⊗k k) (Note that a fundamental group and the corresponding covering transformation group are opposite groups to each other). Thus, tD (σ)IO tD (σ)−1∗gives an inertia subgroup over σN(QD,N ) = σ(QD,N ). On the other hand, by def−1
inition, we have tD (σ)zO tD (σ)= tD (σ)sO (σ)−1 sO (σ)zO sO (σ)−1 sO (σ)tD (σ)−1 =χcyc (σ)ηD (σ)zO
ηD (σ)−1 for a generator zO of IO , hence tD (σ)IO tD (σ)−1 is an iner[N ]tia subgroup over νN (ηD (σ)−1 )(QD,N ), where νN : ∆J  Aut((J \ J[N ]) ⊗k k →(J \ {O}) ⊗k k)opp (Here, we write (−)opp for the opposite group. Note that a fundamental group and the corresponding covering transformation group are opposite groups−1
to each other). Therefore,( we have σ(QD,N ) = νN (ηD (σ) ))(QD,N ). By noting the nat[N ]ural isomorphism Aut (J \ J[N ]) ⊗k k → (J \ {O}) ⊗k k∼
= J[N ] given by γ 7→ γ(O),)
11(−[D]) = −νN (ηD (σ))(O) +(−[D]) .
σN
N()
Hence we have σ N1 [D] − N1 [D] = νN (ηD (σ))(O). This gives us the claim. The
assertion (2) follows from this claim.

we obtain that(

80

Go Yamashita(3): We have the following commutative diagram:0/ H 1 (Gk , H 0 (∆cusp-cent ))/ H 1 (Πcusp-cent )U/ H 0 (Gk , H 1 (∆cusp-cent ))0/ H 1 (Gk , H 0 (∆U ))
/ H 1 (ΠU )/ H 0 (Gk , H 1 (∆U )),U

U

where the horizontal sequences are exact, and we abbreviate the coeﬃcient µZb (ΠU ) by
the typological reason. Here, we have
H 1 (Gk , H 0 (∆U , µZb (ΠX ))) = H 1 (Gk , µZb (ΠX )) = H 1 (Gk , H 0 (∆cusp-cent, µZb (ΠX ))),
Uandcusp-cent01
H 0 (Gk , H 1 (∆U , µZb (ΠX ))) = H 0 (Gk , ∆ab, µZb (ΠX ))).
U ) = H (Gk , H (∆U

Thus by combining these, the assertion (3) is proved.
(4): By the exact sequence
0 → H 1 (Gk , H 0 (∆X , µZb (ΠX ))) → H 1 (ΠX , µZb (ΠX )) → H 0 (Gk , H 1 (∆X , µZb (ΠX ))) (∼= H 0 (Gk , ∆abX )),and H 1 (Gk , H 0 (∆X , µZb (ΠX ))) = H 1 (Gk , µZb (ΠX )) ∼= (k × )∧ , it suﬃces to show thatab Gk ∼H 0 (Gk , ∆ab= T (J)Gk = 0, since ∩N N J(k) = 0 by
X ) = 0. This follows from (∆X )the assumption that k is Kummer-faithful (Here, we write T (J) for the Tate module ofJ, and J[N ] is the group of N -torsion points of J).
∏(5) is trivial by noting H 1 (ΠX , H 0 ( x∈S Ix , µZb (ΠX ))) = H 1 (ΠX , µZb (ΠX )) ∼= (k × )∧by (4).
(6) is trivial.
We write k NF for the algebraic closure of Q in k (Here, NF stands for “number
field”). If Xk is defined over k NF , we say that X is an NF-curve. For an NF-curve
X, points of X(k) (resp. rational functions on Xk , constant rational functions (i.e.,k ⊂ k(X))) which descend to k NF , we shall refer to them as NF-points (resp. NFrational functions, NF-constants) on Xk .
Lemma 3.16.
([AbsTopIII, Proposition 1.8]) Assume that k is Kummer-faithful.
Let ∅ 6= U ⊂ X be an open subscheme, and write S := X \ U . We also assume that U isan NF-curve (hence X is also an NF-curve). We write PU ⊂ H 1 (ΠU , µZb (ΠX )) for the⊕⊕b via the homomorphism H 1 (ΠU , µb (ΠX )) →inverse image of PU0 ⊂ x∈S Z (⊂ x∈S Z)
Z⊕
b constructed in Lemma 3.15.
Zx∈S(1) an element η ∈ PU is the Kummer class of a non-constant NF-rational function
if and only if there exist a positive integer n and two NF-points x1 , x2 ∈ U (k 0 )a proof of the abc conjecture after Mochizuki

81with a finite extension k 0 of k such that the restrictions (nη)|xi := s∗xi (nη) ∈H 1 (Gk0 , µZb (ΠX )), where sxi : Gk0 → ΠU is the section corresponding to xi for
i = 1, 2, satisfy (in the additive expression) (nη)|x1 = 0 and (nη)|x2 6= 0 (i.e., = 1
and 6= 1 in the multiplicative expression).
×(2) Assume that there exist non-constant NF-rational functions in Γ(U, OU). Then an1× ∧∼element η ∈ PU ∩ H (Gk , µZb (ΠX )) = (k ) is the Kummer class of an NF-constant×in k × if and only if there exist a non-constant NF-rational function f ∈ Γ(U, OU)
00and an NF-point x ∈ U (k ) with a finite extension k of k such that κU (f )|x = η|xin H 1 (Gk0 , µZb (ΠX )).

Proof. Let XNF be a model of Xk over k NF . Then any non-constant rational
function on XNF determines a morphism XNF → P1k , which is non-constant i.e.,
XNF (k NF ) → P1kNF

NF(k NF ) is surjective. Then the lemma follows from the definitions.

Theorem 3.17.
(Mono-anabelian Reconstruction of NF-Portion, [AbsTopIII,
Theorem 1.9]) Assume that k is sub-p-adic, and that X is a hyperbolic orbicurve ofstrictly Belyi type. Let X be the canonical smooth compactification of X. From theextension 1 → ∆X → ΠX → Gk → 1 of profinite groups, we can functorially grouptheoretically reconstruct the NF-rational function field k NF (X) and NF-constant fieldk NF as in the following. Here, the functoriality is with respect to open injective homomorphisms of extension of profinite groups (cf. Remark 3.13.1), as well as with respect
to homomorphisms of extension of profinite groups arising from a base change of thebase field.
(Step 1) By Belyi cuspidalisation (Theorem 3.8), we group-theoretically reconstruct the setof surjections {ΠU  ΠX }U for open sub-NF-curves ∅ 6= U ⊂ X and the decomposition groups Dx in ΠX of NF-points x. We also group-theoretically reconstruct theinertia subgroup Ix := Dx ∩ ∆U .
(Step 2) By cyclotomic rigidity for inertia subgroups (Proposition 3.14 and Remark 3.14.1),∼
we group-theoretically obtain isomorphism Ix → µZb (ΠX ) for any x ∈ X(k), where
Ix is group-theoretically reconstructed in (Step 1).
(Step 3) By the inertia subgroups Ix reconstructed in (Step 1), we group-theoretically reconstruct the restriction homomorphism H 1 (ΠU , µZb (ΠX )) → H 1 (Ix , µZb (ΠX )). By thecyclotomic rigidity isomorphisms in (Step 2), we have an isomorphism H 1 (Ix , µZb (ΠX )) ∼=b
Z. Therefore, we group-theoretically obtain an exact sequence
1 → (k × )∧ → H 1 (ΠU , µZb (ΠX )) →

⊕
x∈S

b
Z

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Go Yamashita

in Lemma 3.15 (5) (Note that, without the cyclotomic rigidity Proposition 3.14, web × -indeterminacies on each direct summand of ⊕bwould have Z
x∈S Z, and that thereconstruction algorithm in this theorem would not work). By the characterisationof principal cuspidal divisors (Lemma 3.15 (2), and the decomposition groups in(Step 1)), we group-theoretically reconstruct the subgroup
PU ⊂ H 1 (ΠU , µZb (ΠU ))of principal cuspidal divisors.
(Step 4) Note that we already group-theoretically reconstructed the restriction map η|xi in
Lemma 3.16 by the decomposition group Dxi reconstructed in (Step 1). By the characterisations of non-constant NF-rational functions and NF-constants in Lemma 3.16(1), (2) in PU reconstructed in (Step 3), we group-theoretically reconstruct the subgroups (via Kummer maps κU ’s in Lemma 3.15)
×k NF ⊂ k NF (X)× ⊂ lim H 1 (ΠU , µZb (ΠX )),−→U

where U runs through the open sub-NF-curves of X ×k k 0 for a finite extension k 0
of k.
(Step 5) In (Step 4), we group-theoretically reconstructed the datum k NF (X)× in Proposition 3.12 (a). Note that we already reconstructed the data ordx ’s in Proposition 3.12⊕b re(b) as the component at x of the homomorphism H 1 (ΠU , µZb (ΠX )) → x∈S Z
constructed in (Step 3). Note also that we already group-theoretically reconstructed
the evaluation map f 7→ f (x) in Proposition 3.12 as the restriction map to thedecomposition group Dx reconstructed in (Step 1). Thus, we group-theoretically obtain the data Uv ’s in Proposition 3.12 (c). Therefore, we can apply Uchida’s Lemma(Proposition 3.12), and we group-theoretically reconstruct the additive structures on
×

k NF ∪ {0},k NF (X)× ∪ {0}.

Proof. The theorem immediately follows from the group-theoretic algorithms referred in the statement of the theorem. The functoriality immediately follows from thedescribed constructions.
Remark 3.17.1. The input data of Theorem 3.17 is the extension 1 → ∆X →
ΠX → Gk → 1 of profinite groups. If k is a number field or a non-Archimedean local
field, then we need only the profinite group ΠX as an input datum by Proposition 2.2(1), and Corollary 2.4. (Note that we have a group-theoretic characterisation of cuspidal
decomposition groups for the number field case as well by Remark 2.9.2.)

a proof of the abc conjecture after Mochizuki

83

Remark 3.17.2. (Elementary Birational Analogue, [AbsTopIII, Theorem 1.11])
We write ηX for the generic point of X. If k is l-cyclotomically full for some l, thenwe have the characterisation of the cuspidal decomposition groups in ΠηX at (not only
NF-points but also) all closed points of X (cf. Remark 2.9.2). Therefere, under theassumption that k is Kummer-faithful (cf. also Lemma 3.2 (2)), if we start not fromthe extension 1 → ∆X → ΠX → Gk → 1, but from the extension 1 → ∆ηX → ΠηX →Gk → 1, then the same group-theoretic algorithm (Step 2)-(Step 5) works without using
Belyi cuspidalisation (Theorem 3.8) or (GC) (cf. Theorem B.1), and we can obtain (notonly the NF-rational function field k NF (X) but also) the rational function field k(X)and (not only the NF-constant field k NF but also) the constant field k (Note also that
we do not use the results in Section 3.2, hence we have no circular arguments here).
Remark 3.17.3. (Slimness of Gk for Kummer-Faithful k, [AbsTopIII, a part ofTheorem 1.11]) By using the above Remark 3.17.2 (Note that we do not use the resultsin Section 3.2 to show Remark 3.17.2, hence we have no circular arguments here), we
can show that Gk := Gal(k/k) is slim for any Kummer-faithful field k as follows (cf.
also [pGC, Lemma 15.8]): Let Gk0 ⊂ Gk be an open subgroup, and take g ∈ ZGk (Gk0 ).
∼Assume that g 6= 1. Then we have a finite Galois extension K of k 0 such that g : K → K
is not an identity on K. We have K = k 0 (α) for some α ∈ K. Let E be an elliptic curve
over K with j-invariant α. Write X := E \ {O}, where O is the origin of E. Write∼also X g := X ×K,g K i.e., the base change by g : K → K. The conjugate by g defines∼
an isomorphism ΠX → ΠX g . This isomorphism is compatible to the quotients to GK ,since g is in ZGk (Gk0 ). Thus, by the functoriality of the algorithm in Remark 3.17.2,∼
this isomorphism induces an K-isomorphism K(X) → K(X g )(= K(X) ⊗K,g K) of
function fields. Therefore, we have g(α) = α by considering the j-invariants. This is acontradiction.
Remark 3.17.4.
(cf. also [AbsTopIII, Remark 1.9.5 (ii)], and [IUTchI, Remark
4.3.2]) The theorem of Neukirch-Uchida (which is a bi-anabelian theorem) uses thedata of the decomposition of primes in extensions of number fields. Hence it has nofunctoriality with respect to the base change from a number field to non-Archimedean
local fields. On the other hand, (mono-anabelian) Theorem 3.17 has the functoriality
with respect to the base change of the base fields, especially from a number field to
non-Archimedean local fields. This is crucial for the applications to inter-universal
Teichmüller theory (For example, see the beginning of 10, Example 8.12 etc.). cf. also
[IUTchI, Remark 4.3.2 requirements (a), (b), and (c)].
In inter-universal Teichmüller theory, we will treat local objects (i.e., objects over
local fields) which a priori do not come from a global object (i.e., an object over anumber field), in fact, we completely destroy the above data of “the decomposition ofprimes” (Recall also the “analytic section” of Spec OK  Spec OFmod ). Therefore, it is84

Go Yamashita

crucial to have a mono-anabelian reconstruction algorithm (Theorem 3.17) in a purely
local situation for the applications to inter-universal Teichmüller theory. It also seems
worthwhile to give a remark that such a mono-anabelian reconstruction algorithm in apurely local situation got available by the fact that the bi-anabelian theorem in [pGC]
was proved for a purely local situation, unexpectedly at that time to many people froma point of view of analogy with Tate conjecture!
Definition 3.18.
µQ/Z (Gk ) :=Let k be a finite extension of Qp . We definelim−→(H ab )tors ,H⊂Gk : openµZb (Gk ) := Hom(Q/Z, µQ/Z (Gk )),where the transition maps are given by Verlangerung (or transfer) maps (cf. also theproof of Proposition 2.1 (6) for the definition of Verlangerung map). We shall refer tothem as the cyclotomes of Gk .
Remark 3.18.1. Similarly as Remark 3.13.1, in this subsection, by the functoriality of cohomology with µQ/Z (G(−) )-coeﬃcients for an open injective homomorphism of
1on the homomorphism
profinite groups Gk0 ⊂ Gk , we always mean multiplying [Gk :G
k0 ]between the cyclotomes of Gk and Gk0 (cf. also [AbsTopIII, Remark 3.2.2]). Note that
we have a commutative diagram∼=

H 2 (Gk , µQ/Z (Gk ))1·restriction[Gk :G 0 ]
k

∼
=

/ Q/Z
=


2
0
H (Gk , µQ/Z (Gk0 ))∼=
/ Q/Z,

where the horizontal arrows are the isomorphisms given in Proposition 2.1 (7).
Corollary 3.19.
(Mono-anabelian Reconstruction over an MLF, [AbsTopIII,
Corollary 1.10, Proposition 3.2 (i), Remark 3.2.1]) Assume that k is a non-Archimedean
local field, and that X is a hyperbolic orbicurve of strictly Belyi type. From the profinitegroup ΠX , we can group-theoretically reconstruct the following in a functorial mannerwith respect to open injections of profinite groups:(1) the set of the decomposition groups of all closed points in X,(2) the function field k(X) and the constant field k, and(3) a natural isomorphism(Cyc. Rig. LCFT)∼µZb (Gk ) → µZb (O (ΠX )),×

×

where we put µZb (O (ΠX )) := Hom(Q/Z, κ(k NF )) for κ : k NF ,→ limU H 1 (ΠU , µZb (ΠX )).
−→a proof of the abc conjecture after Mochizuki

85

We shall refer to the isomorphism (Cyc. Rig. LCFT) as the cyclotomic rigidity via
LCFT or classical cyclotomic rigidity (LCFT stands for “local class field theory”).
Proof. (1) is just a restatement of Corollary 3.9.
(2): By Theorem 3.17 and Corollary 2.4, we can group-theoretically reconstruct the∼fields k NF (X) and k NF . On the other hand, by the natural isomorphism H 2 (Gk , µZb (Gk )) →b group-theoretically constructed in Proposition 2.1 (7) (with Hom(Q/Z, −)) and theZ∼
cup product, we group-theoretically construct isomorphisms H 1 (Gk , µZb (Gk )) →b Z)b ∼Hom(H 1 (Gk , Z),= Gab
k . We also have group-theoretic constructions of a surjecabababab ∼ btion Gk  Gk /Im(Ik → Gabk ) and an isormorphism Gk /Im(Ik → Gk ) = Z by
Proposition 2.1 (4a) and Proposition 2.1 (5) respectively (cf. also Remark 2.1.1). Henceb We have an isomorwe group-theoretically obtain a surjection H 1 (Gk , µZb (Gk ))  Z.
b × . Then this induces
phism µZb (Gk ) ∼= µZb (ΠX ) well-defined up to multiplication by Zb well-defined up to multiplication by Z
b × . We groupa surjection H 1 (Gk , µZb (ΠX ))  Z
theoretically reconstruct the field k as the completion of the field (H 1 (Gk , µZb (ΠX )) ∩××

k NF )∪{0} (induced by the field structure of k NF ∪{0}) with respect to the{ valuation de}×11b
termined by the subring of (H (Gk , µZb (ΠX ))∩k NF )∪{0} generated by ker H (Gk , µZb (ΠX ))  Z ∩×
k NF . The reconstructed object is independent of the choice of an isomorphism µZb (Gk ) ∼=µZb (ΠX ). By taking the inductive limit of this construction with respect to open subgroups of Gk , we group-theoretically reconstruct k. Finally, we group-theoreticallyreconstruct k(X) by k(X) := k ⊗kNF k NF (X).
(3): We put µQ/Z (O (ΠX )) := µZb (O (ΠX )) ⊗Zb Q/Z. We group-theoretically reconstruct Gur = Gal(k ur /k) by Proposition 2.1 (4a). Then by the same way as Proposition 2.1 (7), we have group-theoretic constructions of isomorphisms:∼×∼

H 2 (Gk , µQ/Z (O (ΠX ))) → H 2 (Gk , κ(k )) ←− H 2 (Gur , κ((k ur )× ))∼∼∼→ H 2 (Gur , Z) ←− H 1 (Gur , Q/Z) = Hom(Gur , Q/Z) → Q/Z.
∼

Thus, by taking Hom(Q/Z, −), we obtain a natural isomorphism H 2 (Gk , µZb (O (ΠX ))) →b By imposing the compatibility of this isomorphism with the group-theoretically conZ.
∼ bstructed isomorphism H 2 (Gk , µZb (Gk )) → Z
in (2), we obtain a natural isomorphism∼µZb (Gk ) → µZb (O (ΠX )).
Remark 3.19.1. ([AbsTopIII, Corollary 1.10 (c)]) Without assuming that X is∼of strictly Belyi type, we can construct an isomorphism µZb (Gk ) → µZb (ΠX ) (cf. Corollary 3.19 (3)). However, the construction needs technically lengthy reconstruction algorithms of the graph of special fiber ([profGC, §1–5], [AbsAnab, Lemma 2.3]. cf. also
[SemiAnbd, Theorem 3.7, Corollary 3.9] Proposition 6.6 for the reconstruction without
Galois action in the case where a tempered structure is available) and the “rational positive structure” of H 2 (cf. also [AbsAnab, Lemma 2.5 (i)]), where we need Raynaud’s

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Go Yamashita

theory on “ordinary new part” of Jacobians (cf. also [AbsAnab, Lemma 2.4]), though
it has an advantage of no need of [pGC]. cf. also Remark 6.12.2.
Remark 3.19.2. ([AbsTopIII, Proposition 3.2, Proposition 3.3]) For a topologicalmonoid (resp. topological group) M with continuous Gk -action, which is isomorphic
×
to Ok (resp. k ) compatible with the Gk -action, we write µZb (M ) := Hom(Q/Z, M × ))and µQ/Z (M ) := µZb (M ) ⊗Zb Q/Z. We shall refer to them as the cyclotome of aur
topological monoid M . We also write M ur := M ker(G→G ) . We can canonically takethe generator of M ur /M × ∼= N (resp. the generator of M ur /M × up to {±1}) to obtainan isomorphism (M ur )gp /(M ur )× ∼= Z (resp. an isomorphism (M ur )gp /(M ur )× ∼= Z
well-defined up to {±1}). Then by the same way as Corollary 3.19 (3), we have∼∼

H 2 (Gk , µQ/Z (M )) → H 2 (Gk , M gp ) ←− H 2 (Gur , (M ur )gp )
∼

∼

∼

∼

→ H 2 (Gur , (M ur )gp /(M ur )× ) → H 2 (Gur , Z) ←− H 1 (Gur , Q/Z) = Hom(Gur , Q/Z) → Q/Z,(∗)∼

where the isomorphism H 2 (Gur , (M ur )gp /(M ur )× ) → H 2 (Gur , Z) is canonically defined(∗)(resp. well-defined up to {±1}), as noted above. Then we have a canonical isomorphism(resp. an isomorphism well-defined up to {±1})∼µZb (Gk ) → µZb (M ),(Cyc. Rig. LCFT2)

by the same way as in Corollary 3.19 (3). We shall also refer to the isomorphism(Cyc. Rig. LCFT2) as the cyclotomic rigidity via LCFT or classical cyclotomic
rigidity. We also obtain a canonical homomorphism (resp. a homomorphism welldefined up to {±1})
M ,→

lim
−→

J⊂G: open

H 1 (J, µZb (M )) ∼
=

lim
−→

J⊂G: open

H 1 (J, µZb (Gk )),by the above isomorphism, where the first injection is the canonical injection (The notation  in Ok = Ok× · (uniformiser)N indicates that the “direction” N (∼= (uniformiser)N )×
of Z (∼= (uniformiser)Z ) (or a generator of Z) is chosen, compared to k = O× ·k(uniformiser)Z , which has {±1}-indeterminacy of choosing a “direction” or a generatorof Z (∼= (uniformiser)Z ). In the non-resp’d case (i.e., the O -case), the above canonicalinjection induces an isomorphismKum∼

M → Ok (ΠX ),where we write Ok (ΠX ) for the ind-topological monoid determined by the ind-topologicalfield reconstructed by Corollay 3.19. We shall refer to this isomprhism as the Kummer
isomorphism for M .

a proof of the abc conjecture after Mochizuki

87

We can also consider the case where M is an topological group with Gk -action,
which is isomorphic to Ok× compatible with the Gk -action. Then in this case, we have∼

an isomorphism µZb (Gk ) → µZb (M ) and an injection M ,→ limJ⊂G: open H 1 (J, µZb (Gk )),−→×bwhich are only well-defined up to Z -multiple (i.e., there is no rigidity).

It seems important to give a remark that we use the value group portion (i.e., weuse O , not O× ) in the construction of the cyclotomic rigidity via LCFT. In interuniversal Teichmüller theory, not only the existence of reconstruction algorithms, butalso the contents of reconstruction algorithms are important, and whether or not we
use the value group portion in the algorithm is crucial for the constructions in the
final multiradial algorithm in inter-universal Teichmüller theory. cf. also Remark 9.6.2,
Remark 11.4.1, Propositin 11.5, and Remark 11.11.1.
§ 3.5.

On the Philosophy of Mono-analyticity and Arithmetic
Holomorphicity.

In this subsection, we explain Mochizuki’s philosophy of mono-analyticity and arithmetic holomorphicity, which is closely related to inter-universality.
Let k be a finite extension of Qp , k an algebraic closure of k, and k 0 (⊂ k) a finite
extension of Qp . It is well-known that, at least for p 6= 2, the natural map(nonGC for MLF)
Isomtopological fields (k/k, k/k 0 ) ,→ Isomprofinite groups (Gal(k/k 0 ), Gal(k/k))(scheme theory)(group theory)is not bijective (cf. [NSW, Chap. VII, §5, p.420–423]. cf. also [AbsTopI, Corollary 3.7]).
This means that there exists an automorphism of Gk := Gal(k/k) which does not comefrom an isomorphism of topological fields (i.e., does not come from a scheme theory).
In this sense, by treating Gk as an abstract topological group, we can go outside of ascheme theory. (A part of) Mochizuki’s philosophy of arithmetically holomorphicity andmono-analyiticity is to consider the image of the map (nonGC for MLF) as arithmetically holomorphic, and the right-hand side of (nonGC for MLF) as mono-analytic(Note that this is a bi-anabelian explanation, not a mono-anabelian explanation (cf.
Remark 3.4.4) for the purpose of the reader’s easy getting the feeling. We will see monoanabelian one a little bit later). The arithmetic holomorphicity versus mono-analyticity
is an arithmetic analougue of holomorphic structure of C versus the undeyling analytic
strucutre of R2 (∼= C).
Note that Gk has cohomological dimension 2 like C is two-dimensional as a topological manifold. It is well-known that this two-dimensionality comes from the exactbb
sequence 1 → Ik → Gk → ZFrob
k → 1 and that both Ik and ZFrobk have cohomological dimension 1. In the abelianisation, these groups correspond to the unit group

88

Go Yamashita

and the value group respectively via the local class field theory. Proposition 2.1 (2d)says that we can group-theoretically reconstruct the multiplicative group k × from theabstract topological group Gk . This means that we can see the multiplicative structure of k in any scheme theory, in other words, the multiplicative structure of k isinter-universally rigid. However, we cannot group-theoretically reconstruct the field kfrom the abstract topological group Gk , since there exists a non-scheme-theoretic automorphism of Gk as mentioned above. In other words, the additive structure of k isinter-universally non-rigid. Proposition 2.1 (5) also says that we can group-theoreticallybreconstruct Frobenius element Frobk in ZFrob
k ( Gk ) from the abstract topologicalbgroup Gk , and the unramified quotient ZFrobk corresponds to the value group via thelocal class field theory. This means that we can detect the Frobenius element in anybscheme theory. In other words, the unramified quotient ZFrob
k and the value group×Z( k ) are inter-universally rigid. However, there exists automorphisms of the topological group Gk which do not preserve the ramification filtrations (cf. also [AbsTopIII,
Remark 1.9.4]), and the ramification filtration (with upper numberings) corresponds tothe filtration (1 + mnk )n of the unit group via the local class field theory, where we write
mk for the maximal ideal of Ok . In other words, the inertia subgroup Ik and the unitgroup Ok× are inter-universally non-rigid (We can also directly see that the unit group
Ok× is non-rigid under the automorphism of topological group k × without the class field
theory). In summary, one dimension of Gk or k × (i.e., the unramified quotient and thevalue group) is inter-universally rigid, and the other dimension (i.e., the inertia subgroupand the unit group) is not. Thus, Mochizuki’s philosophy of arithmetic holomorphicity
and mono-analyticity regards a non-scheme-theoretic automorphism of Gk as a kindof an arithmetic analogue of the Teichmüller dilation of the undeyling analytic
strucutre of R2 (∼= C) (cf. also [Pano, Fig. 2.1] instead of the poor picture below):↑↑→−−−−−−−−−→

Note that it is a theatre of encounter of the anabelian geometry, the Teichmüller point of view, the diﬀerential over F1 (cf. Remark 1.6.1 and Lemma 1.9)
and Hodge-Arakelov theory (cf. Appendix A), which gives rise a Diophantine
consequence!
Note also that [Qp GC, Theorem 4.2] says that if an automorphisms of Gk preserves
the ramification filtration, then the automorphism arises from an automorphism of k/k.
This means that when we rigidify the portion corresponding to the unit group (i.e.,
non-rigid dimension of Gk ), then it becomes arithmetically holomorphic i.e., [Qp GC,
Theorem 4.2] supports the philosophy. Note also that we have C× ∼= S1 × R>0 , where
we write S1 := OC× ⊂ C× (cf. Section 0.2), and that the unit group S1 is rigid and the“value group” R>0 is non-rigid under the automorphisms of the topological group C×

a proof of the abc conjecture after Mochizuki

89(Thus, the rigidity and non-rigidity for unit group and “value group” in the Archimedean
case are opposite to the non-Archimedean case).
Let X be a hyperbolic orbicurve of strictly Belyi type over a non-Archimedean local
field k. Corollary 3.19 says that we can group-theoretically reconstruct the field k fromthe abstract topological group ΠX . From this mono-anabelian reconstruction theorem,
we obtain one of the fundamental observations of Mochizuki: ΠX or equivalently theouter action Gk → Out(∆X ) (and the actions ΠX y k, Ok , Ok , Ok× ) is arithmeticallyholomorphic, and Gk (and the actions Gk y Ok , Ok× on multiplicative monoid andmultiplicative group) is mono-analytic (thus, taking the quotient ΠX 7→ Gk is a “monoanalyticisation”) (cf. Section 0.2 for the notation Ok ). In other words, the outer actionof Gk on ∆X rigidifies the “non-rigid dimension” of k × . We can also regard X asa kind of “tangent space” of k, and it rigidifies k × . Note also that, in the p-adic
Teichmüller theory (cf. [pOrd] and [pTeich]), a nilpotent ordinary indigenous bundle overa hyperbolic curve in positive characteristic rigidifies the non-rigid p-adic deformations.
In the next section, we study an Archimedean analogue of this rigidifying action. In interuniversal Teichmüller theory, we study number field case by putting together the local
ones. In the analogy between p-adic Teichmüller theory and inter-universal Teichmüller
theory, a number field corresponds to a hyperbolic curve over a perfect field of positive
characteristic, and a once-punctured elliptic curve over a number field corresponds toa nilpotent ordinary indigenous bundle over a hyperbolic curve over a perfect field ofpositive characteristic. We will deepen this analogy later such that log-link corresponds
to a Frobenius endomorphism in positive characteristic, a vertical line of log-theta-latticecorresponds to a scheme theory in positive characteristic, Θ-link corresponds to a mixed
characteristic lifting of ring of Witt vectors pn /pn+1 ; pn+1 /pn+2 , a horizontal line oflog-theta-lattice corresponds to a deformation to mixed characteristic, and a log-thetalattice corresponds to a canonical lifting of Frobenius (cf. Section 12.1).
In short, we obtain the following useful dictionaries:

rigid

b
ZFrob
k

value group

multiplicative structure of k

S1 (⊂ C× )non-rigid

Ik

unit group

additive structure of k

R>0 (⊂ C× )C

field kΠXΠX y k, Ok , Ok , Ok×arith. hol.

R2 (∼
= C)

multiplicative group k ×

Gk

Gk y Ok , Ok×

mono-an.

90

Go Yamashita

inter-universal Teich.

p-adic Teich.

number field

hyperbolic curve of pos. char.

onece-punctured ell. curvenilp. ord. indigenous bundle

log-link

Frobenius in pos. char.

vertical line of log-theta-lattice

scheme theory in pos. char.

Θ-linklifting pn /pn+1 ; pn+1 /pn+2

horizontal line of log-theta-lattice

deformation to mixed. char.

log-theta-lattice

canonical lift of Frobenius

cf. also [AbsTopIII, §I.3] and [Pano, Fig. 2.5]. Finally, we give a remark that
separating additive and multiplicative structures is also one of the main themes of interuniversal Teichmüller theory (cf. Section 10.4 and Section 10.5).
§ 4.

The Archimedean Theory — Formulated Without Reference to aSpecific Model C.

In this section, we introduce a notion of Aut-holomorphic space to avoid a specific fixed local referred model of C (i.e., “the C”) for the formulation of holomorphicity, i.e., “model-implicit” approach. Then we study an Archimedean analogue monoanabelian reconstruction algorithms of Section 3, including elliptic cuspidalisation, andan Archimedean analogue of Kummer theory.
§ 4.1.
Definition 4.1.

Aut-Holomorphic Spaces.

([AbsTopIII, Definition 2.1])(1) Let X, Y be Riemann surfaces.
(a) We write AX for the assignment, which assigns to any connected open subset∼
U ⊂ X the group AX (U ) := Authol (U ) := {f : U → U holomorphic} ⊂∼Aut(U top ) := {f : U → U homeomorphic}.

a proof of the abc conjecture after Mochizuki

91(b) Let U be a set of connected open subset of X such that U is a basis of the
topology of X and that for any connected open subset V ⊂ X, if V ⊂ U ∈ U ,then V ∈ U. We shall refer to U as a local structure on the underlying
topological space X top .
(c) We shall refer to a map f : X → Y between Riemann surfaces as an RCholomorphic morphism if f is holormophic or anti-holomorphic at any pointx ∈ X (Here, RC stands for “real complex”).
(2) Let X be a Riemann surface, and U a local structure on X top .
(a) The Aut-holomorphic space associated to X is a pair X = (Xtop , AX ), where
Xtop := X top the underlying topological space of X, and AX := AX .
(b) We shall refer to AX as the Aut-holomorphic structure on X top .
(c) We shall refer to AX |U as a U-local pre-Aut-holomorphic structure on
X top .
(d) If X is biholomorphic to an open unit disc, then we shall refer to X as an
Aut-holomorphic disc.
(e) If X is a hyperbolic Riemann surface of finite type, then we shall refer to X as
hyperbolic of finite type.
(f) If X is a hyperbolic Riemann surface of finite type associated to an ellipticallyadmissible hyperbolic curve over C, then we shall refer to X as ellipticallyadmissible.
(3) Let X, Y be Aut-holomorphic spaces arising from Riemann surfaces X, Y respectively. Let U, V be local structures of X top , Y top respectively.
(a) A (U, V)-local morphism φ : X → Y of Aut-holomorphic spaces is a local
isomorphism φtop : Xtop → Ytop of topological spaces suth that, for any U ∈ U∼
with φtop : U → V ∈ V (homeomorphism), the map AX (U ) → AY (V ) obtainedby the conjugate by φtop is bijective.
(b) If U, V are the set of all connected open subset of X top , Y top respectively, thenwe shall refer to φ as a local morphism of Aut-holomorphic spaces.
(c) If φtop is a finite covering space map, then we shall refer to φ as finite étale.
(4) Let Z, Z 0 be orientable topological surfaces.
(a) Let p ∈ Z, and we define Orn(Z, p) := limp∈W ⊂Z: connected, open π1 (W \ {p})ab ,←−which is non-canonically isomorphic to Z. Note that after taking the abelianisation, there is no indeterminacy of inner automorphisms arising from the choiceof a basepoint in (the usual topological) fundamental group π1 (W \ {p}).

92

Go Yamashita(b) The assignment p 7→ Orn(Z, p) is a trivial local system, since Z is orientable.
We write Orn(Z) for the abelian group of global sections of this trivial local
system, which is non-canonically isomorphic to Zπ0 (Z) .
(c) Let α, β : Z → Z 0 be local isomorphisms. We say that α and β are co-oriented
if the induced homomorphisms α∗ , β∗ : Orn(Z) → Orn(Z 0 ) of abelian groups
coincide.
(d) A pre-co-orientation ζ : Z → Z 0 is an equivalence class of local isomorphisms
Z → Z 0 of orientable topological surfaces with respect to being co-oriented.
(e) The assignment which assigns to the open sets U in Z the sets of pre-coorientations U → Z 0 is a presheaf. We shall refer to a global section ζ : Z → Z 0of the sheafification of this presheaf as a co-orientation.
(5) Let X, Y be Aut-holomorphic spaces arising from Riemann surfaces X, Y respectively. Let U, V be local structures of X top , Y top respectively.
(a) (U, V)-local morphisms φ1 , φ2 : X → Y of Aut-holomorphic spaces is called
co-holomorphic, if φtop
and φtop
are co-oriented.
12(b) A pre-co-holomorphicisation ζ : X → Y is an equivalence class of (U, V)local morphisms X → Y of Aut-holomorphic spaces with respect to being
co-holomorphic.
(c) The assignment which assigns to the open sets U in Xtop the sets of pre-coholomorphicisation U → Y is a presheaf. We shall refer to a global section
ζ : X → Y of the sheafification of this presheaf as a co-holomorphicisation.
By replacing “Riemann surface” by “one-dimensional complex orbifold”, we caneasily extend the notion of Aut-holomorphic space to Aut-holomorphic orbispace.
Proposition 4.2.
([AbsTopIII, Proposition 2.2]) Let X, Y be Aut-holomorphic
discs arising from Riemann surfaces X, Y respectively. We equip the group Aut(X top )of homeomorphisms with the compact-open topology. We write AutRC-hol (X) (⊂ Aut(X top ))for the subgroup of RC-holomorphic automorphisms of X. We regard Authol (X) andAutRC-hol (X) as equipped with the induced topology by the inclusions
Authol (X) ⊂ AutRC-hol (X) ⊂ Aut(X top ).
(1) We have isomorphisms
Authol (X) ∼= PSL2 (R),AutRC-hol (X) ∼= PGL2 (R)as topological groups, Authol (X) is a subgroup in AutRC-hol (X) of index 2, andAutRC-hol (X) is a closed subgroup of Aut(X top ).

a proof of the abc conjecture after Mochizuki

93(2) AutRC-hol (X) is commensurably terminal (cf. Section 0.2) in Aut(X top ).
∼(3) Any isomorphism X → Y of Aut-holomorphic spaces arises from an RC-holomorphic∼
isomorphism X → Y .
Proof. (1) is well-known (the last assertion follows from the fact of complex analysis that the limit of a sequence of holomorphic functions which uniformly converges oncompact subsets is also holomorphic).
(2) It suﬃces to show that CAut(X top ) (Authol (X)) = AutRC-hol (X) (cf. Section 0.2).
Let α ∈ CAut(X top ) (Authol (X)). Then Authol (X) ∩ αAuthol (X)α−1 is a closed subgroup of finite index in Authol (X), hence an open subgroup in Authol (X). Since
Authol (X) is connected, we have Authol (X) ∩ αAuthol (X)α−1 = Authol (X). Thus,α ∈ NAut(X top ) (Authol (X)) (cf. Section 0.2). Then by the conjugation, α gives an automorphism of Authol (X). The theorem of Schreier-van der Waerden ([SvdW]) saysthat Aut(PSL2 (R)) ∼= PGL2 (R) by the conjugation. Hence we have α ∈ AutRC-hol (X).
(Without using the theorem of Schreier-van der Waerden, we can directly show it as follows: By Cartan’s theorem (a homomorphism as topological groups between Lie groups
is automatically a homomorphism as Lie groups, cf. [Serre1, Chapter V, §9, Theorem2]), the automorphism of Authol (X) given by the conjugate of α is an automorphism of
Lie groups. This induces an automorphism of Lie algebra sl2 (C) with sl2 (R) stabilised.
Hence α is given by an element of P GL2 (R). cf. also [AbsTopIII, proo of Proposition
2.2 (ii)], [QuConf, the proof of Lemma1.10].)(3) follows from (2) since (2) implies that AutRC-hol (X) is normally terminal.
The followoing corollary says that the notions of “holomorphic structure”, “Autholomorphic structure”, and “pre-Aut-holomorphic structure” are equivalent.
Corollary 4.3.
(a sort of Bi-Anabelian Grothendieck Conjecture in the Archimedean
Theory, [AbsTopIII, Corollary 2.3]) Let X, Y be Aut-holomorphic spaces arising from
Riemann surfaces X, Y respectively. Let U, V be local structures of X top , Y top respectively.
(1) Any (U, V)-local isomorphism φ : X → Y of Aut-holomorphic spaces arises from aunique étale RC-holomorphic morphism ψ : X → Y . If X and Y are connected,
then there exist precisely 2 co-holomorphicisations X → Y, corresponding to theholomorphic and anti-holomorphic local isomorphisms.
(2) Any pre-Aut-holomorphic structure on Xtop extends to a unique Aut-holomorphic
structure on Xtop .
Proof. (1) follows from Proposition 4.2 (3).

94

Go Yamashita(2) follows by applying (1) to automorphisms of the Aut-holomorphic spaces determined by the connected open subsets of Xtop which determine the same co-holomorphicisation
as the identity automorphism.
§ 4.2.

Elliptic Cuspidalisation and Kummer Theory in the Archimedean
Theory.

Lemma 4.4.
([AbsTopIII, Corollary 2.4]) Let X be a hyperbolic Aut-holomorphic
orbispace of finite type, arising from a hyperbolic orbicurve X over C. Only from the
Aut-holomorphic orbispace X, we can determine whether or not X admits C-core, andin the case where X admits C-core, we can construct the Aut-holomorphic orbispaceassociated to the C-core in a functorial manner with respect to finite étale morphisms
by the following algorithms:(1) Let Utop → Xtop be any universal covering of Xtop . Then we reconstruct the
topological fundamental group π1 (Xtop ) as the opposite group Aut(Utop /Xtop )opp
of Aut(Utop /Xtop ).
(2) Let U be the local structure of Utop consisting of connected open subsets of Utop
which map isomorphically onto open sub-orbispaces of Xtop . We construct a natural
U-local pre-Aut-holomorphic structure on Utop by restricting Aut-holomorphic structure of X on Xtop and by transporting it to Utop . By Corollary 4.3 (2), this gives
us a natural Aut-holomorphic structure AU on Utop . We write U := (Utop , AU ).
Thus, we obtain a natural injection π1 (Xtop )opp = Aut(Utop /Xtop ) ,→ Aut0 (U) ⊂Aut(U) ∼= PGL2 (R), where we write Aut0 (U) for the connected component of theidentity of Aut(U), and the last isomorphism is an isomorphism as topologicalgroups (Here, we regard Aut(U) as a topological space by the compact-open topology).
(3) X admits C-core if and only if Im(π1 (Xtop )opp ) := Im(π1 (Xtop )opp ⊂ Aut0 (U)) isof finite index in Πcore := CAut0 (U) (Im(π1 (Xtop )opp )). If X admits C-core, then thequotient X top  Xcore := Utop //Πcore in the sense of stacks is the C-core of X. The
restriction of the Aut-holomorphic structure of U to an appropriate local structure on
U and transporting it to Xcore give us a natural Aut-holomorphic structure AXcore of
Xcore , hence the desired Aut-holomorphic orbispace (X )Xcore := (Xcore , AXcore ).
Proof. Assertions follow from the described algorithms. cf. also [CanLift, Remark
2.1.2].
Proposition 4.5.
(Elliptic Cuspidalisation in the Archimedean Theory, [AbsTopIII,
Corollary 2.7], cf. also [AbsTopIII, Proposition 2.5, Proposition 2.6]) Let X be an elliptically admissible Aut-holomorphic orbispace arising from a Riemann orbisurface X.

a proof of the abc conjecture after Mochizuki

95

By the following algorithms, only from the holomorphic space X, we can reconstruct thesystem of local linear holomorphic structures on Xtop in the sense of (Step 10) below ina functorial manner with respect to finite étale morphisms:(Step 1) By the definition of elliptical admissibility and Lemma 4.4 (2), we construct X →Xcore , where Xcore arises from the C-core Xcore of X, and Xcore is semi-elliptic (cf.
Section 3.1). There is a unique double covering E → Xcore by an Aut-holomorphic
space (not orbispace), i.e., the covering corresponding to the unique torsion-freesubgroup of index 2 of the group Πcore of Lemma 4.4. Here, E is the Aut-holomorphic
space associated to a onec-punctured elliptic curve E \ {O} over C.
(Step 2) We consider elliptic cuspidalisation diagrams E  EN ,→ E (cf. also the portion of“E \ {O}  E \ E[N ] ,→ E \ {O}” in the diagram (EllCusp) of Section 3.2), whereEN  E is an abelian finite étale coveing which is also unramified at the uniquepunctured point, Etop ,→ (EN )top is an open immersion, and EN ,→ E, EN  E are
co-holomorphic. By these diagrams, we can reconstruct the torsion points of theelliptic curve E as the points in E \ EN . We also reconstruct the group structureon the torsion points induced by the group structure of the Galois group Gal(EN /E),i.e., σ ∈ Gal(EN /E) corresponds to “+[P ]” for some P ∈ E[N ].
(Step 3) Since the torsion points constructed in (Step 2) are dense in Etop , we reconstruct
the group structure on Etop as the unique topological group structure extending
the group structure on the torsion points constructed in (Step 2). In the subsequent
steps, we take a simply connected open non-empty subset U in Etop .
(Step 4) Let p ∈ U . The group structure constructed in (Step 3) induces a local additive
structure of U at p, i.e., a +p b := (a − p) + (b − p) + p ∈ U for a, b ∈ U , whenever
it is defined.
(Step 5) We reconstruct the line segments of U by one-parameter subgroups relative to the
local additive structures constructed in (Step 4). We also reconstruct the pairs ofparallel line segments of U by translations of line segments relative to the local
additive structures constructed in (Step 4). For a line segment L, write ∂L to be
the subset of L consisting of points whose complements are connected, we shall refer
to an element of ∂L as an endpoint of L.
(Step 6) We reconstruct the parallelograms of U as follows: We define a pre-∂-parallelogram
A of U to be L1 ∪ L2 ∪ L3 ∪ L4 , where Li (i ∈ Z/4Z) are line segments (constructedin (Step 5)) such that (a) for any p1 6= p2 ∈ A, there exists a line segment L
constructed in (Step 5) with ∂L = {p1 , p2 }, (b) Li and Li+2 are parallel line segments constructed in (Step 5) and non-intersecting for any i ∈ Z/4Z, and (c)96

Go Yamashita

Li ∩ Li+1 = (∂Li ) ∩ (∂Li+1 ) with #(Li ∩ Li+1 ) = 1. We reconstruct the parallelograms of U as the interiors of the unions of the line segments L of U such that
∂L ⊂ A for a pre-∂-parallelogram A. We define a side of a parallelogram in U to
be a maximal line segment contained in P \ P for a parallelogram P of U , where wewrite P for the closure of P in U .
(Step 7) Let p ∈ U . We define a frame F = (S1 , S2 ) to be an ordered pair of intersecting
sides S1 6= S2 of a parallelogram P of U constructed in (Step 6), such that S1 ∩S2 ={p}. If a line segment L of U have an infinite intersection with P , then we shallrefer to L as being framed by F . We reconstruct an orientation of U at p (ofwhich there are precisely 2) as an equivalence class of frames of U top at p relative
to the equivalence relation of frames F = (S1 , S2 ), F = (S10 , S20 ) of U at p generated
by the relation that S10 is framed by F and S2 is framed by F 0 .
(Step 8) Let V be the Aut-holomorphic space determined by a parallelogram Vtop ⊂ U constructed in (Step 7). Let p ∈ Vtop . Let S be a one-parameter subgroup of the
topological group AV (Vtop )(∼= PSL2 (R)) and a line segment L in U constructed in(Step 5) such that one of the endpoints (cf. (Step 5)) of L is equal to p. Note that
one-parameter subgroups are characterised by using topological (not diﬀerentiable)group structure as the closed connected subgroups for which the complement of someconnected open neighbourhood of the identity element is not connected. We say that
L is tangent to S·p at p if any pairs of sequences of points of L\{p}, (S·p)\{p} converge to the same element of the quotient space Vtop \ {p}  P(V, p) determined by
identifying positive real multiples of points of Vtop \ {p} relative to the local additive
structure constructed in (Step 4) at p (i.e., projectivification). We can reconstruct
the orthogonal frames of U as the frames consisting of pairs of line segments
L1 , L2 having p ∈ U as an endpoint that are tangent to the orbits S1 · p, S2 · p of
one-parameter subgroups S1 , S2 ⊂ AV (Vtop ) such that S2 is obtained from S1 by
conjugating S1 by an element of order 4 (i.e., “±i”) of a compact one-parameter
subgroup of AV (Vtop ).
(Step 9) For p ∈ U , let (V )p∈V ⊂U be the projective system of connected open neighbourhoodsof p in U , and put{}
Ap := f ∈ Aut((V )p∈V ⊂U ) f satisfies (LAS), (Orth), and (Ori) ,where(LAS): compatibility with the local additive structures of V (⊂ U ) at p constructed in(Step 4),
(Orth): preservation of the orthogonal frames of V (⊂ U ) at p constructed in (Step 8),
anda proof of the abc conjecture after Mochizuki

97(Ori): preservation of the orientations of V (⊂ U ) at p constructed in (Step 7)(cf. also Section 0.2 for the Hom for a projective system). We equip Ap with the
topology induced by the topologies of the open neighbourhoods of p that Ap acts
on. The local additive structures of (Step 4) induce an additive structure on Ap :=Ap ∪{0}. Hence we have a natural topological field structure on Ap . Tha tautological∼action of C× on C ⊃ U induces a natural isomorphism C× → Ap of topological∼groups, hence a natural isomorphism C → Ap of topological fields. In this manner,
we reconstruct the local linear holomorphic structure “C× at p” of U at p as
the topological field Ap with the tautological action of Ap (⊂ Ap ) on (V )p∈V ⊂U .
∼(Step 10) For p, p0 ∈ U , we construct a natural isomorphism Ap → Ap0 of topological fields asfollows: If p0 is suﬃciently close to p, then the local additive structures constructedin (Step 4) induce homeomorphism from suﬃciently small neighbourhoods of p onto
suﬃciently small neighbourhoods of p0 by the translation (=the addition). These∼
homeomorphisms induce the desired isomorphism Ap → Ap0 . For general p, p0 ∈ U ,∼we can obtain the desired isomorphism Ap → Ap0 by joining p0 to p via a chainof suﬃciently small open neighbourhoods and composing the isomorphisms on local
linear holomorphic structures. This isomorphism is independent of the choice of∼such a chain. We shall refer to ((Ap )p , (Ap → Ap0 )p,p0 ) as the system of local
linear holomorphic structures on Etop or Xtop . We identify (Ap ⊂ Ap )’s for
p’s via the above natural isomorphisms and we write AX ⊂ AX for the identified
ones.
Proof. The assertions immedeately follow from the described algorithms.
Hence the formulation of “Aut-holomorphic structure” succeeds to avoid a specificfixed local referred model of C (i.e., “the C”) in the above sense too, unlike the usual
notion of “holomorphic structure”. This is also a part of “mono-anabelian philosophy” of Mochizuki. cf. also Remark 3.4.4 (3), and [AbsTopIII, Remark 2.1.2, Remark
2.7.4].
Let k be a CAF (cf. Section 0.2). We recall (cf. Section 0.2) that we write Ok ⊂ C
for the subset of elements with | − | ≤ 1 in k, Ok× ⊂ Ok for the group of units i.e.,elements with | − | = 1, and Ok := Ok \ {0} ⊂ Ok for the multiplicative monoid.
Definition 4.6.

([AbsTopIII, Definition 4.1])(1) Let X be an elliptically admissible Aut-holomorphic orbispace. A model Kummer∼
structure κk : k → AX (resp. κO× : Ok× ,→ AX , resp. κk× : k × ,→ AX , resp.
kκOk : Ok ,→ AX ) on X is an isomorphism of topological fields (resp. its restriction
to Ok× , resp. its restriction to k × , resp. its restriction to Ok ). An isomorphism

98

Go Yamashita∼κM : M → AX of topological fields (resp. an inclusion κM : Ok× ,→ AX of topologicalgroups, resp. an inclusion κM : k × ,→ AX of topological groups, resp. an inclusion
κM : Ok ,→ AX of topological monoids) is called a Kummer structure on X∼
if there exist an automorphicm f : X → X of Auto-holomorphic spaces, and an∼∼
isomorphism g : M → k of topological fields (resp. an isomorphism g : M → Ok× of∼
topological groups, resp. an isomorphism g : M → k × of topological groups, resp.
∼
an isomorphism g : M → Ok of topological monoids) such that f ∗ ◦ κk = κM ◦ g(resp. f ∗ ◦ κO× = κM ◦ g resp. f ∗ ◦ κk× = κM ◦ g resp. f ∗ ◦ κOk = κM ◦ g), where
∼

k

∼

∼

∼

f ∗ : AX → AX (resp. f ∗ : AX → AX , resp. f ∗ : AX → AX , resp. f ∗ : AX → AX ) isκ
the automorphism induced by f . We often abbreviate it as X x M .
κ1

κ2(2) A morphism φ : (X1 x M1 ) → (X2 x M2 ) of elliptically admissible Autholomorphic orbispaces with Kummer structures is a pair φ = (φX , φM ) ofa finite étale morphism φX : X1 → X2 and a homomorphism φM : M1 → M2 of
topological monoids, such that the Kummer structures κ1 and κ2 are compatible
with φM : M1 → M2 and the homomorphism (φX )∗ : AX1 → AX2 arising from thefunctoriality of the algorithms in Proposition 4.5.
The reconstruction()
XX
X 7→ X, X x A ⊂ A (with field str.) tautological Kummer structure
described in Proposition 4.5 is an Archimedean analogue of the reconstruction
)
(
Π 7→

Π, Π y k (with field str.) ⊃ k

× Kummer map

,→

lim
−→

J⊂Π: open

H 1 (J, µZb (Π)) ,described in Corollary 3.19 for non-Archimedean local field k. Namely, the reconstruction in Corollary 3.19 relates the base field k to ΠX via the Kummer theory, and thereconstruction in Proposition 4.5 relates the base field AX (∼= C) to X, hence it is a kindof Archimedean Kummer theory.
Definition 4.7.

(cf. also [AbsTopIII, Definition 5.6 (i), (iv)])−→(1) We say that a pair G = (C, C ) of a topological monoid C and a topological sub−→−→monoid C ⊂ C is a split monoid, if C is isomorphic to OC , and C ,→ C deter→− ∼mines an isomorphism C × × C → C of topological monoids (Note that C × andlog−→
C are necessarily isomorphic to S1 and (0, 1] ∼= R≥0 respectively). A morphism
−
→
→
−
∼
of split monoids G1 = (C1 , C 1 ) → G2 = (C2 , C 2 ) is an isomorphism C1 → C2
→
− ∼ −
→
of topological monoids which induce an isomorphism C 1 → C 2 of the topologicalsubmonoids.

a proof of the abc conjecture after Mochizuki

99

Remark 4.7.1.
We omit the definition of Kummer structure of split monoids([AbsTopIII, Definition 5.6 (i), (iv)]), since we do not use them in inter-universal Teichmüller theory (Instead, we consider split monoids for mono-analytic Frobenius-like−→objects). In [AbsTopIII], we consider a split monoid G = (C, C ) arising from arithholomorphic “OC ” via the mono-analyticisation, and consider a Frobenius-like object→−
M and k ∼ (G) = C ∼ × C ∼ (cf. Proposition 5.4 below) for G = (C, C ). On the other
hand, in inter-universal Teichmüller theory, we consider k ∼ (G) = C ∼ × C ∼ directlyfrom “OC ” (cf. Proposition 12.2 (4)). When we consider k ∼ (G) directly from “OC ”,
then the indeterminacies are only {±1} × {±1} (i.e., Archimedean (Indet →)); however,−→when we consider a Frobenius-like object for G = (C, C ), then we need to consider→−
the synchronisation of k1 and k2 via group-germs, and need to consider C up to R>0(i.e., we need to consider the category TB in [AbsTopIII, Definition 5.6 (i)]). cf. also
[AbsTopIII, Remark 5.8.1 (i)].
→ −We write GX = (OA
X , O AX ) for the split monoid associated to the topological field
−
→



AX , i.e., the topological monoid OA
X , and the splitting OAX ←- OAX ∩ R>0 =: O AX of
κ
×




OA
X  OAX /OAX and X x Ok . For a Kummer structure X x Ok of an elliptically−−→admissible Aut-holomorphic orbispace, we pull back OAX via the Kummer structure−→−→∼  ×Ok /Ok . We
Ok ,→ AX , we obtain a decomposition of Ok as Ok× × Ok , where Ok =consider this assignment−→(X x Ok ) 7→ (GX x Ok× × Ok )as a mono-analytification.

§ 4.3.

On the Philosophy of Étale-like and Frobenius-like Objects.

We further consider the similarities between the reconstruction algorithms in Corollary 3.19 and Proposition 4.5, and then, we explain Mochizuki’s philosophy of the
dichotomy of étale-like objects and Frobenius-like objects.
Note also that the tautological Kummer structure X x AX rigidifies the non-rigid“R>0 ” (cf. Secton 3.5) in AX (∼= C× ) in the exact sequence 0 → S1 → C× → R>0 → 0(cf. also [AbsTopIII, Remark 2.7.3]). In short, we have the following dictionary:

100

Go Yamashita

Arith. Hol.
non-Arch. k/Qp : fin.

ΠX ,ΠX y Ok rigidifies Ok×b→0
0 → Ok× → k × → Z(rigid)“k”

can be reconstructedArch. k (∼
= C)

X,

X x Ok rigidifies “R>0 ”

0 → S1 (rigid) → C× → R>0 → 0“C”

can be reconstructedMono-analytic
Gk ,

−
→
Gk y Ok× × Ok
Ok× : non-rigid

GX ,

−
→
GX x Ok× × Ok
“R>0 ”: non-rigid

We consider profinite groups ΠX , Gk , categories of the finite étale coverings overhyperbolic curves or spectra of fields, and the objects reconstructed from these as étalelike objects, and we consider, on the other hand, abstract topological monoids (withactions of ΠX , Gk ), the categories of line bundles on finite étale coverings over hyperbolic
curves, the categories of arithmetic line bundles on finite étale coverings over spectraof number fields, as Frobenius-like objects, i.e., when we reconstruct ΠX y Ok or
X x Ok , then these are regarded as étale-like objects whenever we remember that therelations with ΠX and X via the reconstruction algorithms; however, if we forget therelations with ΠX and X via the reconstruction algorithms, and we consider them as an
abstract topological monoid with an action of ΠX , and an abstract topological monoid
with Kummer structure on X, then these objects are regarded as Frobenius-like objects(cf. also [AbsTopIII, Remark 3.7.5 (iii), (iv), Remark 3.7.7], [FrdI, §I4], [IUTchI, §I1]).
Note that if we forget the relations with ΠX and X via the reconstruction algorithms,
then we cannot obtain the functoriality with respect to ΠX or X for the abstract objects.
We have the dichotomy of étale-like objects and Frobenius-like objects both onarithmetically holomorphic objects and mono-analytic objects, i.e., we can consider 4kinds of objects – arithmetically holomorphic étale-like objects (indicated by D), arithmetically holomorphic Frobenius-like objects (indicated by F), mono-analytic étale-likeobjects (indicated by D` ), and mono-analytic Frobenius-like objects (indicated by F ` )(Here, as we can easily guess, the symbol ` means “mono-analytic”). The types andstructures of prime-strips (cf. Section 10.3) and Hodge theatres reflect this classificationof objects (cf. Section 10).
Note that the above table also exhibits these 4 kinds of objects. Here, we consider
Gk y Ok× × (Ok /Ok× ) and GX x Ok× × (Ok /Ok× ) as the mono-analyticisations of arithmetically holomorphic objects Πk y Ok , and X x Ok respectively. cf. the following

101

a proof of the abc conjecture after Mochizuki

diagrams:
Frobenius-like(base with line bundle)

forgetforget/ étale-like/ étale-like
Frobenius-like(base) (base with line bundle)(base)

ΠX y Ok 
_

/ ΠX
_

X x Ok 
_

/X
_


−
→
Gk y Ok× × Ok / Gk ,

−→
GX x Ok× × Ok 
/ GX .

arith. hol.
mono-anlyticisation
mono-an.

The composite of the reconstruction algorithms Theorem 3.17 and Proposition 4.5 with“forgetting the relations with the input data via the reconstruction algorithms” areforgetthe canonical “sections” of the corresponding functors Frobenius-like −→ étale-like(Note also that, by Proposition 2.1 (2c), the topological monoid Ok can be grouptheoretically reconstructed from Gk ; however, we cannot reconstruct Ok as a submonoidof a topological field k, which needs an arithmetically holomorphic structure).
In inter-universal Teichmüller theory, the Frobenius-like objects are used to construct links (i.e., log-links and Θ-links). On the other hand, some of étale-like objects
are used (a) to construct shared objects (i.e., vertically coric, horizontally coric, andbi-coric objects) in both sides of the links, and (b) to exchange (!) both sides of aΘ-link (which is called étale-transport. cf. also Remark 9.6.1, Remark 11.1.1, andTheorem 13.12 (1)), after going from Frobenius-like picture to étale-like picture, whichis called Kummer-detachment (cf. also Section 13.2), by Kummer theory and byadmitting indeterminacies (Indet →), (Indet ↑), and (Indet xy). (More precisely, étalelike ΠX and Gk are shared in log-links. The mono-analytic Gk is also (as an abstract
topological group) shared in Θ-links; however, arithmetically holomorphic ΠX cannot
be shared in Θ-links, and even though Ok× /tors’s are Frobenius-like objects, Ok× /tors’s(not Ok ’s because the portion of the value group is dramatically dilated) are sharedb × -indeterminacies.) cf. also Theorem 12.5.
after admitting Zétaleobjects reconstructed from

Galois categoryindiﬀerent to order-likeΠX , Gk , X, GX

coverings

can be shared, can be exchanged

Frobenius−→abstract ΠX y Ok , Gk y Ok× × Ok ,Frobenioids

order-conscious

-like

−→
X x OC , GX x OC× × OCline bundles

can make links

102

Go Yamashita§ 4.4.

Mono-anabelian Reconstruction Algorithms in the Archimedean
Theory.

The following theorem is an Archimedean analogue of Theorem 3.17.
Proposition 4.8.
(Mono-anabelian Reconstruction, [AbsTopIII, Corollary 2.8])
Let X be a hyperbolic curve of strictly Belyi type over a number field k. Let k be an algebraic closure of k, and ΠX the arithmetic fundamental group of X for some basepoint.
From the topological group ΠX , we group-theoretically reconstruct the field k = k NF by
the algorithm in Theorem 3.17 (cf. Remark 3.17.1). Let v be an Archimedean place ofk. By the following group-theoretic algorithm, from the topological group ΠX and theArchimedean place v, we can reconstruct the Aut-holomorphic space Xv associated to
Xv := X ×k kv in a functorial manner with respect to open injective homomorphisms of
profinite groups which are compatible with the respective choices of Archimedean valuations:(Step 1) We reconstruct NF-points of Xv as conjugacy classes of decomposition groups of NFpoints in ΠX by in Theorem 3.17. We also reconstruct non-constant NF-rational
functions on Xk by Theorem 3.17 (Step 4) (or Lemma 3.16). Note that we alsogroup-theoretically obtain the evaluation map f 7→ f (x) at NF-point x as the restriction to the decomposition group of x (cf. Theorem 3.17 (Step 4), (Step 5)),and that the order function ordx at NF-point x as the component at x of the homob in Theorem 3.17 (Step 3) (cf. Theorem 3.17
morphism H 1 (ΠU , µZb (ΠX )) → ⊕y∈S Z(Step 5)).
(Step 2) Define a Cauchy sequence {xj }j∈N of NF-points to be a sequence of NF-points xj
such that there exists an exceptional finite set of NF-points S satisfying the following
conditions:• xj 6∈ S for all but finitely many j ∈ N, and• For any non-constant NF-rational function f on Xk , whose divisor of poles
avoids S, the sequence of values {f (xj ) ∈ kv }j∈N forms a Cauchy sequence (inthe usual sense) in kv .
For two Cauchy sequences {xj }j∈N , {yj }j∈N of NF-points with common exceptional
set S, we call that these are equivalent, if for any non-constant NF-rational function f on Xk , whose divisor of poles avoids S, the Cauchy sequences {f (xj ) ∈kv }j∈N , {f (yj ) ∈ kv }j∈N in kv converge to the same element of kv .

a proof of the abc conjecture after Mochizuki

103(Step 3) For an open subset U ⊂ kv and a non-constant NF-rational function f on Xv ,write N (U, f ) to be the set of Cauchy sequences of NF-points {xj }j∈N such thatf (xj ) ∈ U for all j ∈ N. We reconstruct the topological space Xtop = Xv (kv ) as
the set of equivalence classes of Cauchy sequences of NF-points, equipped with the
topology defined by the sets N (U, f ). A non-contant NF-rational function extends
to a function on Xtop , by taking the limit of the values.
(Step 4) Let UX ⊂ Xtop , Uv ⊂ kv be connected open subsets, and f a non-constant NFrational function on Xk , such that the function defined by f on UX gives us a∼homeomorphism fU : UX → Uv . We write Authol (Uv ) for the group of homeomor∼
phisms f : Uv → Uv (⊂ kv ), which can locally be expressed as a convergent power
series with coeﬃcients in kv with respect to the topological field structure of kv .
(Step 5) Write AX (UX ) := fU−1 ◦Authol (Uv )◦fU ⊂ Aut(UX ). By Corollary 4.3, we reconstruct
the Aut-holomorphic structure AX on Xtop as the unique Aut-holomorphic structure
which extends the pre-Aut-holomorphic structure defined by the groups AX (UX ) in
(Step 4).
Proof. The assertions immediately follow from the described algorithms.
We can easily generalise the above theorem to hyperbolic orbi curves of strictly
Belyi type over number fields.
Lemma 4.9.
(Compatibility of Elliptic Cuspidalisation in Archimedean Placewith Galois Theoretic Belyi Cuspidalisation, [AbsTopIII, Corollary 2.9]) In the situationof Proposition 4.8, suppose further that X is elliptically admissible. From the topological group ΠX , we group-theoretically reconstruct the field k = k NF by Theorem 3.17
(cf. Remark 3.17.1), i.e., via Belyi cuspidalisation. Let v be an Archimedean place ofk(ΠX ). Let X = (Xtop , AX ) be the Aut-holomorphic space constructed from the topological group ΠX and the Archimedean valuation v in Proposition 4.8, i.e., via Cauchy
sequences. Let AX be the field constructed in Proposition 4.5, i.e., via elliptic cuspidalisation. By the following group-theoretically algorithm, from the topological group∼
ΠX and the Archimedean valuation v, we can construct an isomorphism AX → kv of
topological fields in a functorial manner with respect to open injective homomorphismsof profinite groups which are compatible with the respective choices of Archimedean valuations:(Step 1) As in Proposition 4.8, we reconstruct NF-points of Xv , non-constant NF-rational
functions on Xk , the evaluation map f 7→ f (x) at NF-point x, and the order function
ordx at NF-point x. We also reconstruct Etop and the local additive structures on
it in Proposition 4.5.

104

Go Yamashita(Step 2) The local additive structures of Etop determines the local additive structures of
Xtop . Let x be an NF-point of Xv (kv ), ~v an element of a suﬃciently small neighbourhood UX ⊂ Xtop of x in Xtop which admits such a local additive structure.
For each NF-rational function f which vanishes at x, the assignment (~v , f ) 7→
limn→∞ nf (n ·x ~v ) ∈ kv , where “ ·x 00 is the operation induced by the local additive
structure at x, depends only on the image df |x ∈ ωx of f in the Zariski cotangent
space ωx to Xv . It determines an embedding UX ,→ Homkv (ωx , kv ) of topologicalspaces, which is compatible with the local additive structures.
(Step 3) Varying the neighbourhood UX of x, the embeddings in (Step 2) give us an isomor∼
phism Ax → kv of topological fields by the compatibility with the natural actionsof Ax , kv× respectively. As x varies, the isomorphisms in (Step 3) are compati∼
ble with the isomorphisms Ax → Ay in Proposition 4.5. This gives us the desired∼
isomorphism AX → kv .
Remark 4.9.1.
An importance of Proposition 4.5 lies in the fact that the algorithm starts in a purely local situation, since we will treat local objects (i.e., objects
over local fields) which a priori do not come from a global object (i.e., an object over anumber field) in inter-universal Teichmüller theory. cf. also Remark 3.17.4.
Proof. The assertions immediately follow from the described algorithms.
§ 5.

Log-volumes and Log-shells.

In this section, we construct a kind of “rigid containers” called log-shells bothfor non-Archimedean and Archimedean local fields. We also reconstruct the local logvolume functions. By putting them together, we reconstruct the degree functions ofarithmetic line bundles.
§ 5.1.

Non-Archimedean Places.

Let k be a finite extension of Qp , and k an algebraic closure of k. Let X be a
hyperbolic orbicurve over k of strictly Belyi type. Write k ∼ := (Ok× )pf ( Ok× ) the
perfection of Ok× (cf. Section 0.2). The p-adic logarithm logk induces an isomorphism∼

logk : k ∼ → kof topological monoids, which is compatible with the actions of ΠX . We equip k ∼ withthe topological field structure by transporting it from k via the above isomorphism logk .
Then we have the following diagram, which is called a log-link:
(
)gp
(
)gp
(Log-link (non-Arch))
Ok ⊃ Ok×  k ∼ = Ok∼:= Ok∼∪ {0} ← Ok∼ ,a proof of the abc conjecture after Mochizuki

105

which is compatible with the action of ΠX (this will mean that ΠX is vertically core. cf.
Proposition 12.2 (1), Remark 12.3.1, and Theorem 12.5 (1)). Note that we can construct
the sub-diagram Ok ⊃ Ok× → k ∼ , which is compatible with the action of Gk , only fromthe topological monoid Ok (i.e., only from the mono-analytic structure); however, weneed the topological field k (i.e., need the arithmetically holomorphic structure) toequip k ∼ a topological field structure and to construct the remaining diagram k ∼ =gp(Ok∼ ) ← Ok∼ .
Definition 5.1.
(

OkΠ∼X ⊂)

We put

)
1 ∗(
Ik :=
Ik ⊂ (k ∼ )ΠX ,2p

where Ik∗ := Im

{
}
( )pf
×
×
∼
Ok → Ok=kwhere we write (−)ΠX for the fixed part of the action of ΠX , and we shall refer to Ik
as a Frobenius-like holomorphic log-shell.
On the other hand, from ΠX , we can group-theoretically reconstruct an isomorphk(ΠX ) of the ind-topological field k by Theorem 3.19, and we can construct a logshell I(ΠX ) by using k(ΠX ), instead of k. Then we shall refer to I(ΠX ) as theétale-like holomorphic log-shell for ΠX . By the cyclotomic rigidity isomorphism(Cyc. Rig. LCFT2), the Kummer homomorphism gives us a Kummer isomorphism×∼×(ΠX y k ) → (ΠX y k (ΠX )) (⊂ lim H 1 (ΠU ), µZb (ΠX ))−→U

×for k (ΠX ) (cf. (Step 4) of Theorem 3.17, and Remark 3.19.2), hence obtain a Kummer
isomorphism(Kum (non-Arch))∼

Ik → I(ΠX )for Ik . In inter-universal Teichmüller theory, we will also use the Kummer isomorphismof log-shells via the cyclotomic rigidity of mono-theta environments in Theorem 7.23
(1) cf. Proposition 12.2.
Note that we have important natural inclusions(Upper Semi-Compat. (non-Arch))
Ok× , logk (Ok× ) ⊂ Ik and Ok× (ΠX ), logk(ΠX ) (Ok× (ΠX )) ⊂ I(ΠX ),which will be used for the upper semi-compatibility of log-Kummer correspondence(cf. Proposition 13.7 (2)). Here, we write Ok× (ΠX ) := Ok (ΠX )× , Ok (ΠX ) := Ok (Π)ΠX ,and Ok (ΠX ) is the ring of integers of the ind-topological field k(Π).
Proposition 5.2.
(Mono-analytic Reconstruction of Log-shell and Local Logvolume in non-Archimedean Places, [AbsTopIII, Proposition 5.8 (i), (ii), (iii)]) Let G

106

Go Yamashita

be a topological group, which is isomorphic to Gk . By the following algorithm, from G,
we can group-theoretically reconstruct the log-shell “ Ik ” and the (non-normalised) local
log-volume function “µlogk ” (cf. Section 1.3) in a functorial manner with respect to open
homomorphisms of topological groups:
×(Step 1) We reconstruct p, f (k), e(k), k , Ok , and Ok× by Proposition 2.1 (1), (3b), (3c),(2a), (2c), and (2b) respectively. To indicate that these are reconstructed from G,×
we write pG , fG , eG , k (G), Ok (G) and Ok× (G) for them respectively (From nowGbe the number of elements
on, we use the notation (−)(G) in this sense). Let pm
G×of k (G)G of pG -power orders, where we write (−)G for the fixed part of the action
of G.
{
}
(Step 2) We reconstruct the log-shell “ Ik ” as I(G) := 2p1G Im Ok× (G)G → k ∼ (G) := Ok× (G)pf .
Note that, by the canonical injection Q ,→ End(k ∼ (G)) (Here, End means the endomorphisms as (additive) topological groups), the multiplication by 2p1G canonicallymakes sense. We shall refer to I(G) as the étale-like mono-analytic log-shell.
×(Step 3) Write Rnon (G) := (k (G)/Ok× (G))∧ , where we write (−)∧ for the completion withrespect to the order structure determined by the image of Ok (G)/Ok× (G). By the
canonical isomorphism R ∼= End(Rnon (G)), we consider Rnon (G) as an R-module.
It is also equipped with a distinguished element, i.e., the image F(G) ∈ Rnon (G)of the Frobenius element (constructed in Proposition 2.1 (5)) of Ok (G)G /Ok× (G)G
via the composite Ok (G)G /Ok× (G)G ⊂ Ok (G)/Ok× (G) ⊂ Rnon (G). By sending∼fG log pG ∈ R to F(G) ∈ Rnon (G), we have an isomorphism R → Rnon (G) of Rmodules. By transporting the topological field structure from R to Rnon (G) via thisbijection, we consider Rnon (G) as a topological field, which is isomorphic to R.

(Step 4) We write M(k ∼ (G)G ) for the set of compact open non-empty subsets of the topological additive group k ∼ (G)G . We can reconstruct the local log-volume functionµlog (G) : M(k ∼ (G)G ) → Rnon (G) by using the following characterisation properties:(a) (additivity) For A, B ∈ M(k ∼ (G)G ) with A ∩ B = ∅, we have exp(µlog (G)(A ∪B)) = exp(µlog (G)(A)) + exp(µlog (G)(B)), where we use the topological fieldstructure of Rnon (G) to define exp(−),(b) (+-translation invariance) For A ∈ M(k ∼ (G)G ) and a ∈ k ∼ (G)G , we have
µlog (G)(A + a) = µlog (G)(A),(c) (normalisation)
log

µ

)
(
mG
+ G eG fG F(G),(G)(I(G)) = −1 −fG

where we write G to be 1 if pG 6= 2, and to be 2 if pG = 2.

107

a proof of the abc conjecture after Mochizuki

Moreover, if a field structure on k := k ∼ (G)G is given, then we have the p-adic
logarithm logk : Ok× → k on k (where we can see k both on the domain and thecodomain), and we have
(5.1)

µlog (G)(A) = µlog (G)(logk (A))∼

for an open subset A ⊂ Ok× such that logk induces a bijection A → logk (A).
Remark 5.2.1. Note that, we cannot normalise µlog (G) by “µlog (G)(OkG∼ ) = 0”,
since “OkG∼ ” needs arithmetically holomorphic structure to reconstruct (cf. [Qp GC]).
Remark 5.2.2. The formula (5.1) will be used for the compatibility of log-links
with log-volume functions (cf. Proposition 13.10 (4)).
Proof. To lighten the notation, write p := pG , e := eG , f := fG , m := mG ,  := G .
log
×f
Then we have µlog
k (Ik ) = ef log p + µk (log(O( k )) )= (ef − m) log p − log(p − 1) +µlog
(Ok× ) = (ef −m) log p−log(pf −1)+log 1 − p1f +µlog
k (Ok ) = (ef −m−f ) log p =(k)−1 + e − mf log p.
f
§ 5.2.

Archimedean Places.

Let k be a CAF (cf. Section 0.2). Let X be an elliptically admissible Aut-holomorphic∼
orbispace, and κk : k → AX a Kummer structure. Note that k (resp. k × , Ok× ) and AX
have natural Aut-holomorphic structures, and κk determines co-holomorphicisations between k (resp. k × , Ok× ) and AX . Let k ∼  k × be the universal covering of k × , whichis uniquely determined up to unique isomorphism, as a pointed topological space (Itis well-known that it can be explicitly constructed by the homotopy classes of pathson k × ). The topological group structure of k × induces a natural topological groupstructure of k ∼ . The inverse (i.e., the Archimedean logarithm) of the exponential mapk  k × induces an isomorphism∼

logk : k ∼ → kof topological groups. We equip k ∼ (resp. Ok∼ ) with the topological field structure(resp. the topological multiplicative monoid structure) by transporting it from k via∼
the above isomorphism logk . Then κk determines a Kummer structure κk∼ : k ∼ → AX
(resp. κOk∼ : Ok∼ ,→ AX ) which is uniquely characterised by the property thatthe co-holomorphicisation determined by κk∼ (resp. κOk∼ ) coincides with the co∼holomorphicisation determined by the composite of k ∼ → k and the co-holomorphicisation
determined by κk . By definition, the co-holomorphicisations determined by κk , and κk∼(resp. κOk∼ ) are compatible with logk (This compatibility is an Archimedean analogue

108

Go Yamashitaof the compatibility of the actions of ΠX in the non-Archimedean situation). We havethe following diagram, which is called a log-link:(Log-link (Arch))

(
)gp
(
)gp
Ok ⊂ k ×  k ∼ = Ok∼:= Ok∼∪ {0} ← Ok∼ ,which is compatible with the co-holomorphicisations determined by the Kummer structures (This will mean X is vertically core. cf. Proposition 12.2 (1)). Note that we canconstruct the sub-diagram Ok ⊂ k ×  k ∼ only from the topological monoid Ok (i.e.,
only from the mono-analytic structure); however, we need the topological field k (i.e.,need the arithmetically holomorphic structure) to equip k ∼ a topological field structure
gp
and to construct the remaining diagram k ∼ = (Ok∼ ) ← Ok∼ .
Definition 5.3.

We put()
1
Ok∼ = Ik ⊂ Ik := Ok×∼ · Ik∗ (⊂ k ∼ ) ,π

where Ik∗ is the the uniquely determined “line segment” (i.e., closure of a connectedpre-compact open subset of a one-parameter subgroup) of k ∼ which is preserved bymultiplication by ±1 and whose endpoints diﬀer by a generator of ker(k ∼  k × ) (i.e.,
Ik∗ is the interval between “−πi” and “πi”, and Ik is the closed disk with redius π).
Here, a pre-compact subset means a subset contained in a compact subset, and see
Section 0.2 for π. We shall refer to Ik as a Frobenius-like holomorphic log-shell.
On the other hand, from X, we can group-theoretically reconstruct an isomorphk(X) := AX of the field k by Proposition 4.5, and we can construct a log-shell I(X) byusing k(X), instead of k. Then we shall refer to I(X) as the étale-like holomorphic
log-shell for X. The Kummer structure κk gives us a Kummer isomorphism(Kum (Arch))∼

Ik → I(X)

for Ik .
Note that we have important natural inclusions(Upper Semi-Compat. (Arch))
Ok∼ ⊂ Ik , Ok× ⊂ expk (Ik ) and Ok∼ (X) ⊂ I(X), Ok× (X) ⊂ expk(X) (I(X))
which will be used for the upper semi-compatibility of log-Kummer correspondence(cf. Proposition 13.7 (2)). Here, we write Ok× (X) := Ok (X)× , and Ok (X) (cf. also
Section 0.2) is the subset of elements of absolute value ≤ 1 for the topological field k(X)(or, if we do not want to use absolute value, the topological closure of the subset ofelements x with limn→∞ xn = 0), and expk (resp. expk(X) ) is the exponential function
for the topological field k (resp. k(Π)).

a proof of the abc conjecture after Mochizuki

109

Note also that we use Ok×∼ to define Ik in the above, and we need the topologicalfield structure of k to construct Ok×∼ ; however, we can construct Ik as the closure of
the union of the images of Ik∗ via the finite order automorphisms of the topological(additive) group k ∼ , thus, we need only the topological (multiplicative) group structure×of k (not the topological field structure of k) to construct Ik .
Proposition 5.4.
(Mono-analytic Reconstruction of Log-shell and Local Logvolumes in Archimedean Places, [AbsTopIII, Proposition 5.8 (iv), (v), (vi)]) Let G =−→(C, C ) be a split monoid. By the following algorithm, from G, we can group-theoreticallyreconstruct the log-shell “ IC ”, the (non-normalised) local radial log-volume function
log
“µlog
C ” and the (non-normalised) local angular log-volume function “µ̆C ” in a functorial
manner with respect to morphisms of split monoids (In fact, the constructions do not→−depend on C , which is “non-rigid” portion. cf. also [AbsTopIII, Remark 5.8.1]):(Step 1) Let C ∼  C × be the (pointed) unversal covering of C × . The topological groupstructure of C × induces a natural topological group structure on C ∼ . We regardC ∼ as a topological group (Note that C × and C ∼ are isomorphic to S1 and the
additive group R respectively). Writek ∼ (G) := C ∼ × C ∼ ,k × (G) := C × × C ∼ .

(Step 2) Let Seg(G) be the equivalence classes of compact line segments on C ∼ , i.e., compact subsets which are either equal to the closure of a connected open set or aresets of one element, relative to the equivalence relation determined by translation
on C ∼ . Forming the union of two compact line segments whose intersection is aset of one element determines a monoid structure on Seg(G) with respect to whichSeg(G) ∼= R≥0 (non-canonical isomorphism). Thus, this monoid structure determines a topological monoid structure on Seg(G) (Note that the topological monoid
structure on Seg(G) is independent of the choice of an isomorphism Seg(G) ∼= R≥0 ).
(Step 3) We have a natural homomorphism k ∼ (G) = C ∼ × C ∼  k × (G) = C × × C ∼ of two
dimensional Lie groups, where we equip C ∼ , C × with the diﬀerentiable structure bychoosing isomorphisms C ∼ ∼= R, C × ∼= R× (the diﬀerentiable structures do notdepend on the choices of isomorphisms). We reconstruct the log-shell “ IC ” as}{I(G) := (ax, bx) | x ∈ IC∗ ∼ ; a, b ∈ R; a2 + b2 = 1 ⊂ k ∼ (G),
where we write IC∗ ∼ ⊂ C ∼ for the unique compact line segment on C ∼ which isinvariant with respect to the action of {±1}, and maps bijectively, except for its
endpoints, to C × . Note that, by the canonical isomorphism R ∼= End(C ∼ ) (Here,End means the endomorphisms as (additive) topological groups), ax for a ∈ R and110

Go Yamashitax ∈ IC∗ ∼ canonically makes sense. We shall refer to I(G) as the étale-like monoanalytic log-shell.
(Step 4) We write Rarc (G) := Seg(G)gp (Note that Rarc (G) ∼= R as (additive) topologicalgroups). By the canonical isomorphism R ∼= End(Rarc (G)), we consider Rarc (G) as
an R-module. It is also equipped with a distinguished element, i.e., (Archimedean)∗
Frobenius element F(G) ∈ Seg(G) ⊂ Rarc (G) determined by IC∼ . By sending 2π ∈∼R to F(G) ∈ Rarc (G), we have an isomorphism R → Rarc (G) of R-modules. Bytransporting the topological field structure from R to Rarc (G) via this bijection, we
consider Rarc (G) as a topological field, which is isomorphic to R.
(Step 5) By the same way as I(G), we put{}
Ok×∼ (G) := (ax, bx) | x ∈ ∂IC∗ ∼ ; a, b ∈ R; a2 + b2 = π −2 ⊂ k ∼ (G),
where ∂IC∗ ∼ is the set of endpoints of the line segment IC∗ ∼ (i.e., the points whose
complement are connected. cf. Proposition 4.5). Then we have a natural isomorphism R>0 × Ok×∼ (G) ∼ k ∼ (G) \ {(0, 0)}, where (a, x) is sent to ax (Note thatax makes sense by the canonical isomorphism R ∼= End(C ∼ ) as before). We
write prrad : k ∼ (G) \ {(0, 0)}  R>0 , prang : k ∼ (G) \ {(0, 0)}Ok×∼ (G) for the
first and second projection via the above isomorphism. We extend the map prrad :k ∼ (G) \ {(0, 0)}  R>0 to a map prrad : k ∼ (G) → R.
(Step 6) Let M(k ∼ (G)) be the set of non-empty compact subsets A ⊂ k ∼ (G) such that
A projects to a (compact) subset prrad (A) of R which is the closure of its interior in R. For any A ∈ M(k ∼ (G)), by taking the length µ(G)(A) of prrad (A) ⊂R with respect to the usual Lebesgues measure on R. By taking the logarithmµlog (G)(A) := log(µ(G)(A)) ∈ R ∼= Rarc (G), where we use the canonical identification R ∼= Rarc (G), we reconstruct the desired local radial log-volume functionµlog (G) : M(k ∼ (G)) → Rarc (G). This also satisfiesµlog (G)(I(G)) =

log π
F(G)
2π

by definition.
(Step 7) We write M̆(k ∼ (G)) for the set of non-empty compact subsets A ⊂ k ∼ (G) \ {(0, 0)}such that A projects to a (compact) subset prang (A) of Ok×∼ (G) which is the closureof its interior in Ok×∼ (G). We reconstruct the local angular log-volume functionµ̆log (G) : M̆(k ∼ (G)) → Rarc (G) by taking the integration µ̆(G)(A) of prang (A) ⊂Ok×∼ (G) on Ok×∼ (G) with respect to the diﬀerentiable structure induced by the one in(Step 1), taking the logarithm µ̆log (G)(A) := log(µ̆(G)(A)) ∈ R ∼= Rarc (G), wherea proof of the abc conjecture after Mochizuki

111

we use the canonical identification R ∼= Rarc (G), and the normalisationµ̆log (G)(Ok×∼ (G)) =log 2πF(G).
2πMoreover, if a field structure on k := k ∼ (G) is given, then we have the exponential map
expk : k → k × on k (where we can see k both on the domain and the codomain), andwe haveµlog (G)(A) = µ̆log (G)(expk (A))

(5.2)

for a non-empty compact subset A ⊂ k with expk (A) ⊂ Ok× , such that prrad and expk∼∼
induce bijections A → prrad (A), and A → expk (A) respectively.
Remark 5.4.1.
The formula (5.2) will be used for the compatibility of log-links
with log-volume functions (cf. Proposition 13.10 (4)).
Proof. Proposition immediately follows from the described algorithms.
§ 6.

Preliminaries on Tempered Fundamental Groups.

In this section, we collect some prelimiraries on tempered fundamental groups, andwe show a theorem on “profinite conjugates vs. tempered conjugates”, which plays animportant role in inter-universal Teichmüller theory.
§ 6.1.

Some Definitions.

From this section, we use André’s theory of tempered fundamental groups ([A1])for rigid-analytic spaces (in the sense of Berkovich) over non-Archimedean fields. We
give a short review on it here. He introduced the tempered fundamental groups to
obtain a fundamental group of “reasonable size” for rigid analytic spaces: On one
hand, the topological fundamental groups π1top for rigid analytic spaces are too small(e.g., π1top (P1Cp \ {0, 1, ∞}, x) = {1}. If X is a proper curve with good reduction,then π1top (X an , x) = {1}). On the other hand, the étale fundamental groups π1ét for
rigid analytic spaces aree too big (e.g., By the Gross-Hopkins period mappings ([GH1],[GH2]), we have a surjection π1ét (P1Cp , x)  SL2 (Qp ). cf. also [A2, II.6.3.3, and Remark
after III Corollary 1.4.7]). André’s tempered fundamental group π1temp is of reasonablesize, and it comparatively behaves well at least for curves. An étale covering Y  X
of rigid analytic spaces is called tempered covering if there exists a commutative
diagram
//T
Z

Y


//X

112

Go Yamashitaof étale coverings, where T  X is a finite étale covering, and Z  T is a possibly
inifinite topological covering. When we define a class of coverings, then we can define
the fundamental group associated to the class. In this case, π1temp (X, x) classifies allbtempered pointed coverings of (X, x). For example, we have π1temp (P1Cp \ {0, ∞}) = Z,b×Z
b if
and for an elliptic curve E over Cp with j-invariant jE , we have π1temp (E) ∼=Z
b if |j|p > 1 ([A1, §4.6]). Here, Z corresponds to the|j|p ≤ 1, and π temp (E) ∼= Z×Z
1universal covering of the graph of the special fiber. The topology of π1temp is a little bitcomplicated. In general, it is neither discrete, profinite, nor locally compact; however,
it is pro-discrete. For a (log-)orbicurve X over an MLF, we write B temp (X) for the
category of the (log-)tempered coverings over the rigid analytic space associated with
X. For a (log-)orbicurve X over a field, we also write B(X) for the Galois category of
the finite (log-)étale coverings over X.
Definition 6.1.

([SemiAnbd, Definition 3.1 (i), Definition 3.4])(1) If a topological group Π can be written as an inverse limit of an inverse system ofsurjections of countable discrete topological groups, then we shall refer to Π as atempered group (Note that any profinite group is a tempered group).
(2) Let Π be a tempered group. We say that Π is temp-slim if we have ZΠ (H) = {1}for any open subgroup H ⊂ Π.
(3) Let f : Π1 → Π2 be a continuous homomorphism of tempered groups. We say Π1is relatively temp-slim over Π2 (via f ) if we have ZΠ2 (Im{H → Π2 }) = {1} forany open subgroup H ⊂ Π1 .
(4) ([IUTchI, §0]) For a topological group Π, we write B temp (Π) (resp. B(Π)) forthe category whose objects are countable discrete sets (resp. finite sets) with acontinuous Π-action, and whose morphisms are morphisms of Π-sets. A category
C is called a connected temperoid, (resp. a connected anabelioid) if C is
equivalent to B temp (Π) (resp. B(Π)) for a tempered group Π (resp. a profinite group
Π). Note that, if C is a connected temperoid (resp. a connected anabelioid), thenC is naturally equivalent to (C 0 )> (resp. (C 0 )⊥ ) (cf. Section 0.2 for (−)0 , (−)> and(−)⊥ ). If a category C is equivalent to B temp (Π) (resp. B(Π)) for a tempered groupΠ with countable basis (resp. a profinite group Π), then we can reconstruct the
topological group Π, up to inner automorphism, by the same way as Galois category(resp. by the theory of Galois category). (Note that in the anabelioid/profinite
case, we have no need of condition like “having countable basis”, since “compact
set arguments” are available in profinite topology.) We write π1 (C) for it. We also
write π1 (C 0 ) := π1 ((C 0 )> ) (resp. π1 (C 0 ) := π1 ((C 0 )⊥ )) for C a connected temperoid(resp. a connected anabelioid).

a proof of the abc conjecture after Mochizuki

113(5) For connected temperoids (resp. anabelioids) C1 , C2 , a morphism C1 → C2 oftemperoids (resp. a morphism C1 → C2 of anabelioids) is an isomorphismclass of functors C2 → C1 which preserves finite limits and countable colimits (resp.
finite colimits) (This is definition in [IUTchI, §0] is slightly diﬀerent from the one
in [SemiAnbd, Definition 3.1 (iii)]). We also define a morphism C10 → C20 to be amorphism (C10 )> → (C20 )> (resp. (C10 )⊥ → (C20 )⊥ ).
Note that if Π1 , Π2 are tempered groups with countable basis (resp. profinitegroups), then there are natural bijections among• the set of continuous outer homomorphisms Π1 → Π2 ,• the set of morphisms Btemp (Π1 ) → B temp (Π2 ) (resp. B(Π1 ) → B(Π2 )), and• the set of morphisms Btemp (Π1 )0 → B temp (Π2 )0 (resp. B(Π1 )0 → B(Π2 )0 ).
(cf. also [IUTchI, Remark 2.5.3].)
Let K be a finite extension of Qp .
Lemma 6.2.
Let X be a hyperbolic curve over K. We write ∆temp⊂ Πtemp
XX
temp
for the geometric tempered fundamental group π1 (X, x) and the arithmetic tempered
fundamental group π1temp (X, x) for some basepoint x, respectively. Then we have agroup-theoretic characterisation of the closed subgroup ∆temp
in Πtemp.
X
X
Remark 6.2.1.

By remark 2.4.1, pro-Σ version of Lemma 6.2 holds as well.

Proof. Note that the homomorphisms ∆temp→ ∆X := (∆temp)∧ and Πtemp→
X
X
X
temp ∧
ΠX := (ΠX ) to the profinite completions are injective respectively, since the homomorphism from a (discrete) free group to its profinite completion is injective (Freegroups and surface groups are residually finite (cf. also Proposition C.5)). Then byusing the group-theoretic characterisation of ∆X in ΠX (Corollary 2.4), we obtain agroup-theoretic characterisation of ∆temp
as ∆temp= Πtemp∩ ∆X .
X
X
X
Let K be an algebraic closure of K. We write k and k for the residue field of K
and K respectively (k is an algebraic closure of k).
Definition 6.3.
(1) Let X be a pointed stable curve over k with marked points D. Write X := X \ D.
Then we associate a dual semi-graph (resp. dual graph) GX to X as follows:
We set the set of the vertices of GX to be the set of the irreducible componentsof X, the set of the closed edges of GX to be the set of the nodes of X, and the114

Go Yamashita

set of the open edges of GX to be the set of the divisor of infinity of X (i.e., themarked points D of X). To avoid confusion, we write Xv and νe for the irreducible
component of X and the node of X corresponding to a vertex v and an closed edgee respectively. A closed edge e connects vertices v and v 0 (we may allow the case ofv = v 0 ), if and only if the node νe is the intersection of two branches corresponding
to Xv and Xv0 . An open e connects a vertex v, if and only if the marked point
corresponding to e lies in Xv .
(2) (cf. [AbsAnab, Appendix]) We contitue the situation of (1). Let Σ be a set ofprime numbers. A finite étale covering of curves is called of Σ-power degree if any
prime number dividing the degree is in Σ. We also associate a (pro-Σ) semi-graphΣ) of anabelioids to X, such that the underlying semi-graph is GX as
GX (= GX
follows: Write X 0 := X \ {nodes}. For each vertex v of GX , let Gv be the Galois
category (or a connected anabelioid) of the finite étale coverings of Σ-power degree of
Xv0 := Xv ×X X 0 which are tamely ramified along the nodes and the marked points.
For the branches νe (1) and νe (2) of the node νe corresponding to a closed edge e of
GX , we consider the scheme-theoretic interstion Xν0 e (i) of the completion along thebranch νe (i) at the node νe of X 0 for i = 1, 2 (Note that Xν0 e (i) is non-canonicallyisomorphic to Spec k((t))). We fix a k-isomorphism Xν0 e (1) ∼= Xν0 e (2) , we identify
these, and we write Xe0 for the identified object. Let Ge be the Galois category(or a connected anabelioid) of the finite étale coverings of Σ-power degree of Xe0
which are tamely ramified along the node. For each open edge ex corresponding toa marked point x, write Xx0 to be the scheme-theoretic interstion of the completionof X at the marked point x with X 0 (Note that Xx0 is non-canonically isomorphic
to Spec k((t))). Let Gex be the Galois category (or a connected anabelioid) of thefinite étale coverings of Σ-power degree of Xx0 which are tamely ramified along themarked point. For each edge e connecting vertices v1 and v2 , we have naturalfunctors Gv1 → Ge , Gv2 → Ge by the pull-backs. For an open edge e connected toa vertex v, we have a natural functor Gv → Ge by the pull-backs. Then the dataΣ
GX (= GX
) := {Gv ; Ge ; Gv → Ge } defines a semi-graph of anabelioids.
(3) (cf. [SemiAnbd, Definition 2.1]) For a (pro-Σ) semi-graph G(= G Σ ) = {Gv ; Ge ; Gv →Ge } of anabelioids with connected underlying semi-graph G, we define a category
B(G)(= B(G Σ )) as follows: An object of B(G)(= B(G Σ )) is data {Sv , φe }v,e , wherev (resp. e) runs over the vertices (resp. the edges) of G, such that Sv is an object∼of Gv , and φe : e(1)∗ Sv1 → e(2)∗ Sv2 is an isomorphism in Ge , where e(1) and e(2)are the branches of e connecting v1 and v2 respectively (Here, e(i)∗ : Gvi → Ge isa given datum of G). We define a morphism of B(G) in the evident manner. Then
B(G) itself is a Galois category (or a connected anabelioid). In the case of G = GXin (2), the fundamental group associated to B(G)(= B(G Σ )) is called the (pro-Σ)a proof of the abc conjecture after Mochizuki

115

admissible fundamental group of X.
(4) (cf. [SemiAnbd, paragraph before Definition 3.5 and Definition 3.5]) Let G(= G Σ ) ={Gv ; Ge ; Gv → Ge } be a (pro-Σ) semi-graph of anabelioids such that the underlying semi-graph G is connected and countable. We define a category Bcov (G)(=B cov (G Σ )) as follows: An object of B cov (G)(= BΣ,cov (G)) is data {Sv , φe }v,e , wherev (resp. e) runs over the vertices (resp. the edges) of G, such that Sv is an object∼of (Gv0 )> (cf. Section 0.2 for (−)0 and (−)> ), and φe : e(1)∗ Sv1 → e(2)∗ Sv2 is an
isomorphism in (Ge0 )> , where e(1) and e(2) are the branches of e connecting v1 andv2 respectively (Here, e(i)∗ : Gv → Ge is a given datum of G). We define a morphism of Bcov (G) in the evident manner. We can extend the definition of B cov (G)to a semi-graph of anabelioids such that the underlying semi-graph G is countable;
however, is not connected. We have a natural full embedding B(G) ,→ B cov (G). We
write (B(G) ⊂) B temp (G)(= B temp (G Σ )) ⊂ B cov (G) for the full subcategory whose
objects {Sv , φe }v,e are as follows: There exists an object {Sv0 , φ0e } of B(G) such thatfor any vertex or edge c, the restriction of {Sv0 , φ0e } to Gc splits the restriction of{Sv , φe } to Gc i.e., the fiber product of Sv0 (resp. φ0e ) with Sv (resp. φe ) over theterminal object (resp. over the identity morphism of the terminal object) in (Gv0 )>(resp. (Ge0 )> ) is isomorphic to the coproduct of a countable number of copies of Sv0(resp. φ0e ) for any vertex v and any edge e. We shall refer to B temp (G)(= B temp (G Σ ))as the (pro-Σ) (connected) temperoid associated with G(= G Σ ).
(Σ),temp

We can associate the fundamental group ∆temp(= ∆G) := π1 (Btemp (G)) (=G
π1 (Btemp (G Σ ))) of B temp (G)(= B temp (G Σ )) (after taking a fiber functor) by the same(Σ)
way as a Galois category. We write ∆G (= ∆G ) for the profinite completion of(Σ),temp(Σ)∆G. (Note that ∆G (= ∆G ) is not the maximal pro-Σ quotient of π1 (B(G Σ ))since the profinite completion of the “graph covering portion” is not pro-Σ). By(Σ),temp(Σ)
definition, ∆temp(= ∆G) and ∆G are tempered groups (Definition 6.1 (1),G
cf. also [SemiAnbd, Proposition 3.1 (i)]).
Remark 6.3.1. (cf. [SemiAnbd, Example 3.10]) Let X be a smooth log-curve over
K. The special fiber of the stable model of X determines a semi-graph G of anabelioids.
We can relate the tempered fundamental group ∆temp:= π1temp (X) of X with a systemXof admissible fundamental groups of the special fibers of the stable models of coveringsof X as follows: Let · · · ⊂ Ni ⊂ · · · ⊂ ∆temp(i ≥ 1) be an exhausitive sequence ofX. Then Ni determines a finite
open characteristic subgroups of finite index of ∆tempX
log-étale covering of X whose special fiber of the stable model gives us a semi-graph Giof anabelioids, on which ∆temp/Ni acts faithfully. Then we obtain a natural sequenceXtemp
of functors · · · ← B(Gi ) ← · · · ← B temp (G) which are compatible with the actions of∆temp/Ni . Hence this gives us a sequence of surjections of tempered groups ∆temp

X
X

116

Go Yamashita
out

out

· · ·  π1 (B temp (Gi )) o (∆temp/Ni )  · · ·  π1 (B temp (Gj )) o (∆temp/Nj )  · · · 
X
X
temp
π1 (B
(G)). Then by construction, we have()
temp out
temp
temp ∼/ker(Ni  ∆temp).
(6.1)∆X = lim ∆Gi o (∆X /Ni ) = lim ∆temp
Gi
←−
←− X
i

We also have
(6.2)

i

(
)
out
∼
c
ci  ∆G ),∆X = lim ∆Gi o (∆X /Ni ) = lim ∆X /ker(N
i
←−
←−
i

i

ci for the closure of Ni in ∆X . By these expressions of ∆temp and ∆X in
where we write N
X
terms of ∆temp
’s
and
∆
’s,
we
can
reducesomeproperties
ofthetemperedfundamental
Gi
Gi
temp
group ∆X of the generic fiber to some properties of the admissible fundamental groups(Σ),tempof the special fibers (cf. Lemma 6.4 (5), and Corollary 6.10 (1)). We write ∆Xforthe fundamental group associated to the category of the tempered coverings dominatedby coverings which arise as a graph covering of a finite étale Galois covering of X over K(Σ)(Σ)
of Σ-power degree, and ∆X its profinite completion (Note that ∆X is not the maximal
pro-Σ quotient of ∆tempor ∆X since the profinite completion of the “graph coveringX
portion” is not pro-Σ). If p 6∈ Σ, then we have(Σ),temp ∼∆X(Σ),temp

= ∆G

and

(Σ)
(Σ)
∆X ∼
= ∆G ,

since Galois coverings of Σ-power degree are necessarily admissible (cf. [Hur, §3], [SemiAnbd,
Corollary 3.11]).
§ 6.2.

Profinite Conjugates vs. Tempered Conjugates.

Lemma 6.4.
(special case of [SemiAnbd, Proposition 2.6, Corollary 2.7 (i), (ii),
Proposition 3.6 (iv)] and [SemiAnbd, Example 3.10]) Let X be a smooth hyperbolic log:= π1temp (X). We write
curve over K. Write ∆temp:= π1temp (X ×K K) and Πtemp
X
X
G temp (= G Σ,temp ) for the temperoid determined by the special fiber of the stable modelof X ×K K and a set Σ of prime numbers, and write ∆temp:= π1 (G temp ) (for someGbase point). Let H be a connected sub-semi-graph containing a vertex of the underlingsemi-graph G of G temp . We assume that H is stabilised by the natural action of GK
on G. We write Htemp for the temperoid over H obtained by the restriction of G temp
to H. Write ∆temp:= π1 (Htemp )(⊂ ∆temp). We write ∆G and ∆H for the profinite
H
G
temp
temp
completion of ∆Gand ∆H respectively.
(1) ∆H ⊂ ∆G is commensurably terminal,(2) ∆H ⊂ ∆G is relatively slim (resp. ∆temp⊂ ∆tempis relatively temp-slim),HG(3) ∆H and ∆G are slim (resp. ∆tempand ∆temp
are temp-slim),HG

a proof of the abc conjecture after Mochizuki

117(4) inertia subgroups in ∆temp
of cusps are commensurably terminal, andG(5) ∆tempand Πtemp
are temp-slim.
X
X
Proof. (1) can be shown by the same manner as in Proposition 2.7 (1a) (i.e.,
consider coverings which are connected over H and totally split over a vertex outside H).
(3) for ∆: We can show that ∆H and ∆G are slim in the same way as in Proposition 2.7.
(2): ∆H ⊂ ∆G is relatively slim, by (1), (3) for ∆ and Lemma 2.6 (2). Then theinjectivity (which comes from the residual finiteness of free groups and surface groups (cf.
also Proposition C.5)) of ∆temp,→ ∆H and ∆temp,→ ∆G implies that ∆temp⊂ ∆tempH
G
H
G
is relatively temp-slim. (3) for ∆temp : It follows from (2) for ∆temp in the same way as
in Proposition 2.6 (2). (4) can also be shown by the same manner as in Proposition 2.7(2c). (5): By the isomorphism (6.1) in Remark 6.3.1 and (3) for ∆temp , it follows that∆temp
is temp-slim (cf. [SemiAnbd, Example 3.10]). Hence Πtemp
is also temp-slim by
X
X
Proposition 2.7 (1c).
Definition 6.5.

Let G be a semi-graph of anabelioids.

) for a vertex v as(1) We shall refer to a subgroup of the form ∆v := π1 (Gv ) (⊂ ∆temp
Ga verticial subgroup.
b Σ\{p} := ∏(2) We shall refer to a subgroup of the form ∆e := π1 (Ge ) (∼=Zl∈Σ\{p} Zl )(⊂temp∆G ) for a closed edge e as an edge-like subgroup.
Proposition 6.6.
([SemiAnbd, Theorem 3.7 (iv)]) Let X be a smooth hyperbolic log-curve over K. We write G temp (= G Σ,temp ) for the temperoid determined by
the special fiber of the stable model of X and a set Σ of prime numbers, and write:= π1 (G temp ) (for some base point). For a vertex v (resp. an edge e) of∆tempGthe underlying sub-semi-graph G of G temp , we write ∆v := π1 (Gv )(⊂ ∆temp) (resp.
Gtemp∆e := π1 (Ge )(⊂ ∆G )) to be the profinite group corresponding to Gv (resp. Ge ) (Notethat we are not considering open edges here). Then we have the followng group-theoretic
characterisations of ∆v ’s and ∆e ’s.
(1) The maximal compact subgroups of ∆temp
are precisely the verticial subgroups of
G
temp
∆G .
(2) The nontrivial intersection of two maximal compact subgroups of ∆temp
are preciselyG
temp
the edge-like subgroups of ∆G .
Remark 6.6.1. Proposition 6.6 reconstructs the dual graph (not the dual semigraph) of the special fiber from the tempered fundamental group without using theaction of the Galois group of the base field. In Corollary 6.12 below, we reconstruct

118

Go Yamashita

the inertia subgroups, hence open edges as well, using the Galois action. However,
we can reconstruct the open edges without Galois action, by more delicate method in[SemiAnbd, Corollary 3.11] (i.e., by constructing a covering whose fiber at a cusp underconsideration contains a node).
We can also reconstruct the dual semi-graph of the special fiber from the profinite
fundamental group by using the action of the Galois group of the base field (cf. [profGC]).
Proof. We write ∆G for the profinite completion of ∆temp. First, note that it folG0lows that ∆v ∩∆v0 has infinite index in ∆v for any vertices v 6= v by the commensurable
terminality of ∆temp(Lemma 6.4 (1)). Next, we take an exhausitive sequence of openv
characteristic subgroups · · · ⊂ Ni ⊂ · · · ⊂ ∆temp
of finite index, and let Gi (→ G) be theGtemp
covering corresponding to Ni (⊂ ∆G ). We write G∞i for the universal graph coveringof the underlying semi-graph Gi of Gi .
Let H ⊂ ∆temp
be a compact subgroup, then H acts continuously on G∞
i for eachGi ∈ I, thus its action factors through a finite quotient. Hence H fixes a vertex or anedge of G∞i (cf. also [SemiAnbd, Lemma 1.8 (ii)]), since an action of a finite group on
a tree has a fixed point by [Serre2, Chapter I, §6.5, Proposition 27] (Note that a graphin [Serre2] is an oriented graph; however, if we split each edge of G∞
i into two edges,
then the argument works). Since the action of H is over G, if H fixes an edge, thenit does not change the branches of an edge. Therefore, H fixes at least one vertex. If,
for some cofinal subset J ⊂ I, H fixes more than or equal to three vertices of G∞
j foreach j ∈ J, then by considering paths connecting these vertices (cf. [Serre2, Chapter I,§2.2, Proposition 8]), it follows that there exists a vertex having (at least) two closed
edges in which H fixes the vertex and the closed edges (cf. also [SemiAnbd, Lemma 1.8(ii)]). Since each Gj is finite semi-graph, we can choose a compatible system of such avertex having (at least) two closed edges on which H acts trivially. This implies that H
is contained in (some conjugate in ∆G of) the intersection of ∆e and ∆e0 , where e and
e0 are distinct closed edges. Hence H should be trivial. By the above arguments also
show that any compact subgroup in ∆temp
is contained in ∆v for precisely one vertex
Gv or in ∆v , ∆v0 for precisely two vertices v, v 0 , and, in the latter case, it is contained in∆e for precisely one closed edge e.
Proposition 6.7.
([IUTchI, Proposition 2.1]) Let X be a smooth hyperbolic logcurve over K. We write G temp (= G Σ,temp ) for the temperoid determined by the special:= π1 (G temp ),fiber of the stable model of X and a set Σ of prime numbers. Write ∆temp
G
and we write ∆G for the profinite completion of ∆temp(Note that the “profinite portion”
Gremains pro-Σ, and the “combinatorial portion” changes from discrete to profinite). LetΛ ⊂ ∆temp
be a nontrivial compact subgroup, γ ∈ ∆G an element such that γΛγ −1 ⊂G
∆temp
. Then γ ∈ ∆temp.
GG

a proof of the abc conjecture after Mochizuki

119b (resp. Γtemp ) be the “profinite semi-graph” (resp. “pro-semi-graph”)
Proof. Let Γ
associated with the universal profinite étale (resp. tempered) covering of G temp . Thenb We shall refer to a pro-vertex in Γb in the image
we have a natural inclusion Γtemp ,→ Γ.
−1of this inclusion as a tempered vertex. Since Λ and γΛγ
are compact subgroups
temp
0of ∆G , there exists vertices v, v of G (here we write G for the underlying semiby Proposition 6.6 (1) forgraph of G temp ) such that Λ ⊂ ∆temp
and γΛγ −1 ⊂ ∆temp
v
v0
temp
temp
for this base points correspond to temperedsome base points. Here, ∆v
and ∆v0temp−1
0
γ −1 is, and γ∆temp
γ −1 ∩ ∆temp
vertices ve, ve ∈ Γ. Now, {1} 6= γΛγ⊂ γ∆tempvv
v0
also a fundamental group of Gvtemp with the base point obtained by conjugating thebase point under consideration above by γ. This correponding to a tempered vertex
veγ ∈ Γtemp . Hence for the tempered vertices veγ and ve0 , the associated fundamentalgroup has nontrivial intersection.
By replacing Πtempby an open covering, we may assume that each irreducible
G
component has genus ≥ 2, any edge of G abuts to two distinct vertices, and that, for
any two (not necessarily distinct) vertices w, w0 , the set of edges e in G such that e abuts
to a vertex w00 if and only if w00 ∈ {w, w0 } is either empty of of cardinality ≥ 2. In thecase where Σ = {2}, then by replacing Πtempby an open covering, we may assume thatG
the last condition “cardinality ≥ 2” is strongthen ot the condition “even cardinality”.
If veγ is not equal to ve0 nor veγ is adjacent to ve0 , then we can construct the coveringover Xv (here Xv is the irreducible component corresponding to v), such that theramification indices at the nodes and cusps of Xv are all equal (Note that such a covering
exists by the assumed condition on G in the last paragraph), then we extend this coveringover the irreducible components which adjacent to Xv , finally we extend the covering toa split covering over the rest of X (cf. also [AbsTopII, Proposition 1.3 (iv)] or [NodNon,
Proposition 3.9 (i)]). This implies that there exist open subgroups J ⊂ ∆temp
which
G
temp
temp −1
contain ∆v0
and determine arbitrarily small neighbourhoods γ∆v γ ∩ J of {1}.
This is a contradiction. Therefore, veγ is equal to ve0 , or veγ is adjacent to ve0 . In particular,veγ is tempered since ve0 is tempered. Hence both ve and veγ are tempered. Thus, we haveγ ∈ ∆temp, as desired.
G
Corollary 6.8.

([IUTchI, Proposition 2.2]) Let ∆tempand ∆temp
be as in Lemma 6.4.
GH(1) ∆temp⊂ ∆G is commensurably terminal, andG
is also⊂ ∆G is commensurably terminal. In particular, ∆temp⊂ ∆temp(2) ∆temp
G
H
H
commensurably terminal as well.
Proof. (1): Let γ ∈ ∆G be an element such that ∆temp∩ γ∆temp
γ −1 is finite index
GG
in ∆temp. Let ∆v ⊂ ∆temp
be a verticial subgroup, and write Λ := ∆v ∩ γ∆temp
γ −1 ⊂GG
G∆v ⊂ ∆temp. Since [∆v : Λ] = [∆temp: ∆temp∩ γ∆temp
γ −1 ] < ∞, the subgroup Λ is
G
G
G
G

120

Go Yamashita

open in the compact subgroup ∆v , so, it is a nontrivial compact subgroup of ∆temp
.
G
temp
temp
temp
−1
−1
−1
Now, γ Λγ = γ ∆v γ ∩∆G⊂ ∆G . Since Λ, γ Λγ ⊂ ∆G
and Λ is a nontrivial
temp−1
by Proposition 6.7. Thus γ ∈ ∆temp, ascompact subgroup, we have γ∈ ∆GGdesired.
(2): We have ∆temp⊂ C∆temp (∆temp) ⊂ C∆G (∆temp) ⊂ C∆G (∆H ) by definiH
H
H
G

) =
tion. By Lemma 6.4 (1), we have C∆G (∆H ) = ∆H . Thus, we have C∆G (∆temp
H
temp
temp
temp
C∆H (∆H ) combining these. On the other hand, by (1) for ∆H , we have C∆H (∆H ) =∆temp. By combining these, we have ∆temp⊂ C∆G (∆temp) = C∆H (∆temp) = ∆temp
, as
H
H
H
H
H
desired.
Corollary 6.9.
([IUTchI, Corollary 2.3]) Let ∆X , ∆temp, ∆temp, H, ∆G , ∆H
G
H
temp
temp
temp
temp
be as in Lemma 6.4. Write ∆X,H := ∆X ×∆temp ∆H (⊂ ∆X ), and ∆X,H :=G∆X ×∆G ∆H (⊂ ∆X ).
temp(1) ∆temp(resp. ∆X,H ⊂ ∆X ) is commensurably terminal.
X,H ⊂ ∆X

(2) The closure of ∆temp
X,H in ∆X is equal to ∆X,H .
(3) We have ∆X,H ∩ ∆temp= ∆tempX
X,H (⊂ ∆X ).
(4) Let Ix ⊂ ∆temp(resp. Ix ⊂ ∆X ) be a cusp x of X. Write xe for the cusp in theXtemp
stable model corresponding to x. Then Ix lies in a ∆X -(resp. ∆X -)conjugate of∆temp
e meets an irreducible component of the special
X,H (resp. ∆X,H ) if and only if x
fiber of the stable model which is contained in H.
(5) Suppose that p ∈/ Σ, and there is a prime number l 6∈ Σ ∪ {p}. Then ∆X,H is slim.
In particular, we can define
out

temp
Πtemp
X,H := ∆X,H o GK ,out

ΠX,H := ∆X,H o GK

by the natural outer actions of GK on ∆temp
X,H and ∆X,H respectively.
temp(6) Suppose that p 6∈ Σ, and there is a prime number l 6∈ Σ ∪ {p}. ΠtempandX,H ⊂ ΠX
ΠX,H ⊂ ΠX are commensurably terminal.

Proof. (1) follows from Lemma 6.4 (1) and Corollary 6.8 (2). Next, (2) and (3) are
trivial. (4) follows by noting that an inertia subgroup of a cusp is contained in precisely
one verticial subgroup. We can show this, (possibly after replacing G by a finite étale
covering) for any vertex v which is not abuted by the open edge e corresponding to
the inertia subgroup, by constructing a covering which is trivial over Gv and nontrivial
over Ge ([CombGC, Proposition 1.5 (i)]). (6) follows from (5) and (1). We show (5)(The following proof is a variant of the proof of Proposition 2.7 (2a)). Let J ⊂ ∆X be

a proof of the abc conjecture after Mochizuki

121

an open normal subgroup, and write JH := J ∩ ∆X,H . We write J  J Σ∪{l} for theΣ∪{l}maximal pro-Σ ∪ {l} quotient, and JH:= Im(JH → J Σ∪{l} ). Suppose α ∈ ∆X,H
commutes with JH . Let v be a vertex of the dual graph of the geometric special fiber of astable model XJ of the covering XJ of XK corresponding to J. We write Jv ⊂ J for thedecomposition group of v, (which is well-defined up to conjugation in J), and we write
Σ∪{l}
Σ∪{l}
Σ∪{l}
Jv
:= Im(Jv → J Σ∪{l} ). First, we show a claim that Jv∩ JH
is infinite and
Σ∪{l}
Σ∪{l}
non-abelian. Note that Jv ∩ JH , hence also Jv∩ JH, surjects onto the maxmall
pro-l quotient Jv of Jv , since the image of the homomorphism Jv ⊂ J ⊂ ∆X  ∆G
is pro-Σ, and we have ker(Jv ⊂ J ⊂ ∆X  ∆G ) ⊂ Jv ∩ JH , and l 6∈ Σ. Now, Jvl isthe pro-l completion of the fundamental group of hyperbolic Riemann surface, hence
is infinite and non-abelian. Therefore, the claim is proved. Next, we show (5) from
Σ∪{l}
Σ∪{l}
the claim. We consider various ∆X -conjugates of Jv∩ JH
in J Σ∪{l} . Then
Σ∪{l}
Σ∪{l}
by Proposition 6.6, it follows that α fixes v, since α commutes with Jv
∩ JH
.
Σ∪{l}
Σ∪{l}
l
Moreover, since the conjugation by α on Jv ( Jv∩ JH) is trivial, it follows
that α not only fixes v, but also acts trivially on the irreducible component of thespecial fiber of XJ corresponding to v (Note that any nontrivial automorphism of anirreducible component of the special fiber induces a nontrivial outer automorphism of
the tame pro-l fundamental group of the open subscheme of this irreducible component
given by taking the complement of the nodes and cusps). Then α acts on (J Σ∪{l} )ab asa unipotent automorphism of finite order, since v is arbitrary, hence α acts trivially on(J Σ∪{l} )ab . Then we have α = 1, as desired since J is arbitrary.
Corollary 6.10.
([IUTchI, Proposition 2.4 (i), (iii)]) We continue to use the(Σ),temp ∼same notation as above. We assume that p 6∈ Σ (which implies that ∆temp
 ∆X=X(Σ),temp
(Σ) ∼
(Σ)
temp
∆G
= ∆G
and ∆X  ∆X = ∆G = ∆G ).
(1) Let Λ ⊂ ∆temp
be a nontrivial pro-Σ compact group, γ ∈ ΠX an element such that
X
temp
−1
γΛγ ⊂ ∆X . Then we have γ ∈ Πtemp.
X(2) ([A1, Corollary 6.2.2]) ∆temp⊂ ∆X (resp. Πtemp⊂ ΠX ) is commensurably termiXX
nal.
Remark 6.10.1. By Corollary 6.10 (2) and Theorem B.1, we can show a tempered
version of Theorem B.1:∼

dense in an open subgp. of fin. index
Homdom(Πtemp, Πtemp)/Inn(∆temp)
K (X, Y ) → HomGKXYY(For a homomorphism, up to inner automorphisms of ∆temp, in the right-hand side,Yconsider the induced homomorphism on the profinite completions. Then it comes froma morphism in the left-hand side by Theorem B.1, and we can reduce the ambiguity of inner automorphisms of the profinite completion of ∆temp
to the one of innerY
temp
automorphisms of ∆Y
by Corollary 6.10 (2)). cf. also [SemiAnbd, Theorem 6.4].

122

Go Yamashita

Proof. (1): Let γe ∈ Πtemp
 GK be a lift of the image of γ ∈ ΠX  GK . By
X−1
replacing γ by γ(eγ ) ∈ ∆X , we may assume that γ ∈ ∆X . For an open characteristic
temp
b for the closure of N in ∆X , and we write GN for thesugroup N ⊂ ∆X , we write N(pro-Σ) semi-graph of anabelioids determined by the stable model of the covering of
X ×K K corresponding to N . By the isomorphisms (6.1) and (6.2) in Remark 6.3.1,, the image ofit suﬃces to show that for any open characteristic subgroup N ⊂ ∆temp
X
temp
temp
b 
b
γ ∈ ∆X  ∆X /ker(N  ∆GN ) comes from ∆X /ker(N  ∆GN ) ,→ ∆X /ker(N∆GN ). Let N be such an open characteristic subgroup of ∆temp. Since N is of finite
X
temp
temp
b . We take a lift γ
index in ∆X , we have ∆X /N ∼e ∈ ∆temp ∆temp/N ∼= ∆X /N
=
X
X
−1
b
b
b
∆X /N of the image γ ∈ ∆X  ∆X /N . By replacing γ by γ(eγ)∈ N , we may
tempb
assume that γ ∈ N . Note that ΛN := Λ ∩ N (⊂ N ⊂ ∆X ) is a nontrivial open
compact subgroup, since N is of finite index in ∆temp. Since ΛN is a pro-Σ subgroup in
X(Σ),temp
temp
temp. Hence the image∆X , it is sent isomorphically to the image by ∆X  ∆X(Σ),temp ∼(Σ),temp
temp
temp
temp
ΛN ⊂ ∆Gof ΛN by ∆X  ∆X
= ∆G
is also nontrivial open
= ∆Gcompact subgroup (Here we need the assumption p 6∈ Σ. If p ∈ Σ, then we only have(Σ),temp(Σ),temp
a surjection ∆X ∆G, and the image of ΛN might be trivial). Note that
temp
temp
ΛN is in ∆GN = Im(N ⊂ ∆X ∆temp). Consider the following diagram, whereG
the horizontal sequences are exact:
1

1/ ∆tempG N_/ ∆temp /ker(N  ∆temp )XGN_/ ∆temp /N


/ ∆G





N

b  ∆temp )/ ∆X /ker(N
GN

X

/1

∼
=

b
/ ∆X /N/1b , the image γ of γ ∈ ∆X  ∆X /ker(Nb  ∆G ) lands in ∆G . Since
Since γ is in N
N
N
temp
temp
−1
ΛN (⊂ ∆GN ) is a nontrivial open compact subgroup, and γΛN γ ⊂ ∆GN by assumption, we conclude γ ∈ ∆temp
by Proposition 6.7, as desired. (2) follows from (1) by theGNsame way as in Corollary 6.8 (1).
The following theorem is technically important for inter-universal Teichmüller theory :Theorem 6.11.
(Profinite Conjugates vs. Tempered Conjugates, [IUTchI, Corollary 2.5]) We continue to use the same notation as above. We assume that p 6∈ Σ. Then
(1) Any inertia subgroup in ΠX of a cusp of X is contained in Πtemp
if and only if itXtemp
is an inertia subgroup in ΠXof a cusp of X, and(2) A ΠX -conjugete of Πtemp
contains an inertia subgroup in Πtemp
of a cusp of X if
XX
temp
and only if it is equal to ΠX .

123

a proof of the abc conjecture after Mochizuki

Remark 6.11.1.

In inter-universal Teichmüller theory,(1) we need to use tempered fundamental groups, because the theory of the étale theta
function (cf. Section 7) plays a crucial role, and(2) we also need to use profinite fundamental groups, because we need hyperbolic orbicurve over a number field for the purpose of putting “labels” for each places ina consistent manner (cf. Proposition 10.19 and Proposition 10.33). Note also thattempered fundamental groups are available only over non-Archimedean local fields,
and we need to use profinite fundamental groups for hyperbolic orbicurve over anumber field.
Then in this way, the “Profinite Conjugates vs. Tempered Conjugates” situation as in
Theorem 6.11 naturally arises (cf. Lemma 11.9). The theorem says that the profinite
conjugacy indeterminacy is reduced to the harmless tempered conjugacy indeterminacy.
b be an inertia subgroup of a cusp x. By applying Corollary 6.10
Proof. Let Ix (∼= Z)
to the unique pro-Σ subgroup of Ix , it follows that a ΠX -conjugate of Ix is containedin Πtemp
if and only if it is a Πtemp-conjugate of Ix , and that a ΠX -conjugate of Πtemp
X
X
X
temp
containes Ix if and only if it is equal to ΠX
Corollary 6.12.
Let X be a smooth hyperbolic log-curve over K, an algebraic
closure K of K. Then we can group-theoretically reconstruct the inertia subgroups andthe decomposition groups of cusps in Πtemp:= π1temp (X).
X
Remark 6.12.1. By combining Corollary 6.12 with Proposition 6.6, we can grouptheoretically reconstruct the dual semi-graph of the special fiber (cf. also Remark 6.6.1).
Proof. By Lemma 6.2 (with Remark 6.2.1) we have a group-theoretic reconstruction of the quotient Πtemp
 GK from Πtemp. We write ∆X and ΠX for the profinite
X
X
temp
temp
,→ ∆X
completions of ∆X and ΠX respectively. By using the injectivity of ∆temp
X
temp
and ΠX ,→ ΠX (i.e., residual finiteness (cf. also Proposition C.5)), we can reconstruct
inertia subgroups I of cusps by using Corollary 2.9, Remark 2.9.2, and Theorem 6.11
(Note that the reconstruction of the inertia subgroups in ∆X has ∆X -conjugacy indeterminacy; however, by using Theorem 6.11, this indeterminacy is reduced to ∆temp
X
conjugacy indeterminacy, and it is harmless). Then we can group-theoretically reconstruct the decomposition groups of cusps, by taking the normaliser NΠtemp (I), since I
is normally terminal in ∆temp
by Lemma 6.4 (4).
X

X

Remark 6.12.2. (a little bit sketchy here, cf. [AbsAnab, Lemma 2.5], [AbsTopIII,
Theorem 1.10 (c)]) By using the reconstruction of the dual semi-graph of the special
fiber (Remark 6.12.1), we can reconstruct

124

Go Yamashitab(1) a positive rational structure on H 2 (∆X , µZb (GK ))∨ := Hom(H 2 (∆X , µZb (GK )), Z),(2) hence a cyclotomic rigidity isomorphism:(Cyc. Rig. via Pos. Rat. Str.)∼µZb (GK ) → µZb (ΠX )(We shall refer to this as the cyclotomic rigidity isomorphism via positive
rational structure and LCFT.)
as follows (cf. also Remark 3.19.1):(1) By taking finite étale covering of X, it is easy to see that we may assume thatthe normalisation of each irreducible component of the special fiber of the stable model X of X has genus ≥ 2, and that the dual semi-graph ΓX of the special fiber is non-contractible (cf. [profGC, Lemma 2.9, the first two paragraphs ofthe proof o Theorem 9.2]). By Remark 6.12.1, we can group-theoretically reconstruct the quotient ∆temp ∆comb
corresponding to the coverings of graphs (NoteX
X
that, in [AbsAnab], we reconstruct the dual semi-graph of the special fiber fromprofinite fundamental group, i.e., without using tempered structure, via the reconstruction algorithms in [profGC]. cf. also Remark 6.6.1). We write ∆X for the
profinite completion of ∆temp, and write V := ∆ab
X . Note that the abelianisation
X
sing
comb
comb ab ∼V:= (∆X ) = H1 (ΓX , Z)(6= 0) is a free Z-module. By using a theoremof Raynaud (cf. [AbsAnab, Lemma 2.4], [Tam, Lemma 1.9], [Ray, Théorème 4.3.1]),
after replacing X by a finite étale covering (whose degree depends only on p andthe genus of X), and K by a finite unramified extension, we may assume that the“new parts” of the Jacobians of the irreducible components of the special fiber are
all ordinary, hence we obtain a GK -equivariant quotient V  V new , such that wehave an exact sequence
0 → V mult → VZnew:= V new ⊗Zb Zp → V ét → 0,pwhere V ét is an unramified GK -module, and V mult is the Cartier dual of an unramb = 0). We write (−)−
ified GK -module, and that V new  VZbcomb := V comb ⊗Z Z(6(like VZnew, VZbcomb ) for the tensor product in this proof. Then the restriction of thepnon-degenerate group-theoretic cup productb
V ∨ ⊗Zb V ∨ ⊗Zb µZb (GK ) → M := H 2 (∆, µZb (GK )) (∼= Z),
b to (V new )∨where (−)∨ := Hom(−, Z),b(V new )∨ ⊗Zb (V new )∨ ⊗Zb µZb (GK ) → M (∼
= Z)

125

a proof of the abc conjecture after Mochizuki

is still non-degenerate since it arises from the restriction of the polarisation givenby the theta divisor on the Jacobian of X to the “new part” of X (i.e., it gives us
an ample divisor). Then we obtain an inclusion(VZbcomb )∨ ⊗Zb µZb (GK )⊗Zb M ∨ ,→ (V new )∨ ⊗Zb µZb (GK )⊗Zb M ∨ ,→ ker(V new  VZbcomb ) ⊂ V new ,where the second last inclusion comes from µZb (GK )GK = 0.
By the Riemann hypothesis for abelian varieties over finite fields, the (ker(V ét ) ⊗Zp Qp )GK = 0, where we write (−)GK) ⊗Zp Qp )GK = ((ker(V ét  VZcomb
VZcombpp
for the GK -coinvariant quotient (Note that ker(V ét  VZcomb) arises from the pp
divisible group of an abelian variety over the residue field). Thus, the surjection
V ét  V comb ⊗Zb Zp has a unique GK -splitting VZcomb,→ V ét ⊗Qp . Similarly, byptaking Cartier duals, the injection (VZbcomb )∨ ⊗Zb µZb (GK ) ⊗ M ∨ ⊗Zb Zp ,→ V mult also
has a unique GK -splitting V mult  (VZbcomb )∨ ⊗Zb µZb (GK ) ⊗ M ∨ ⊗Zb Qp . By these
splittings, the GK -action on V new ⊗Zp gives us a p-adic extension classηZp ∈ ((VQcomb)∨ )⊗2 ⊗M ∨ ⊗H 1 (K, µZb (GK ))/Hf1 (K, µZb (GK )) = ((VQcomb)∨ )⊗2 ⊗M ∨ :pp0/ V new/ V mult
Qp
O
?

Qp

/ V ét

Qp b

/0

 .

~~

(VQcomb)∨ ⊗ µZb (GK ) ⊗ M ∨p

VQcomb.
p

Next, ker(VZbnew
 VZbcomb) is an unramified GK -module, since it arises from l(6= p)00
b 0 :=
divisible group of a semi-abelian variety over the residue field, where we write Z∏
l6=p Zl . Again by the Riemann hypothesis for abelian varieties over finite fields,
the injection (VZbcomb)∨ ⊗ µZb (GK ) ⊗ M ∨ ,→ ker(VZbnew
 VZbcomb) of unramified GK 000modules splits uniquely over Q. Then we can construct a prime-to-p-adic extension
classηZb0 ∈ ((VZbcomb)∨ )⊗2 ⊗M ∨ ⊗H 1 (K, µZb (GK ))/Hf1 (K, µZb (GK ))⊗Q = ((VZbcomb)∨ )⊗2 ⊗M ∨ ⊗Q :000

/ ker(V new  V comb )
b0 ⊗Q
b0 ⊗Q
Z
Z
O
?

/ V newb0 ⊗QZ/ V comb
b0 ⊗Q
Z

/0

~~

(VZbcomb)∨ ⊗ µZb (GK ) ⊗ M ∨ .
0 ⊗QThen combining p-adic extension class and prime-to-p-adic extension class, we obtain an extension classηZb ∈ ((VZbcomb )∨ )⊗2 ⊗M ∨ ⊗H 1 (K, µZb (GK ))/Hf1 (K, µZb (GK ))⊗Q = ((VZbcomb )∨ )⊗2 ⊗M ∨ ⊗Q.

126

Go Yamashita

Therefore, we obtain a bilinear form
(VZbcomb )⊗2 → M ∨ ⊗Zb Q,
and the image of (V comb )⊗2 ⊂ (VZbcomb )⊗2 gives us a positive rational structure(i.e., Q>0 -structure) on M ∨ ⊗Zb Q (cf. [AbsAnab, Lemma 2.5]).
(2) By the group-theoretically reconstructed homomorphisms∼abab ∼ bb Z)b ∼H 1 (GK , µZb (GK )) → Hom(H 1 (GK , Z),= Gab
K  GK /Im(IK → GK ) = Z

in the proof of Corollary 3.19 (2), we obtain a natural surjection
H 1 (GK µZb (ΠX ))  Hom(µZb (GK ), µZb (ΠX )) ∼= H 2 (∆X , µZb (GK ))∨(Recall the definition of µZb (ΠX )). Then by taking the unique topological generator of Hom(µZb (GK ), µZb (ΠX )) which is contained in the positive rational structue∼of H 2 (∆X , µZb (GK ))∨ , we obtain the cyclotomic rigidity isomorphism µZb (GK ) →µZb (ΠX ).
It seems important to give a remark that we use the value group portion (i.e., we∼use O , not O× ) in the construction of the above surjection H 1 (GK , µZb (GK )) →abab ∼ bb Z)
b ∼Hom(H 1 (GK , Z),= Gab
K  GK /Im(IK → GK ) = Z, hence in the constructionof the cyclotomic rigidity via positive rational structure and LCFT as well. In interuniversal Teichmüller theory, not only the existence of reconstruction algorithms, butalso the contents of reconstruction algorithms are important, and whether or not we
use the value group portion in the algorithm is crucial for the constructions in the final multiradial algorithm in inter-universal Teichmüller theory. cf. also Remark 9.6.2,
Remark 11.4.1, Propositin 11.5, and Remark 11.11.1.
§ 7.

Étale Theta Functions — Three Fundamental Rigidities.

In this sectin, we introduce another (probably the most) important ingredient ofinter-universal Teichmüller theory, that is, the theory of the étale theta functions. In
Section 7.1, we introduce some varieties related to the étale theta function. In Section 7.4, we introduce the notion of mono-theta environment, which plays importantroles in inter-universal Teichmüller theory.
§ 7.1.

Theta-related Varieties.

We introduce some varieties and study them in this subsection. Let K be a finite
extension of Qp , and K an algebraic closure of K. Write GK := Gal(K/K). Let

a proof of the abc conjecture after Mochizuki

127

X → Spf OK be a stable curve of type (1, 1) such that the special fiber is singularand geometrically irreducible, the node is rational, and the Raynaud generic fiber X(which is a rigid-analytic space) is smooth. For the varieties and rigid-analytic spacesin this Section, we also call marked points cusps, we always write log-structure onthem, and we always consider the fundamental groups for the log-schemes and logfor the tempered fundamental group of, ∆temprigid-analytic spaces. We write Πtemp
X
X
X (with log-structure on the marked point) for some basepoint. We have an exact
sequence 1 → ∆X → ΠX → GK → 1. Write ΠX := (Πtemp)∧ , ∆X := (∆temp)∧ to beX
X
the profinite completions of Πtemp, ∆temprespectively. We have the natural surjection
X
X
temp
∆X
 Z corresponding to the universal graph-covering of the dual-graph of theconfiguration of the irreducible components of X. We write Z for this quotient for
b for the profinite
the purpose of distinguish it from other Z’s. We also write ∆X  Z
completion of ∆temp
 Z.
XΘWrite ∆X := ∆X /[∆X , [∆X , ∆X ]], and we shall refer to it as the theta quotient∧2 ab ∼ babof ∆X . We also write ∆Θ :=∆X (= Z(1)), and ∆ell
X := ∆X . We have the following
exact sequences:ell
1 → ∆Θ → ∆Θ
X → ∆X → 1,bb
1 → Z(1)→ ∆ell
X → Z → 1.
We write (∆temp)Θ and (∆temp)ell for the image of ∆temp
via the surjections ∆X  ∆Θ
X
X
X
X
ell
and ∆X  (∆Θ)∆respectively:
X
X
∆O X

/ / ∆Θ
OX/ / ∆ell
X
O

?
∆temp
X

?
/ / (∆temp )Θ

?
/ / (∆temp )ell .

XX

We write (Πtemp)Θ and (Πtemp)ell for the push-out of Πtemp
via the surjections ∆temp

X
X
X
X
temp Θ
temp
temp Θ
temp ell(∆X ) and ∆X  ((∆X ) )(∆X ) respectively:
Πtemp
XO

/ / (Πtemp )Θ
XO/ / (Πtemp )ell
X O

?
∆temp
X

?
/ / (∆temp )Θ

?
/ / (∆temp )ell .

X

X

We have the following exact sequences:
1 → ∆Θ → (∆temp)Θ → (∆temp)ell → 1,XXb
1 → Z(1)→ (∆temp)ell → Z → 1.
X

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Go Yamashita

Let Y  X (resp. Y  X) be the infinite étale covering correspoinding to thekernel Πtempof Πtemp
 Z. We have Gal(Y /X) = Z. Here, Y is an infinite chainYX
of copies of the projective line with a marked point 6= 0, ∞ (which we call a cusp),
joined at 0 and ∞, and each of these points “0” and “∞”is a node in Y. We write(∆temp)Θ , (∆temp)ell (resp. (Πtemp)Θ , (Πtemp)ell ) for the image of ∆temp(resp. Πtemp
)
Y
Y
Y
Y
Y
Y
temp elltemp Θ
temp Θ
temp
temp
 ((∆X ) )(∆X ) (resp.
 (∆X ) and ∆X
via the surjections ∆Xtemp Θ
temp
temp Θ
temp elltempΠX  (ΠX ) and ΠX  ((ΠX ) )(ΠX ) ) respectively:
∆temp
XO

/ / (∆temp )Θ
XO/ / (∆temp )ell
XO

Πtemp
XO

/ / (Πtemp )Θ
XO/ / (Πtemp )ell
X O

?
∆temp
Y

?
/ / (∆temp )Θ

?
/ / (∆temp )ell ,

Πtemp
Y

?

?
/ / (Πtemp )Θ

?
/ / (Πtemp )ell .

Y

Y

Y

Y

We also have a natural exact sequence
1 → ∆Θ → (∆temp)Θ → (∆temp)ell → 1.
YY∼ Z(1)bb ⊕2 ) is abelian.
Note that (∆temp)ell =and that (∆temp)Θ (∼= Z(1)YY
Let qX ∈ OK be the q-parameter of X. For an integer N ≥ 1, set KN :=1/N
K(µN , qX ) ⊂ K. Any decomposition group of a cusp of Y gives us a section GK →(Πtemp)ell of the natural surjection (Πtemp)ell  GK (Note that the inertia subgroupYY
of cusps are killed in the quotient (−)ell ). This section is well-defined up to conju)ell)ell /N (∆temp)ell  (Πtemp)ell . The composite GKN ,→ GK → (Πtemp
gate by (∆temp
YYYY
is injective by the definition of KN , and the image is stable under the conjugate byΠtemp, since GKN acts trivially on 1 → Z/N Z(1) → (∆temp)ell /N (∆temp)ell → Z → 1
X
X
Y
1/N
(whose extension class is given by qX ), by the definition of KN . Thus, the image
GKN ,→ (Πtemp)ell /N (∆temp)ell determines a Galois covering YN  Y . We have natuYY
ral exact sequences:
1 → Πtemp→ Πtemp→ Gal(YN /Y ) → 1,
YNY
1 → (∆temp)ell ⊗ Z/N Z (∼= Z/N Z(1)) → Gal(YN /Y ) → Gal(KN /K) → 1.
Ytemp elltemp elltempΘΘ
We write (∆temp(resp. (Πtemp(resp.
YN ) , (∆YN )YN ) , (ΠYN ) ) for the image of ∆YN
temp
temp
temp Θ
temp
temp Θ
temp ellΠYN ) via the surjections ∆Y (∆Y ) and ∆Y ((∆Y ) )(∆Y )temp Θ
temp ell
temp
temp
temp Θ ((ΠY ) )(ΠY ) ) respectively:
 (ΠY ) and ΠY(resp. ΠY

∆temp
YO

/ / (∆temp )Θ
Y O/ / (∆temp )ell
Y O

Πtemp
YO

/ / (Πtemp )Θ
Y O/ / (Πtemp )ell
Y O

?
∆temp
YN

?
/ / (∆temp )Θ

?
/ / (∆temp )ell ,

Πtemp
YN

?

?
/ / (Πtemp )Θ

?
/ / (Πtemp )ell .

YN

YN

YN

YN

a proof of the abc conjecture after Mochizuki

129

We also have a natural exact sequence
temp
temp
1 → ∆Θ ⊗ Z/N Z (∼= Z/N Z(1)) → (ΠYN )Θ /N (∆Y )Θ → GKN → 1.

Let YN  Y be the normalisation of Y in YN , i.e., write Y and YN as the formal
scheme and the rigid-analytic space associated to OK -algebra A and K-algebra BN
respectively, and take the normalisation AN of A in BN , then YN = Spf AN . Here, YN
is also an infinite chain of copies of the projective line with N marked points 6= 0, ∞(which we call cusps), joined at 0 and ∞, and each of these points “0” and “∞”is anode in Y. The covering YN  Y is the covering of N -th power map on the each copyof Gm obtained by removing the nodes, and the cusps correspond to “1”, since we takea section GK → (Πtemp)ell corresponding to a cusp in the construction of YN . Note also
Y
that if N is divisible by p, then YN is not a stable model over Spf OKN .
We choose some irreducible component of Y as a “basepoint”, then by the naturalaction of Z = Gal(Y /X) on Y, the projective lines in Y are labelled by elements of Z.
The isomorphism class of a line bundle on YN is completely determined by the degreeof the restriction of the line bundle to each of these copies of the projective line. Thus,
these degrees give us an isomorphism∼

Pic(YN ) → ZZ ,i.e., the abelian group of the functions Z → Z. In the following, we consider Cartierdivisors on YN , i.e., invertible sheaves for the structure sheaf OYN of YN . Note that we
can also consider an irreducible component of YN as a Q-Cartier divisor of YN (cf. also
the proof of [EtTh, Proposition 3.2 (i)]) although it has codimension 0 as underlying
topological space in the formal scheme YN . We write LN for the line bundle on YN
correspoinding to the function Z → Z : a 7→ 1 for any a ∈ Z, i.e., it has degree 1 on any
irreducible component. Note also that we have Γ(YN , OYN ) = OKN . In this section,
we naturally identify a line bundle as a locally free sheaf with a geometric object (i.e.,a (log-)(formal) scheme) defined by it.
Write JN := KN (a1/N | a ∈ KN ) ⊂ K, which is a finite Galois extension of KN ,×× N
since KN/(KN
) is finite. Two splitting of the exact sequence
temp Θ
Θ
1 → ∆Θ ⊗ Z/N Z → (Πtemp) → GKN → 1
YN ) /N (∆Y

determines an element of H 1 (GKN , ∆Θ ⊗Z/N Z). By the definition of JN , the restrictionof this element to GJN is trivial. Thus, the splittings coincide over GJN , and the image
temp ΘΘ
GJN ,→ (Πtemp) is stable under the conjugate by Πtemp. Hence the image
YN ) /N (∆Y
Xtemp Θ
temp Θ
GJN ,→ (ΠYN ) /N (∆Y ) determines a finite Galois covering ZN  YN . We havethe natural exact sequences
temp
1 → Πtemp→ Gal(ZN /YN ) → 1,
ZN → ΠYN

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Go Yamashita

1 → ∆Θ ⊗ Z/N Z → Gal(ZN /YN ) → Gal(JN /KN ) → 1.

(7.1)

temp elltemp elltempΘΘ
We write (∆temp(resp. (Πtemp
ZN ) , (∆ZN )ZN ) , (ΠZN ) ) for the image of ∆ZN (resp.
temp
Θ
Θ
ell
 (∆temp
and ∆temp ((∆temp
)(∆temp
Πtemp
ZN ) via the surjections ∆YN
YN )
YN
YN )
YN )
temp
temp ellΘΘ ((Πtemp(resp. Πtemp
 (Πtemp
YN ) and ΠYNYN ) )(ΠYN ) ) respectively:
YN

∆temp
YN
O

/ / (∆temp )Θ
YN
O/ / (∆temp )ell
YN
O

Πtemp
YN
O

/ / (Πtemp )Θ
YN
O/ / (Πtemp )ell
YN
O

?
∆temp
ZN

?
/ / (∆temp )Θ

?
/ / (∆temp )ell ,

Πtemp
ZN

?

?
/ / (Πtemp )Θ

?
/ / (Πtemp )ell .

ZN

ZN

ZN

ZN

Let ZN  YN be the normalisation of Y in ZN in the same sense as in the definition of
YN . Note that the irreducible components of ZN are not isomorphic to the projective
line in general.
A section s1 ∈ Γ(Y, L1 ) whose zero locus is the cusps is well-defined up to an∼
×OK -multiple, since we have Γ(Y, OY ) = OK . Fix an isomorphism L⊗N→ L1 |YN andN
we identify them. A natural action of Gal(Y /X) (∼= Z) on L1 is uniquely determinedby the condition that it preserves s1 . This induces a natural action of Gal(YN /X) on
L 1 | YN .
Lemma 7.1.

([EtTh, Proposition 1.1])(1) The section s1 |YN ∈ Γ(YN , L1 |YN ) = Γ(YN , L⊗NN ) has an N -th root sN ∈ Γ(ZN , LN |ZN )over ZN .
(2) There is a unique action of Πtemp
on the line bundle LN ⊗OKN OJN over YN ×OKN
X
OJN which is compatible with the section sN : ZN → LN ⊗OKN OJN . Furthermore,this action factors through Πtemp Πtemp/ΠtempX
X
ZN = Gal(ZN /X), and the action of
temp
temp
∆X /∆ZN on LN ⊗OKN OJN is faithful.
Proof. Write (YN )JN := YN ×KN JN , and GN to be the group of automorphismsof LN |(YN )JN which is lying over the JN -automorphisms of (YN )JN induced by elementsof ∆temp/∆temp⊂ Gal(YN /X) and whose N -th tensor power fixes the s1 |(YN )JN . ThenX
YN
by definition, we have a natural exact sequence
1 → µN (JN ) → GN → ∆temp/∆temp→ 1.
X
YN
We claim that∼
HN := ker(GN  ∆temp/∆temp ∆temp/∆temp
= Z)
X
YN
X
Y
is an abelian group killed by N , where the above two surjections are natural ones, andthe kernels are µN (JN ) and (∆temp)ell ⊗ Z/N Z (∼= Z/N Z(1)) respectively. Proof ofXa proof of the abc conjecture after Mochizuki

131

the claim (This immediate follows from the structure of the theta group (=Heisenberg
group); however, we include a proof here): Note that we have a natural commutative
diagram1/ µN (JN )1

/ µN (JN )

1

1


/ HN


/ (∆temp )ell ⊗ Z/N Z (∼= Z/N Z(1))

/1


/ GN


/ ∆temp /∆temp

/1

Y

=


temp
∆temp
/∆
X
Y

X

=

YN


/ ∆temp /∆temp (∼
= Z),
X


1

Y


1

whose rows and columns are exact. Let ζ be a primitive N -th root of unity. The functionb m = Spf OK [[U ]] of
whose restriction to every irreducible component minus nodes G
U −1
YN is equal f (U ) := U−ζ represents an element of H which maps to a generator of
temp
temp
∆Y /∆YN , since it changes the pole divisor from 1 to ζ. Then the claim follows from∏U −ζ NU −1 U −ζ
the identity 0≤j≤N −1 f (ζ −j U ) = U−ζ U −ζ 2 · · · U −ζ N −1 = 1. The claim is shown.
Let RN be the tautological Z/N Z(1)-torsor RN (→ YN obtained by) taking an⊗(−j)
N -th root of s1 , i.e., the finite YN -formal scheme Spf ⊕0≤j≤N −1 LN, where the⊗(−N )

algebra structure is defined by the multiplication LN→ OYN by s1 |YN . Then
GN naturally acts on (RN )JN := RN ×OKN JN by the definition of GN . Since s1 |YN
has zero of order 1 at each cusp, (RN )JN is connected and Galois over XJN := X ×K∼
JN , and GN → Gal((RN )JN /XJN ). Since (i) ∆temp/∆tempacts trivially on µN (JN ),X
YNand (ii) HN is killed by N by the above claim, we have a morphism ZN ×OJN K →RN ×OKN OJN over YN ×OKN OJN by the definitions of ∆Θ
X = ∆X /[∆X , [∆X , ∆X ]]and ZN , i.e., geometrically, ZN ×OJN K( YN ×OKN K) has the universality having
properties (i) and (ii) (Note that the domain of the morphism is ZN ×OJN K, not ZN
since we are considering ∆(−) , not Π(−) ). Since we used the open immersion GJN ,→temp Θ
Θ(Πtemp) , whose image is stable under conjugate by Πtemp, to define theYN ) /N (∆Y
Xmorphism ZN  YN , and s1 |YN is defined over KN , the above morphism ZN ×OJN K →∼RN ×OKN OJN factors through ZN , and induces an isomorphism ZN → RN ×OKN OJN by
considering the degrees over YN ×OKN OJN on both sides (i.e., this isomorphism meansthat the covering determined by ∆Θ ⊗ Z/N Z coincides with the covering determined by
an N -th root of s1 |YN ). This proves the claim (1) of the lemma.

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Go Yamashita

Next, we show the claim (2) of the lemma. We have a unique action of Πtempon
X
LN ⊗OKN OJN over YN ×OKN OJN which is compatible with the section sN : ZN →( Gal(YN /X)) on L1 |YN ∼LN ⊗OKN OJN , since the action of Πtemp
preserves
= L⊗N
X
N
temp
s1 |YN , and the action of ΠX
on YN preserves the isomorphism class of LN . This
temp
temp
action factors through ΠX /ΠZN , since sN is defined over ZN . Finally, the action of/ΠtempΠtemp
X
ZN is faithful since s1 has zeroes of order 1 at the cusps of YN , and the action
temp
on YN is tautologically faithful.
of ∆X /∆temp
YN
We setK̈N := K2N ,J¨N := K̈N (a1/N | a ∈ K̈N ) ⊂ K,

ŸN := Y2N ×OK̈ OJ¨N , ŸN := Y2N ×K̈N J¨N , L̈N := LN |ŸN ∼= L⊗2
2N ×OK̈ OJ¨N .
NN¨ roughly expresses “double covering”. Note that we need to consider(The symbol (−)
double coverings of the rigid analytic spaces under consideration to consider a theta
function below.) Let Z̈N be the composite of the coverings ŸN  YN and ZN  YN ,and Z̈N the normalisation of ZN in Z̈N in the same sense as in the definition of YN .
Write alsoŸ := Ÿ1 = Y2 , Ÿ := Ÿ1 = Y2 , K̈ := K̈1 = J¨1 = K2 .
Since Πtemp
acts compatibly on ŸN and YN , and on LN ⊗OKN OJN , and the naturalX
commutative diagram
/ LN
L̈N

ŸN


/ YN

.
is cartesian, we have a natural action of Πtempon L̈N , which factors through Πtemp/ΠtempX
X
Z̈N
Next, we choose an orientation on the dual graph of the configuration of the irre∼
ducible components of Y. Such an orientation gives us an isomorphism Z → Z. We givea label ∈ Z for each irreducible component of Y. This choice of labels also determines alabel ∈ Z for each irreducible component of YN , ŸN . Recall that we can also consider
the irreducible component (ŸN )j of ŸN labelled j as a Q-Cartier divisor of ŸN (cf.
also the proof of [EtTh, Proposition 3.2 (i)]) although it has codimension 0 as underlying topological space in the formal scheme ŸN (Note that (ŸN )j is Cartier, since the
1/2N
completion of ŸN at each node is isomorphic to Spf OJ¨N [[u, v]]/(uv − qX )). Write∑j 2 /2N
DN := j∈Z j 2 (ŸN )j (i.e., the divisor defined by the summation of “qX= 0” onthe irreducible component labelled j with respect to j ∈ Z). We claim that
(7.2)

OŸN (DN ) ∼= L̈N (∼= L⊗2
2N ⊗OK̈ OJ¨N ).
N

133a proof of the abc conjecture after Mochizuki

Proof of the claim: Since Pic(ŸN ) ∼= ZZ , it suﬃces to show that DN .(ŸN )i = 2 forany i ∈ Z, where we write DN .(ŸN )i for the intersection product of DN and (ŸN )i ,∑i.e., the degree of OŸN (DN )|(ŸN )i . We have 0 = ŸN .(ŸN )i = j∈Z (ŸN )j .(ŸN )i =
2 + ((ŸN )i )2 by the configuration of the irreducible components of ŸN (i.e., an infinite
chain of copies of the projective line joined at 0 and ∞). Thus, we obtain ((ŸN )i )2 = −2.
∑
Then we have DN .(ŸN )i = j∈Z j 2 (ŸN )j .(ŸN )i = (i − 1)2 − 2i2 + (i + 1)2 = 2. This
proves the claim.
By the claim, there exists a section
τN : ŸN → L̈N ,well-defined up to an OJ×¨ -multiple, whose zero locus is equal to DN . We shall refer toNτN as a theta trivialisation. Note that the action of Πtemp
on ŸN , L̈N preserves τNYtemp
×up to an OJ¨ -multiple, since the action of ΠY
on ŸN fixes DN .
N
Let M ≥ 1 be an integer which divides N . Then we have natural morphisms
YN  YM  Y, ŸN  ŸM  Y, ZN  ZM  Y, and natural isomorphisms⊗(N/M )⊗(N/M )
L M | YN ∼, L̈M |YN ∼. By the definition of J¨N (= K2N (a1/N | a ∈
= LN
= L̈N
K2N )), we also have a natural diagram
L̈N
O
τN

ŸN

/ / L̈MO
τM/ / ŸM ,which is commutative up to an OJ×¨ -multiple at L̈N , and an OJ×¨ -multiple at L̈M ,N

M

since τN and τM are defined over Y2N and Y2M respectively (Recall that ŸN :=Y2N ×OK̈ OJ¨N ). By the relation Θ̈(−Ü ) = −Θ̈(Ü ) given in Lemma 7.4 (2), (3) below
N(Note that we have no circular argument here), we can choose τ1 so that the naturalaction of Πtempon L̈1 preserves ±τ1 . In summary, by the definition of J¨N , we have theŸfollowing:• By modifying τN ’s by OJ×¨ -multiples, we can assume that τNNpositive integers N and M such that M | N .

N/M= τM for any• In particular, we have a compatible system of actions of Πtemp
on {ŸN }N ≥1 ,Ÿ{L̈N }N ≥1 which preserve {τN }N ≥1 .
• Each of the above actions of Πtemp
on ŸN , L̈N diﬀers from the action determinedŸ
temp
by the action of ΠX
on YN , LN ⊗OKN OJN in Lemma 7.1 (2) by an element ofµN (J¨N ).

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Go Yamashita

Definition 7.2.
We take τN ’s as above. By taking the diﬀerence of the compatible system of the action of Πtemp
on {ŸN }N ≥1 , {L̈N }N ≥1 in Lemma 7.1 determinedŸby {sN }N ≥1 and the compatible system of the action of Πtemp
on {ŸN }N ≥1 , {L̈N }N ≥1Ÿin the above determined by {τN }N ≥1 (Note also that the former actions, i.e., the one; however, thedetermined by {sN }N ≥1 in Lemma 7.1 come from the actions of Πtemp
X
latter actions, i.e., the one determined by {τN }N ≥1 in the above do not come from theactions of Πtemp), we obtain a cohomology classXη̈ Θ ∈ H 1 (Πtemp, ∆Θ ),
Ÿ
via the isomorphism µN (J¨N ) ∼= Z/N Z(1) ∼= ∆Θ ⊗ Z/N Z (Note that we are currently
studying in a scheme theory here, and that the natural isomorphism µN (J¨N ) ∼= ∆Θ ⊗
Z/N Z comes from the scheme theory (cf. also Remark 3.15.1).
Remark 7.2.1.

(cf. also [EtTh, Proposition 1.3])/Πtemp, ∆Θ ⊗(1) Note that η̈ Θ arises from a cohomology class in limN ≥1 H 1 (Πtemp
Z̈N
Ÿ
←−
Z/N Z), and that the restriction, ∆Θ ⊗ Z/N Z)/∆temp, ∆Θ ⊗ Z/N Z) → lim H 1 (∆temp/Πtemplim H 1 (Πtemp
Z̈N
ŸN
Z̈N
Ÿ
←−
←−
N ≥1

N ≥1

∼
, ∆Θ ⊗ Z/N Z)/∆tempHom(∆temp
= lim
Z̈N
ŸN
←−
N ≥1
∼

sends η̈ Θ to the system of the natural isomorphisms {∆temp/∆temp→ ∆Θ ⊗ŸNZ̈N
Z/N Z}N ≥1 .
×
(2) Note also that s2 : Ÿ → L̈1 is well-defined up to an OK̈-multiple, s2N : Z̈N → L̈N×is an N -th root of s2 , τ1 : Ÿ → L̈1 is well-defined up to an OK̈-multiple, andtempΘ
1τN : ŸN → L̈N is an N -th root of τ1 . Thus, η̈ ∈ H (ΠŸ , ∆Θ ) is well-defined
×up to an OK̈-multiple. Hence the set of cohomology classes×OK̈· η̈ Θ ⊂ H 1 (Πtemp, ∆Θ )Ÿ×, ∆Θ )
is independent of the choices of sN ’s and τN ’s, where OK̈
acts on H 1 (Πtemp
Ÿ
×1
via the composite of the Kummer map OK̈ → H (GK̈ , ∆Θ ) and the natural homo, ∆Θ ). We shall refer to any element in the set
morphism H 1 (GK̈ , ∆Θ ) → H 1 (ΠtempŸ×ΘOK̈ · η̈ as the étale theta class.

§ 7.2.

The Étale Theta Function.

bm ∼Let (G= ) U ⊂ Y be the irreducible component labelled 0 ∈ Z minus nodes. We
take the unique cusp of U as the origin. The group structure of the underlying elliptic

a proof of the abc conjecture after Mochizuki

135

curve X, determines a group structure on U. By the orientation on the dual graph ofthe configuration of the irreducible components of Y, we have a unique isomorphismb m over OK . This gives us a multiplicative coordinate U ∈ Γ(U, O× ). This has a
U∼
=G
U
×
square root Ü ∈ Γ(Ü, OÜ ) on Ü := U ×Y Ÿ (Note that the theta function lives in the
double covering. cf. also Lemma 7.4 below).
We recall the section associated with a tangential basepoint. (cf. also [AbsSect,
Definition 4.1 (iii), and the terminology before Definition 4.1]): For a cusp y ∈ Ÿ (L)with a finite extension L of K̈, let Dy ⊂ ΠŸ be a cuspidal decomposition group of y(which is well-defined up to conjugates). We have an exact sequenceb
1 → Iy ( ∼→ Dy → GL → 1,
= Z(1))and the set Sect(Dy  GL ) of splittings of this short exact sequnece up to conjugates by∼b
Iy is a torsor over H 1 (GL , Z(1))= (L× )∧ by the usual way (the diﬀerence of two sections
gives us a 1-cocycle, and the conjugates by Iy yield 1-coboundaries), where (L× )∧ is the
profinite completion of L. We write ωy for the cotangent space to Ÿ at y. For a non-zeroelement θ ∈ ωy , take a system of N -th roots (N ≥ 1) of any local coordinate t ∈ mŸ ,y1/Nb (∼with dt|y = θ, then, this system gives us a Z(1)))N ≥1  Ÿ |∧= Iy )-torsor (Ÿ |∧
y (t
y
∧ 1/N
∼
b
over the formal completion of Ÿ at y. This Z(1) (= Iy )-covering (Ÿ |y (t))N ≥1  Ÿ |∧ybhence it gives us a section
corresponding to the kernel of a surjection Dy  Iy (∼= Z(1)),of the above short exact sequence. This is called the (conjugacy class of ) section
associated with the tangential basepoint θ. In this manner, the structure group(L× )∧ of the (L× )∧ -torsor Sect(Dy  GL ) is canonically reduced to L× , and the L× torsor obtained in this way is canonically identified with the L× -torsor of the non-zeroelements of ωy . Furthermore, noting also that Ÿ comes from the stable model Ÿ, which
gives us the canonical OL -submodule ωby ( ⊂ ωy ) of ωy , the structure group (L× )∧ of the××(L× )∧ -torsor Sect(Dy  GL ) is canonically reduced to OL, and the OL-torsor obtained×in this way is canonically identified with the OL -torsor of the generators of ωby .
Definition 7.3.
We shall refer to this canonical reduction of the (L× )∧ -torsor×
Sect(Dy  GL ) to the canonical OL-torsor as the canonical integral structure of
Dy , and we say that a section s in Sect(Dy  GL ) is compatible with the canonical×
integral structure of Dy , if s comes from a section of the canonical OL-torsor. We
××shall refer to the L -torsor obtained by the push-out of the canonical OL -torsor via
×
b 0 for the maximal
OL→ L× as the canonical discrete structure of Dy . We write Zb and write (O× )0 := Im(O× → (L× )⊗ Zb 0 ). We shall refer to theprime-to-p quotient of Z,
L
L
× 0
×
×
× 0
(OL
) -torsor obtained by the push-out of the canonical OL-torsor via OL
→ (OL
) as
the canonical tame integral structure of Dy (cf. [AbsSect, Definition 4.1 (ii), (iii)]).
We also shall refer to a reduction of the (L× )∧ -torsor Sect(Dy  GL ) to a {±1}-torsor(resp. µ2l -torsor) as {±1}-structure of Dy (resp. µ2l -structure of Dy ). When a {±1}-136

Go Yamashita

structure (resp. µ2l -structure) of Dy is given, we say that a section s in Sect(Dy  GL )is compatible with the {±1}-structure of Dy , (resp. the µ2l -structure of Dy , if
s comes from a section of the {±1}-torsor (resp. the µ2l -torsor).
Lemma 7.4.

([EtTh, Proposition 1.4]) Write−1Θ̈(Ü ) := qX 8

∑

n

1
2

(n+ 12 )(−1) qX

2

Ü 2n+1 ∈ Γ(Ü, OÜ ).

n∈Z

Note that Θ̈(Ü ) extends uniquely to a meromorphic function on Ÿ (cf. a classical complex theta function
(
(
)2
(
)(
))
∑
1
1 2
1
1∑
1
1
θ1,1 (τ, z) :=exp πiτ n ++ 2πi z +n+=(−1)n q 2 (n+ 2 ) Ü 2n+1 ,
2
2
2
i
n∈Z

n∈Z

2

1
n(n+1)(n+ 12 )−1
where q := e2πiτ , and Ü := eπiz ) and that qX 8 qX2= qX 2 is in K.

(1) Θ̈(Ü ) has zeroes of order 1 at the cusps of Ÿ, and there is no other zeroes. Θ̈(Ü )has poles of order j 2 on the irreducible component labelled j, and there is no otherpoles, i.e., the divisor of poles of Θ̈(Ü ) is equal to D1 .
(2) For a ∈ Z, we haveΘ̈(Ü ) = −Θ̈(Ü −1 ), Θ̈(−Ü ) = −Θ̈(Ü ),
( a )
2
−a
Θ̈ qX2 Ü = (−1)a qX 2 Ü −2a Θ̈(Ü ).
××(3) The classes OK̈· η̈ Θ are precisely the Kummer classes associated to an OK̈-multipleof the regular function Θ̈(Ü ) on the Raynaud generic fiber Ÿ . In particular, for a
non-cuspidal point y ∈ Ÿ (L) with a finite extension L of K̈, the restriction of theclasses×∼b
OK̈· η̈ Θ |y ∈ H 1 (GL , ∆Θ ) ∼= H 1 (GL , Z(1))= (L× )∧
×lies in L× ⊂ (L× )∧ , and are equal to OK̈· Θ̈(y) (Note that we are currently studyingbin a scheme theory here, and that the natural isomorphism ∆Θ ∼comes from= Z(1)

the scheme theory (cf. also Remark 3.15.1).
(4) For a cusp y ∈ Ÿ (L) with a finite extension L of K̈, we have a similar statement
as in (3) by modifying as below: Let Dy ⊂ ΠŸ be a cuspidal decomposition group ofy (which is well-defined up to conjugates). Let s : GL ,→ Dy be a section which iscompatible with the canonical integral structure of Dy . Let s comes from a generatorθb ∈ ω
by . Then the restriction of the classes
×
∼
b
OK̈
· η̈ Θ |s(GL ) ∈ H 1 (GL , ∆Θ ) ∼= H 1 (GL , Z(1))= (L× )∧ ,137

a proof of the abc conjecture after Mochizukis×
via GL ,→ Dy ⊂ Πtemp, lies in L ⊂ (L× )∧ , and are equal to OK̈· dθbΘ̈ (y), whereŸdΘ̈b In particular, the(y) is the value at y of the first derivative of Θ̈(Ü ) at y by θ.
θb

×set of the restriction of the classes OK̈· η̈ Θ |s(GL ) is independent of the choice ofthe generator θb ∈ ωby (hence the choice of the section s which is compatible with thecanonical integral structure of Dy ).
×
We also shall refer to the classes in OK̈· η̈ Θ as the étale theta functions in lightof the above relationship of the values of the theta function and the restrictions of theseclasses to GL via points.

Proof. (2):
Θ̈(Ü

−1

∑

−1
) = qX 8(n+ 21 )

1
2

n

2

(−1) qX

Ü

−2n−1

n∈Z
− 18

= −qX

∑

−1
= qX 8

∑

(−1)

−n−1

1
2

(−n−1+ 21 )qX

2Ü 2n+1n∈Z((−1)n qX
1
2

n+ 12

)

2

Ü 2n+1 = −Θ̈(Ü ),

n∈Z

−1

Θ̈(−Ü ) = qX 8

∑

∑
1
1
1
1
−1
2 (n+ 2 )2 (n+ 2 )(−1)n qX(−Ü )2n+1 = −qX 8(−1)n qXÜ 2n+1 = −Θ̈(Ü ),

n∈Z

n∈Z

2

2

( a )
2
∑
∑
1
1 2
1
a
(n+ 12 ) +a(n+ 12 ) 2n+1
−1
−1
2 (n+ 2 )(−1)n qX
(qX2 Ü )2n+1 = qX 8(−1)n qX2
ÜΘ̈ qX2 Ü = qX 8
n∈Z
− 81

= qX

∑

n∈Z

(
(−1)n qX
12

n+a+ 21

)

2

2
− a2

2

−a

Ü 2n+1 = (−1)a qX 2 Θ̈(Ü ).

n∈Za(1): Firstly, note that qX2 Ü is the canonical coordinate of the irreducible component
labelled a, and that the last equality of (2) gives us the translation formula for changing
the irreducible components. The description of the divisor of poles comes from thistranslation formula and Θ̈(Ü ) ∈ Γ(Ü, OÜ ) (i.e., Θ̈(Ü ) is a regular function on Ü). Next,
by putting Ü = ±1 in the first equality of (1), we obtain Θ̈(±1) = 0. Then by the lastequality of (2) again, it suﬃces to show that Θ̈(Ü ) has simple zeroes at Ü = ±1 on Ü.
By taking modulo the maximal ideal of OK̈ , we have Θ̈(Ü ) ≡ Ü − Ü −1 . This shows theclaim.
×(3) is a consequence of the construction of the classes OK̈· η̈ Θ and (1).
(4): For a generator θb ∈ ωby , the corresponding section s ∈ Sect(Dy  GL ) described before this lemma is as follows: Let t ∈ mŸ,y be a system of N -th roots (N ≥ 1)b then, this system gives us a Z(1)b (∼of any local coordinate with dt|y = θ,= Iy )-torsor
∧ 1/N
∧
∼
b
(Ÿ|y (t))N ≥1  Ÿ|y over the formal completion of Ÿ at y. This Z(1) (= Iy )-covering∧ 1/Nb(Ÿ| (t))N ≥1  Ÿ|∧ corresponding to the kernel of a surjection Dy  Iy (∼= Z(1)),
y

y

138

Go Yamashita

hence a section s ∈ Sect(Dy  GL ). For g ∈ GL , take any lift ge ∈ Dy (Πtemp)Ÿ
1/N
1/N −1of GL , then the above description says that s(g) = (eg (t)/t)N ≥1 · ge, where∼b(eg (t1/N )/t1/N )N ≥1 ∈ Z(1)= Iy (Note that the right-hand side does not depend on
the choice of a lift ge). The Kummer class of Θ̈ := Θ̈(Ü ) is given by Πtemp
3 h 7→
Ÿ
s
temp
1/N
1/N
b
(h(Θ̈
)/Θ̈
)N ≥1 ∈ Z(1).
Hence the restriction to GL via GL → Dy ⊂ Πis givenŸby GL 3 g 7→ ((eg (t1/N )/t1/N )−1 ge(Θ̈1/N )/Θ̈1/N )N ≥1 = (eg ((Θ̈/t)1/N )/(Θ̈/t)1/N )N ≥1 ∈b
Z(1).
Since Θ̈(Ü ) has a simple zero at y, we have (eg ((Θ̈/t)1/N )/(Θ̈/t)1/N )N ≥1 =b 1/N )/(dΘ̈/θ)b 1/N )N ≥1 , where dΘ̈/θb is the first derivative dΘ̈ at y by θ.
b Then(g((dΘ̈/θ)θbb 1/N )/(dΘ̈/θ)b 1/N )N ≥1 ∈ Z(1)bGL 3 g 7→ (g((dΘ̈/θ)is the Kummer class of the valuedΘ̈(y) at y.
θb

If an automorphism ιY of Πtemp
is lying over the action of “−1” on the underlyingYelliptic curve of X which fixes the irreducible component of Y labelled 0, then we shallrefer to ιY as an inversion automorphism of Πtemp.
Y
Lemma 7.5.

([EtTh, Proposition 1.5])(1) Both of the Leray-Serre spectral sequences)Θ , ∆Θ ),)ell , H b (∆Θ , ∆Θ )) =⇒ H a+b ((∆temp
E2a,b = H a ((∆temp
ŸŸE20a,b = H a (GK̈ , H b ((∆temp)Θ , ∆Θ )) =⇒ H a+b ((Πtemp)Θ , ∆Θ )ŸŸ
associated to the filtration of closed subgroups∆Θ ⊂ (∆temp)Θ ⊂ (Πtemp)ΘŸŸdegenerate at E2 , and this determines a filtration 0 ⊂ Fil2 ⊂ Fil1 ⊂ Fil0 =)Θ , ∆Θ ) such that we have)Θ , ∆Θ ) on H 1 ((Πtemp
H 1 ((Πtemp
ŸŸb
Fil0 /Fil1 = Hom(∆Θ , ∆Θ ) = Z,b · log(Ü ),Fil1 /Fil2 = Hom((∆temp)Θ /∆Θ , ∆Θ ) = ZŸ∼∼b
Fil2 = H 1 (GK̈ , ∆Θ ) → H 1 (GK̈ , Z(1))→ (K̈ × )∧ .
∼)ell →
Here, we write log(Ü ) for the standard isomorphism (∆temp)Θ /∆Θ = (∆temp
Ÿ
Ÿ
∼
b
Z(1)
→ ∆Θ (given in a scheme theory).
(2) Any theta class η̈ Θ ∈ H 1 (Πtemp, ∆Θ ) arises from a unique class η̈ Θ ∈ H 1 ((Πtemp)Θ , ∆Θ )Ÿ
Ÿ
(Here, we use the same symbol η̈ Θ by abuse of the notation) which maps to theidentity homomorphism in the quotient Fil0 /Fil1 = Hom(∆Θ , ∆Θ ) (i.e., maps tob = Hom(∆Θ , ∆Θ )). We consider O× · η̈ Θ ⊂ H 1 ((Πtemp )Θ , ∆Θ ) additively, and
1∈Z
K̈
Ÿ

139

a proof of the abc conjecture after Mochizuki

temp
temp
×
×
write η̈ Θ + log(OK̈) for it. Then a ∈ Z ∼acts on η̈ Θ + log(OK̈)= Z = ΠX /ΠY
as
a2
×
×
η̈ Θ + log(OK̈) 7→ η̈ Θ − 2a log(Ü ) −log(qX ) + log(OK̈).
2, we have
In a similar way, for any inversion automorphism ιY of Πtemp
Y××ιY (η̈ Θ + log(OK̈)) = η̈ Θ + log(OK̈)××ιY (log(Ü ) + log(OK̈)) = − log(Ü ) + log(OK̈).

∼ Z(1)∼ Z(1)bbb
and Z(1)has cohomological
Proof. (1): Since ∆Θ =and (∆temp)ell =
Ÿ
dimension 1, the first spectral sequence degenerates at E2 , and this gives us a short
exact sequence
0 → H 1 ((∆temp)ell , ∆Θ ) → H 1 ((∆temp)Θ , ∆Θ ) → H 1 (∆Θ , ∆Θ ) → 0.
ŸŸThis is equal tob · log(Ü ) → H 1 ((∆temp )Θ , ∆Θ ) → Z
b → 0.
0→ZŸ
On the other hand, the second spectral sequence gives us an exact sequnece
0 → H 1 (GK̈ , ∆Θ ) → H 1 ((Πtemp)Θ , ∆Θ ) → H 1 ((∆temp)Θ , ∆Θ )GK̈ → H 2 (GK̈ , ∆Θ ) → 0.
ŸŸThen by Remark 7.2.1 (1), the composite)Θ , ∆Θ )GK̈)Θ , ∆Θ ) → H 1 ((∆temp
H 1 ((ΠtempŸŸb)Θ , ∆Θ ) → H 1 (∆Θ , ∆Θ ) = Z⊂ H 1 ((∆tempŸmaps the Kummer class of Θ̈(Ü ) to 1 (Recall also the definition of ZN and the short
exact sequence (7.1)). Hence the second spectral sequence degenerates at E2 , and wehave the description of the graded quotients of the filtration on H 1 ((Πtemp)Θ , ∆Θ ).
Ÿ(2): The first assertion holds by definition. Next, note that the subgroup (∆temp)ell ⊂
Ÿtemp
bb(∆temp)ell corresponds to the subgroup 2Z(1)⊂ Z(1)×Z ∼= (∆X )ell by the theory ofXbTate curves, where Z(1)⊂ (∆temp)ell corresponds to the system of N (≥ 1)-th roots ofXtemp ell∼b
the canonical coordinate U of the Tate curve associated to X, and 2Z(1)
)
= (∆
Ÿ

corresponds to the system of N (≥ 1)-th roots of the canonical coordinate Ü introduced
before (In this sense, the usage of the symbol log(Ü ) ∈ Hom((∆temp)ell , ∆Θ ) is justiŸfied). Then the description of the action of a ∈ Z ∼= Z follows from the last equalityof Lemma 7.4 (2), and the first description of the action of an inversion automorphism
follows from the first equality of Lemma 7.4 (2). The second description of the actionof an inversion automorphism immediately follows from the definition.
The following proposition says that the étale thete function has an anabelian rigidity, i.e., it is preserved under the changes of scheme theory.

140

Go Yamashita

Proposition 7.6.
(Anabelian Rigidity of the Étale Theta Function, [EtTh, The†
orem 1.6]) Let X (resp. X) be a smooth log-curve of type (1, 1) over a finite extension
K (resp. † K) of Qp such that X (resp. † X) has stable reduction over OK (resp O† K ),and that the special fiber is singular, geometrically irreducible, the node is rational. We
use similar notation for objects associated to † X to the notation which was used for
objects associated to X. Let∼γ : Πtemp→ Πtemp†X
X
be any isomorphism of abstract topological groups. Then we have the following:(1) γ(Πtemp) = Πtemp
.
† Ÿ
Ÿ
∼

(2) γ induces an isomorphism ∆Θ → † ∆Θ , which is compatible with the surjections∼∼bb→ (K̈ × )∧  Z
H 1 (GK̈ , ∆Θ ) → H 1 (GK̈ , Z(1))∼∼
bbH 1 (G† , † ∆Θ ) → H 1 (G† , Z(1))→ († K̈ × )∧  Z
K̈K̈determined the valuations on K̈ and † K̈ respectively. In other words, γ induces∼
an isomorphism H 1 (GK̈ , ∆Θ ) → H 1 (G† K̈ , † ∆Θ ) which preserves both the kernel ofb in the quotients.
these surjections and the element 1 ∈ Z
temp
× Θ, ∆Θ ) ∼·η̈(3) The isomorphism γ ∗ : H 1 (Πtemp= H 1 (Π† Ÿ , † ∆Θ ) induced by γ sends OK̈
Ÿ
temp
temp
×
†
†
Θ
to some Z ∼= Π† X /Π† Y -conjugate of O† K̈ · η̈ (This indeterminacy of † Zconjugate inevitably arises from the choice of the irreducible component labelled 0).

Remark 7.6.1.
([EtTh, Remark 1.10.3 (i)]) The étale theta function lives in a
. However, when)Θ , not whole of Πtemp
cohomology group of the theta quotient (Πtemp
XX
we study anabelian properties of the étale theta function as in Proposition 7.6, the theta
quotient (Πtemp)Θ is insuﬃcient, and we need whole of Πtemp.
X
X
Remark 7.6.2.
([IUTchIII, Remark 2.1.2]) Related with Remark 7.6.1, then, howpartial temp
about considering ΠX:= ΠX ×Zb Z instead of Πtemp? (Here, we write ΠX for
X
b is the profinite completion of the natthe profinite fundamental group, and ΠX  Zural surjection Πtemp Z.) The answer is that it does not work in inter-universalX
temp
temp ∼ b
Teichmüller theory since we have NΠX (Πpartial)/Πpartial→ Z/Z (On the other
X
X
temp
temp
hand, NΠX (ΠX ) = ΠX by Cororally 6.10 (2)). The profinite conjugacy indetermitempbnacy on Πpartial
gives rise to Z-translationindeterminacies on the coordinates of theXevaluation points (cf. Definition 10.17). On the other hand, for Πtemp, we can reduceXbindeterminacies to Z-translation indeterminacies by Theorem 6.11 (cf.
the Z-translation
also Lemma 11.9).
Remark 7.6.3.
The statements in Proposition 7.6 are bi-anabelian ones (cf. Remark 3.4.4). However, we can reconstruct the † Z-conjugate class of the theta classes

141

a proof of the abc conjecture after Mochizuki

O†×K̈ · † η̈ Θ in Proposition 7.6 (3) in a mono-anabelian manner, by considering the descriptions of the zero-divisor and the pole-divisor of the theta function.
Proof. (1): Firstly, γ sends ∆temp
to ∆temp, by Lemma 6.2. Next, note that γ
†X
X
temp
temp
to ∆† Y by the discreteness (which is a group-theoretic property) of Z andsends ∆Y†
Z. Finally, γ sends the cuspidal decomposition groups to the cuspidal decompositiongroups by Corollary 6.12. Hence γ sends ΠŸ to Π† Ÿ , since the double coverings Ÿ  Y
bm  Gbmand † Ÿ  † Y are the double covering characterised as the 2-power map [2] : G
on each irreducible component, where the origin of the target is given by the cusps.
(2): We proved that γ(∆temp) = † ∆temp. Then γ(∆Θ ) = † ∆Θ holds, since ∆ΘXX(resp. † ∆Θ ) is group-theoretically defined from ∆temp(resp. † ∆temp). The rest of theX
Xclaim follows from Corollary 6.12 and Proposition 2.1 (5), (6).
∼(3): After taking some Πtemp/Πtemp= Z-conjugate, we may assume that γ :XYtemp ∼temp
ΠŸ→ Π† Ÿ is compatible with suitable inversion automorphisms ιY and † ιY by Theorem B.1 (cf. [SemiAnbd, Theorem 6.8 (ii)], [AbsSect, Theorem 2.3]). Next, note that γb = Hom(∆Θ , ∆Θ ) = Fil0 /Fil1 to 1 ∈ Z
b = Hom(† ∆Θ , † ∆Θ ) =tautologically sends 1 ∈ Z†
b = Hom(∆Θ , ∆Θ ) =Fil0 /† Fil1 . On the other hand, η̈ Θ (resp. † η̈ Θ ) is sent to 1 ∈ Z
0
10
1
b = Hom(† ∆Θ , † ∆Θ ) = † Fil /† Fil ), and fixed by ιY (resp. † ιY )Fil /Fil (resp. 1 ∈ Z
×up to an OK̈-multiple (resp. an O†×K̈ -multiple) by Lemma 7.5 (2). This determinesη̈ Θ (resp. † η̈ Θ ) up to a (K̈ × )∧ -multiple (resp. a († K̈ × )∧ -multiple). Hence it is suf×
ficient to reduce this (K̈ × )∧ -indeterminacy (resp. († K̈ × )∧ -indeterminacy) to an OK̈
×Θ
indeterminacy (resp. an O† K̈ -indeterminacy). This is done by evaluating the class η̈(resp. † η̈ Θ ) at a cusp y of the irreducible component labelled 0 (Note that “labelled 0”
is group-theoretically characterised as “fixed by inversion isomorphism ιY (resp. † ιY )”),
if we show that γ preserves the canonical integral structure of Dy .
(cf. also [SemiAnbd, Corollary 6.11] and [AbsSect, Theorem 4.10, Corollary 4.11] forthe rest of the proof). To show the preservation of the canonical integral structure of Dy
by γ, we may restrict the fundamental group of the irreducible component labelled 0 by
Proposition 6.6 and Corollary 6.12 (cf. also Remark 6.12.1). The irreducible componentb m with marked points (=cusps) {±1} ⊂ G
b m . Then theminus nodes Ü is isomorphic to Gof the geometric fudamental group of the generic fiber isprime-to-p-quotient ∆prime-to-p
UK̈

isomorphic to the prime-to-p-quotient ∆prime-to-pof the one of the special fiber, whereUk̈

we write k̈ for the residue field of K̈. This shows that the reduction of the structure× 0×b 0 ), which isgroup of (K̈ × )∧ -torsor Sect(Dy  GK̈ ) to (OK̈) := Im(OK̈→ K̈ × ⊗ Z
determined the canonical integral strucure (i.e., the canonical tame integral structure),is group-theoretically preserved as follows (cf. [AbsSect, Proposition 4.4 (i)]): The outerprime-to-p
action GK̈ → Out(∆U) canonically factors through Gk̈ → Out(∆prime-to-p), andUK̈

K̈

(prime-to-p)

the geometrically prime-to-p-quotient ΠUk̈of the arithmetic fundamental group

142

Go Yamashita
outof the special fiber is group-theoretically constructed as ∆prime-to-po Gk̈ by using
U
K̈

Gk̈ → Out(∆prime-to-p). Then the decomposition group Dy0 in the geometrically primeUK̈
to-p-quotient of the arithmetic fundamental group of the integral model fits in a short
b 0 → D0 → G → 1, where Iy is an inertia subgroup at y.
exact sequence 1 → (Iy0 :=)Iy ⊗Zyk̈
The set of the splitting of this short exact sequence forms a torsor over H 1 (Gk̈ , Iy0 ) ∼= k̈ × .
These splittings can be regarded as elements of H 1 (Dy0 , Iy0 ) whose restriction to Iy0 is
equal to the identity element in H 1 (Iy0 , Iy0 ) = Hom(Iy0 , Iy0 ). Thus, the pull-back to Dy of× 0
any such element of H 1 (Dy0 , Iy0 ) gives us the reduction of the structure group to (OK̈)
determined by the canonical integral structure.
Then it suﬃces to show that the reduction of the structure group of (K̈ × )∧ -torsor
Sect(Dy  GK̈ ) to K̈ × , which is determined the canonical integral strucure (i.e., thecanonical discrete structure), is group-theoretically preserved since the restriction of thebZb 0 to Z ⊂ Z
b is injective (cf. [AbsSect, Proposition 4.4 (ii)]).
projection Z
Finally, we show that the canonical discrete structure of (K̈ × )∧ -torsor Sect(Dy GK̈ ) is group-theoretically preserved. Let Ü be the canonical cooridnate of Gm K̈ . For
y = ±1, we consider the unit Ü ∓ 1 ∈ Γ(Gm K̈ \ {±1}, OGm K̈ \{±1} ), which is invertible
at 0, fails to be invertible at y, and has a zero of order 1 at y. We consider the exact
sequenceb⊕Zb
1 → (K̈ × )∧ → H 1 (ΠP1 \{0,y} , µb (ΠX )) → Z
Z

constructed in Lemma 3.15 (5). The image of the Kummer class κ(T ∓1) ∈ H 1 (ΠP1 \{0,y} , µZb (ΠX ))b⊕Z
b (i.e., (1, 0)) determines the set (K̈ × )∧ · κ(Ü ∓ 1). The restriction of (K̈ × )∧ ·in Z
κ(Ü ∓ 1) to Dy is the (K̈ × )∧ -torsor Sect(Dy  GK̈ ), since the zero of order ofκ(Ü ∓ 1) at y is 1. On the other hand, κ(Ü ∓ 1) is invertible at 0. Thus, the subset K̈ × · κ(Ü ∓ 1) ⊂ (K̈ × )∧ · κ(Ü ∓ 1) is characterised as the set of elements of(K̈ × )∧ · κ(Ü ∓ 1) whose restriction to the decomposition group D0 at 0 (which liesin (K̈ × )∧ ∼= H 1 (GK̈ , µZb (ΠX )) ⊂ H 1 (D0 , µZb (ΠX )) since κ(Ü ∓ 1) is invertible at 0) infact lies in K̈ × ⊂ (K̈ × )∧ . Thus, we are done by Corollary 6.12 (or Corollary 2.9) (cf.
the proof of [AbsSect, the proof of Theorem 4.10 (i)]).
From now on, we assume that
(1) K̈ = K,
(2) the hyperbolic curve X minus the marked points admits a K-core X  C :=X//{±1}, where the quotient is taken in the sense of stacks, by the natural actionof {±1} determined by the multiplication-by-2 map of the underlying elliptic curveof X (Note that this excludes four exceptional j-invariants by Lemma C.3, and(3)√−1 ∈ K.

a proof of the abc conjecture after Mochizuki

143

We write Ẍ  X for the Galois covering of degree 4 determined by the multiplicationZ
rig Z
by-2 map of the underlying elliptic curve of X (i.e., Grigm /qX  Gm /qX sending therig
2
coordinate U of the Grigm in the codomain to Ü , where Ü is the coordinate of the Gm
in the domain). We write Ẍ  X for its natural integral model. Note that Ẍ  C is
Galois with Gal(Ẍ/C) ∼= (Z/2Z)⊕3 .
√Choose a square root −1 ∈ K of −1. Note that the 4-torsion points of the√ i√ junderlying elliptic curve of Ẍ are Ü = −1 qX 4 ⊂ K for 0 ≤ i, j ≤ 3, and that,√in the irreducible components of Ẍ, the 4-torsion points avoiding nodes are ± −1.
√
We write τ for the 4-torsion point determined by −1 ∈ K. For an étale theta classη̈ Θ ∈ H 1 (Πtemp, ∆Θ ), we writeŸη̈ Θ,Z×µ2 ⊂ H 1 (Πtemp, ∆Θ )
Ÿ
∼
for the Πtemp/Πtemp= Z × µ2 -orbit of η̈ Θ .
XŸDefinition 7.7.

(cf. [EtTh, Definition 1.9])(1) We shall refer to each of two sets of values of η̈ Θ,Z×µ2η̈ Θ,Z×µ2 |τ , η̈ Θ,Z×µ2 |τ −1 ⊂ K ×
as a standard set of values of η̈ Θ,Z×µ2 .
(2) There are two values in K × of maximal valuations of some standard set of values
a √
a2 √
√
a − 2
Θ,Z×µ22(Note that Θ̈(qX −1) = (−1) qX ( −1)−2a Θ̈( −1) by the third
of η̈
a √
a √
equality of Lemma 7.4 (2), and Θ̈(−qX2 −1) = −Θ̈(qX2 −1) by the second equalityof Lemma 7.4 (2)). If they are equal to ±1, then we say that η̈ Θ,Z×µ2 is of standard
type.
Remark 7.7.1.
Double coverings Ẋ  X and Ċ  C are introduced in [EtTh],
and they are used to formulate the definitions of a standard set of values and an étaletheta class of standard type, ([EtTh, Definition 1.9]), the definition of log-orbicurve oftype (1, Z/lZ), (1, (Z/lZ)Θ ), (1, Z/lZ)± , (1, (Z/lZ)Θ )± ([EtTh, Definition 2.5]), and theconstant multiple rigidity of the étale theta function ([EtTh, Theorem 1.10]). However, we avoid them in this survey, since they are not directly used in inter-universal
Teichmüller theory, and it is enough to formulate the above things by modifying in asuitable manner.
Lemma 7.8.
(cf. [EtTh, Proposition 1.8]) Let C = X//{±1} (resp. † C =†X//{±1}) be a smooth log-orbicurve over a finite extension K (resp. † K) of Qp such√√that −1 ∈ K (resp. −1 ∈ † K). We use the notation † (−) for the associated objects∼
with † C. Let γ : Πtemp→ Πtempbe an isomorphism of topological groups. Then γ†CC∼∼temp ∼
temp
induces isomorphisms ΠX → Π† X , Πtemp→ Πtemp, and Πtemp→ Πtemp
.
† Ẍ
† Ÿ
Ẍ
Ÿ

144

Go Yamashita

Proof. (cf. also the proof of Proposition 7.6 (1)). By Lemma 6.2, the isomor∼
phism γ induces an isomorphism γ∆C : ∆temp→ ∆temp. Since ∆temp⊂ ∆temp(resp.
†C
C
X
C
temp
temp
∆† X ⊂ ∆† C ) is characterised as the open subgroup of index 2 whose profinite
completion is torsion-free i.e., corresponds to the geometric fundamental group of ascheme, not a non-scheme-like stack (cf. also [AbsTopI, Lemma 4.1 (iv)]), γ∆C in∼. Then γ∆X induces an isomorphism→ ∆temp
duces an isomorphism γ∆X : ∆temp†X
X
temp elltemp ell ∼temp ell)ell ) is group-theoreticallyγ∆ell: (∆X ) → (∆† X ) , since (∆X ) (resp. (∆temp†X
X
constructed from ∆temp(resp. ∆temp). By the discreteness of Gal(Y /X) ∼= Z (resp.
†X
X
temp
†
†
†
induces an isomorphism γZ : ∆X /∆temp(∼
Gal( Y / X) ∼== Z), the isomorphism γ∆ell
Y
X
∼
temp
temp ∼ †tempZ) → ∆† X /∆† Y (= Z). Thus, by considering the kernel of the action of ΠC(resp. Πtemp) on ∆temp/∆temp(resp. ∆temp/∆temp), the isomorphisms γ and γZ in†C
†X
†Y
X
Y
temp ∼
temp
duce an isomorphism γΠX : ΠX
→ Π† X . Since γΠX preserves the cuspical de∼composition groups by Corollary 6.12, it induces isomorphisms Πtemp→ Πtemp
, and
† Ẍ
Ẍ
temp
temp ∼→ Π† Ÿ .
ΠŸProposition 7.9.
(Constant Multiple Rigidity of the Étale Theta Function, cf.
[EtTh, Theorem 1.10]) Let C = X//{±1} (resp. † C = † X//{±1}) be a smooth log√√
orbicurve over a finite extension K (resp. † K) of Qp such that −1 ∈ K (resp. −1 ∈†K). We assume that C is a K-core. We use the notation † (−) for the associated objects∼
with † C. Let γ : Πtemp→ Πtempbe an isomorphism of topological groups. Note that the†CC∼
isomorphism γ induces an isomorphism Πtemp→ Πtempby Lemma 7.8. Assume that
†X
X
temp
1
Θ,Z×µ2⊂ H (ΠŸ , ∆Θ ) to the subset † η̈ Θ,Z×µ2 ⊂ H 1 (Πtemp, † ∆Θ )γ maps the subset η̈† Ÿ(cf. Proposition 7.6 (3)). Then we have the following:(1) The isomorphism γ preserves the property that η̈ Θ,Z×µ2 is of standard type, i.e.,η̈ Θ,Z×µ2 is of standard type if and only if † η̈ Θ,Z×µ2 is of standard type. This property
uniquely determines this collection of classes.
∼(2) Note that γ induces an isomorphism K × → † K × , where K × (resp. † K × ) is retempgarded a subset of (K × )∧ ∼= H 1 (GK , ∆Θ ) ⊂ H 1 (ΠC , ∆Θ )) (resp. († K × )∧ ∼=temp †
1†
1
H (G† K , ∆Θ ) ⊂ H (Π† C , ∆Θ ))). Then γ maps the standard sets of values ofη̈ Θ,Z×µ2 to the standard sets of values of † η̈ Θ,Z×µ2 .
(3) Assume that η̈ Θ,Z×µ2 (hence † η̈ Θ,Z×µ2 as well by the claim (1)) is of standard type,
and that the residue characteristic of K (hence † K as well) is > 2. Then η̈ Θ,Z×µ2(resp. † η̈ Θ,Z×µ2 ) determines a {±1}-structure (cf. Definition 7.3) on (K × )∧ -torsor(resp. († K × )∧ -torsor) at the unique cusp of C (resp. † C) which is compatible withthe canonical integral structure, and it is preserved by γ.
Remark 7.9.1.
The statements in Proposition 7.9 are bi-anabelian ones (cf. Remark 3.4.4). However, we can reconstruct the set † η̈ Θ,Z×µ2 in Proposition 7.9 (2) anda proof of the abc conjecture after Mochizuki

145(3) in a mono-anabelian manner, by a similar way as Remark 7.6.3.
Proof. The claims (1) and (3) follows from the claim (2). We show the claim(2). Since γ induces an isomorphism from the dual graph of Ÿ to the dual graph of†Ÿ (Proposition 6.6), by the elliptic cuspidalisation (Theorem 3.7), the isomorphism
γ maps the decomposition group of the points of Ÿ lying over τ to the decompositiongroup of the points of † Ÿ lying over τ ±1 . The claim (2) follows from this.
§ 7.3.

l-th Root of the Étale Theta Function.

First, we introduce some log-curves, which are related with l-th root of the étaletheta function. Let X be a smooth log-curve of type (1, 1) over a field K of characteristic 0 (As before, we always write the log-structure associated to the cusp on X,
and consider the log-fundamental group). Note also that we are working in a field of
characteristic 0, not in a finite extension of Qp as in the previous subsections.
Assumption (0): We assume that X admits K-core.
We have a short exact sequence 1 → ∆X → ΠX → GK → 1, where ΠX and ∆X
are the arithmetic fundamental group and the geometric fundamental group (with reab
spect to some basepoints) respectively, and GK = Gal(K/K). Write ∆ell
X := ∆X =2 ell
Θ
∆X /[∆X , ∆X ], ∆ΘX := ∆X /[∆X , [∆X , ∆X ]], and ∆Θ := Im{∧ ∆X → ∆X }. ThenellΘwe have a natural exact sequence 1 → ∆Θ → ∆Θ
X → ∆X → 1. Write also ΠX :=ΠX /ker(∆X  ∆ΘX ).
Let l > 2 be a prime number. Note that the subgroup of ∆ΘX generated by l-thΘpowers of elements of ∆X is normal (Here we use l 6= 2). We write ∆ΘX  ∆X forellΘ
the quotient of ∆X by this normal subgroup. Write ∆Θ := Im{∆Θ → ∆X }, ∆X :=ell∆X /∆Θ , ΠX := ΠX /ker(∆X  ∆X ), and ΠX := ΠX /∆Θ . Note that ∆Θ ∼= (Z/lZ)(1)elland ∆X is a free Z/lZ-module of rank 2.
Let x be the unique cusp of X, and we write Ix ⊂ Dx for the inertia subgroupand the decomposition subgroup at x respectively. Then we have a natural injective
homomorphism Dx ,→ ΠΘX such that the restriction to Ix gives us an isomorphism∼ΘIx → ∆Θ (⊂ ΠX ). Write also Dx := Im{Dx → ΠX }. Then we have a short exact
sequence
1 → ∆Θ → Dx → GK → 1.
ellAssumption (1): We choose a quotient ΠX  Q onto a free Z/lZ-module of rank 1 such
ell

ell

that the restriction ∆X → Q to ∆X remains surjective, and the restriction Dx → Q to
Dx is trivial.

146

Go Yamashita

We writeXX
for the corresponding covering (Note that every cusp of X is K-rational, since therestriction Dx → Q to Dx is trivial) with Gal(X/X) ∼= Q, and we write ΠX ⊂ ΠX ,
ell

ell

∆X ⊂ ∆X , and ∆X ⊂ ∆X for the corresponding open subgroups. We write ιX (resp.
ιX ) for the automorphism of X (resp. X) given by the multiplication by −1 on theunderlying elliptic curve, where the origin is given by the unique cusp of X (resp. a
choice of a cusp of X). Write C := X//ιX , C := X//ιX (Here, //’s mean the quotientsin the sense of stacks). We shall refer to a cusp of C, which arises from the zero (resp.
a non-zero) element of Q, as the zero cusp (resp. a non-zero cusp) of C. We shallrefer to ιX and ιX as inversion automorphisms. We also shall refer to the uniquecusp of X over the zero cusp of C as the zero cusp of X. This X (resp. C) is themain actor for the global additive () portion (resp. global multiplicative () portion)
in inter-universal Teichmüller theory.
Definition 7.10.
([EtTh, Definition 2.1]) A smooth log-orbicurve over K iscalled of type (1, l-tors) (resp. of type (1, l-tors)± ) if it is isomorphic to X (resp.
ellC) for some choice of ΠX  Q (satisfying Assumption (0), (1)).
Note that X → X is Galois with Gal(X/X) ∼= Q; however, C → C is not Galois,
since ιX acts on Q by the multiplication by −1, and any generator of Gal(X/X) does
not descend to an automorphism of C over C (Here we use l 6= 2. cf. [EtTh, Remark
2.1.1]). We write ∆C ⊂ ΠC (resp. ∆C ⊂ ΠC ) for the geometric fundamental groupand the arithmetic fundamental group of C (resp. C) respectively. Write also ΠC :=ΠC /ker(ΠX  ΠX ), (resp. ΠC := ΠC /ker(ΠX  ΠX ),) ∆C := ∆C /ker(∆X  ∆X ),
ell

ell

(resp. ∆C := ∆C /ker(∆X  ∆X ),), and ∆C := ∆C /ker(∆X  ∆X ).
Assumption (2): We choose ιX ∈ ∆C an element which lifts the nontrivial element ofGal(X/C) ∼= Z/2Z.
We consider the conjugate action of ιX on ∆X , which is a free Z/lZ-module of rank 2.
ellThen the eigenspace of ∆X with eigenvalue −1 (resp. +1) is equal to ∆X (resp. ∆Θ ).
Hence we obtain a direct product decompositionell∆X ∼= ∆X × ∆Θ([EtTh, Proposition 2.2 (i)]) which is compatible with the conjugate action of ΠX (since
the conjugate action of ιX commutes with the conjugate action of ΠX ). We write

147

a proof of the abc conjecture after Mochizuki
ell

ell

sι : ∆X ,→ ∆X for the splitting of ∆X  ∆X given by the above direct productdecomposition. Then the normal subgroup Im(sι ) ⊂ ΠX induces an isomorphism∼

Dx → ΠX /Im(sι )over GK .
Assumption (3): We choose any element sA(3) of the H 1 (GK , ∆Θ )(∼= K × /(K × )l )-torsor
Sect(Dx  GK ), where we write Sect(Dx  GK ) for the set of sections of the surjection
Dx  GK .
∼Then we obtain a quotient ΠX  ΠX  ΠX /Im(sι ) → Dx  Dx /sA(3) (GK ) ∼= ∆Θ .
This quotient gives us a covering
XX
with Gal(X/X) ∼= ∆Θ . We write ∆X ⊂ ∆X , ΠX ⊂ ΠX for the open subgroups
elldetermined by X. Note that the composition ∆X ,→ ∆X  ∆X is an isomorphism,
and that ∆X = Im(sι ), ∆X = ∆X · ∆Θ . Since Gal(X/X) = ∆X /∆X = ∆Θ , andIx ∼= ∆Θ  ∆Θ , the covering X  X is totally ramified at the cusps (Note also that theirreducible components of the special fiber of the stable model of X are isomorprhic to
P1 ; however, the irreducible components of the special fiber of the stable model of X are
not isomorphic to P1 ). Note also that the image of ιX in ∆C /∆X is characterised as theunique coset of ∆C /∆X which lifts the nontrivial element of ∆C /∆X and normalises
the subgroup ∆X ⊂ ∆C , since the eigenspace of ∆X /∆X ∼= ∆Θ with eigenvalue 1 isequal to ∆Θ ([EtTh, Proposition 2.2 (ii)]). We omit the construction of “C” (cf. [EtTh,
Proposition 2.2 (iii)]), since we do not use it. This X plays the central role in thetheory of mono-theta environment, and it also plays the central role in inter-universal
Teichmüller theory for places in Vbad .
Definition 7.11.
([EtTh, Definition 2.3]) A smooth log-orbicurve over K isΘcalled of type (1, l-tors ) if it is isomorphic to X (which is constructed under Assumptions (0), (1), (2), and (3)).
The underlines in the notation of X and C indicate “extracting a copy of Z/lZ”,
and the double underlines in the notation of X and C indicate “extracting two copy of
Z/lZ” ([EtTh, Remark 2.3.1]).
Lemma 7.12.
(cf. [EtTh, Proposition 2.4]) Let X (resp. † X) be a smooth logcurve of type (1, l-torsΘ ) over a finite extension K (resp. † K) of Qp . We use thenotation † (−) for the associated objects with † X. Assume that X (resp. † X) has stable reduction over OK (resp. O† K ) whose special fiber is singular and geometrically148

Go Yamashita∼

irreducible, and the node is rational. Let γ : Πtemp→ Πtempbe an isomorphism of
†X
X
∼

∼

topological groups. Then γ induces isomorphisms Πtemp→ Πtemp, Πtemp→ Πtemp
†C
†C
C
C
∼

∼

∼

Πtemp
→ Πtemp, Πtemp, and Πtemp→ Πtemp→ Πtemp
.
†X
†X
† Ÿ
X
X
Ÿ
∼

Proof. By Lemma 6.2, γ induces an isomorphism ∆temp→ ∆temp. By the K†X
X∼, which induces ancoricity, the isomorphism γ induces an isomorphism Πtemp→ Πtemp†CC
temp ∼
temp
isomorphism ∆C→ ∆† C . Then by the same way as in Lemma 7.8, this induces∼∼
temp ∼. Note that ∆X→ Πtempisomorphisms ∆X→ ∆temp, Πtemp→ Πtemp, and Πtemp
†X
†X
† Ÿ
X
Ÿ
†
(resp.
(resp. ∆† X ) and ∆Θ (resp. ∆Θ ) are group-theoretically constructed from ∆temp
X
temp
temp
∆† X ), and that we can group-theoretically reconstruct ∆X ⊂ ∆X
(resp. ∆† X ⊂∆temp) by the image of ∆temp(resp. ∆temp). Hence the above isomorphisms induce
†X
†X
X
∼

an isomorphism ∆X → ∆† X , since ∆X = ∆X · ∆Θ (resp. ∆† X = ∆† X · † ∆Θ ). This∼

isomorphism induces an isomorphism ∆temp→ ∆temp, since ∆temp(reps. ∆temp
) is
†X
†X
X
X
the inverse image of ∆X ⊂ ∆temp) under the natural quotient(resp. ∆† X ⊂ ∆temp
†X
X
∼
temp
temp
→ ∆temp∆X
 ∆X (resp. ∆† X  ∆† X ). The isomorphism ∆temp
induces†XX∼→ Πtemp, since Πtemp(resp. Πtemp) is reconstructed as thean isomprhism Πtemp
†X
†X
X
X
out

out

outer semi-direct product ∆X o GK (resp. ∆† X o G† K ), where the homomorphism
GK → Out(∆X ) (resp. G† K → Out(∆† X )) is given by the above constructions inducedby the action of GK (resp. G† K ).
Remark 7.12.1.
obtain([EtTh, Remark 2.6.1]) Suppose µl ⊂ K. By Lemma 7.12, we

AutK (X) = µl × {±1},

AutK (X) = Z/lZ o {±1},

AutK (C) = {1},where o is given by the natural multiplicative action of {±1} on Z/lZ (Note that C → C
is not Galois, as already remarked after Definition 7.10 (cf. [EtTh, Remark 2.1.1])).
Now, we return to the situation where K is a finite extension of Qp .
Definition 7.13.
([EtTh, Definition 2.5]) Assume that the residue characteristic of K is odd, and that K = K̈. We also make the following two assumptions:ellAssumption (4): We assume that the quotient ΠX  Q factors through the naturalb determined by the quotient Πtemp  Z discussed when we defined Y .
quotient ΠX  Z
XAssumption (5): We assume that the choice of an element of Sect(Dx  GK ) in Assumption (3) is compatible with the {±1}-structure (cf. Definition 7.3) of Proposition 7.9
(3).

149

a proof of the abc conjecture after Mochizuki

A smooth log-orbicurve over K is called of type (1, Z/lZ) (resp. of type (1, (Z/lZ)Θ ),resp. of type (1, Z/lZ)± ), if it is isomorphic to X (resp. X, resp. C) (which is constructed under the Assumptions (0), (1), (2), (3), (4), and (5)).
Note also that the definitions of smooth log-(orbi)curves of type (1, l-tors), of type
(1, l-tors)± , and of type (1, l-torsΘ ) are made over any field of characteristic 0, and thatthe definitions of smooth log-(orbi)curves of type (1, Z/lZ), of type (1, Z/lZ)± and oftype (1, (Z/lZ)Θ ) are made only over finite extensions of Qp .
Let Y  X (resp. Ÿ  X) be the composite of the covering Y  X (resp.
Ÿ  X) with X  X. Note that the coverings Ÿ  Ÿ and Y  Y are of degree l.
We have the following diagramiiiiµ2 iiiii
ii
iiii
i
i
i
tiii ∆Θ (∼=Z/lZ)
/

Y

Ÿ

Y
Z

lZ


X

LLL
L∆LLΘ (∼
=Z/lZ)
LLL
L&
µ2
o
Ÿ

∆Θ (∼
=Z/lZ)/X

Q(∼=Z/lZ)/

ext of Z/2Z

o

by µ2


Ẍ

{±1}

{±1}


C


X

2Z

non-Galois
/
deg=l


C

,

and note that the irreducible components and cusps in the special fibers of X, Ẍ, X,
X, Y , Ÿ , Y , and Ÿ are described as follows (Note that X  X and Y  Y are totally
ramified at each cusp):• X: 1 irreducible component (whose noramalisation ∼= P1 ) and 1 cusp on it.
• Ẍ: 2 irreducible components (∼= P1 ) and 2 cusps on each,• X: l irreducible components (∼= P1 ) and 1 cusp on each,• X: l irreducible components (6∼= P1 ) and 1 cusp on each,• Y : the irreducible components (∼= P1 ) are parametrised by Z, and 1 cusp on each,

150

Go Yamashita• Ÿ : the irreducible components (∼= P1 ) are parametrised by Z, and 2 cusps on each,• Y : the irreducible components (6∼= P1 ) are parametrised by lZ, and 1 cusp on each,• Ÿ : the irreducible components (6∼= P1 ) are parametrised by lZ, and 2 cusps on each.

We have introduced the needed log-curves. Now, we consider the étale theta
functions. By Assumption (4), the covering Ÿ  X factors through X. Hence×the class η̈ Θ ∈ H 1 (Πtemp, ∆Θ ), which is well-defined up to an OK-multiple, and its
Ÿ
temp
temp ∼tempΠX /ΠŸ= Z × µ2 -orbit can be regarded as objects associated to ΠX .
We recall that the element η̈ Θ ∈ H 1 (Πtemp, ∆Θ ⊗ Z/lZ) arises froma an elementŸtemp ΘΘ1η̈ ∈ H ((ΠŸ ) , ∆Θ ⊗ Z/lZ) by the first claim of Lemma 7.5 (2), where we use thesame symbol η̈ Θ by abuse of notation. The natural map Dx → Πtemp→ (Πtemp)ΘŸŸ)Θ , ∆Θ ⊗ Z/lZ) → H 1 (Dx , ∆Θ ⊗ Z/lZ), and theinduces a homomorphism H 1 ((Πtemp
Ÿ
image of η̈ Θ ∈ H 1 ((Πtemp)Θ , ∆Θ ⊗ Z/lZ) in H 1 (Dx , ∆Θ ⊗ Z/lZ) comes from an elementŸη̈ Θ ∈ H 1 (Dx , ∆Θ ⊗Z/lZ), where we use the same symbol η̈ Θ by abuse of notation again,
via the natural map H 1 (Dx , ∆Θ ⊗ Z/lZ) → H 1 (Dx , ∆Θ ⊗ Z/lZ), since we have an exact
sequence
0 → H 1 (Dx , ∆Θ ⊗ Z/lZ) → H 1 (Dx , ∆Θ ⊗ Z/lZ) → H 1 (l∆Θ , ∆Θ ⊗ Z/lZ),
and the image of η̈ Θ in H 1 (l∆Θ , ∆Θ ⊗ Z/lZ) = Hom(l∆Θ , ∆Θ ⊗ Z/lZ) vanishes by
the first claim of Lemma 7.5 (2). On the other hand, for any element s ∈ Sect(Dx GK ), the map Dx 3 g 7→ g(s(g))−1 gives us a 1-cocycle, hence a cohomology class in
H 1 (Dx , ∆Θ ⊗Z/lZ), where we write g for the image of g via the natural map Dx  GK .
In this way, we obtain a map Sect(Dx  GK ) → H 1 (Dx , ∆Θ ⊗ Z/lZ). (cf. the following
diagram:0/ H 1 (Dx , ∆Θ ⊗ Z/lZ)O/ H 1 (Dx , ∆Θ ⊗ Z/lZ)
O

Sect(Dx  GK ))Θ , ∆Θ ⊗ Z/lZ),
H 1 ((Πtemp
Ÿ/ Hom(l∆Θ , ∆Θ ⊗ Z/lZ)

where the horizontal sequence is exact.) We also have a natural exact sequence
0 → H 1 (GK , ∆Θ ⊗ Z/lZ) → H 1 (Dx , ∆Θ ⊗ Z/lZ) → H 1 (∆Θ ⊗ Z/lZ, ∆Θ ⊗ Z/lZ).
The image of η̈ Θ ∈ H 1 (Dx , ∆Θ ⊗ Z/lZ) in H 1 (∆Θ ⊗ Z/lZ, ∆Θ ⊗ Z/lZ) = Hom(∆Θ ⊗Z/lZ, ∆Θ ⊗ Z/lZ) is the identity homomorphism by the first claim of Lemma 7.5 (2)151a proof of the abc conjecture after Mochizuki

again. The image Im(s) ∈ H 1 (Dx , ∆Θ ⊗ Z/lZ) of any element s ∈ Sect(Dx  GK ) via
the above map Sect(Dx  GK ) → H 1 (Dx , ∆Θ ⊗Z/lZ) in H 1 (∆Θ ⊗Z/lZ, ∆Θ ⊗Z/lZ) =Hom(∆Θ ⊗Z/lZ, ∆Θ ⊗Z/lZ) is also the identity homomorphism by the calculation ∆Θ ⊗Z/lZ 3 g 7→ g(s(g))−1 = g(s(1))−1 = g · 1−1 = g. Hence any element in Im{Sect(Dx GK ) → H 1 (Dx , ∆Θ ⊗Z/lZ)} diﬀers from η̈ Θ ∈ H 1 (Dx , ∆Θ ⊗Z/lZ) by an H 1 (GK , ∆Θ ⊗Z/lZ) ∼
= K × /(K × )l -mutiple. Now, we consider the element sA(3) ∈ Sect(Dx  GK )which is chosen in Assumption (3), and we write Im(sA(3) ) ∈ H 1 (Dx , ∆Θ ⊗ Z/lZ)
for its image in H 1 (Dx , ∆Θ ⊗ Z/lZ). By the above discussions, we can modify η̈ Θ ∈H 1 (Dx , ∆Θ ⊗ Z/lZ) by a K × -multiple, which is well-defined up to a (K × )l -multiple,
to make it coincide with Im(sA(3) ) ∈ H 1 (Dx , ∆Θ ⊗ Z/lZ). Note that stronger claim
×also holds, i.e., we can modify η̈ Θ by an OK-multiple, which is well-defined up to an× l(OK ) -multiple, to make it coincide with Im(sA(3) ), since sA(3) ∈ Sect(Dx  GK ), is
compatible with the canonical integral structure of Dx by Assumption (5) (Note thatnow we do not assume that η̈ Θ,Z×µ2 is of standard type; however, the assumption thatsA(3) is compatible with the {±1}-structure in the case where η̈ Θ,Z×µ2 is of standard
type implies that sA(3) is compatible with the canonical integral structure of Dx even
we do not assume that η̈ Θ,Z×µ2 is of standard type). As a conclusion, by modifying
×)Θ , ∆Θ ⊗ Z/lZ) by an OK-multiple, which is well-defined up to anη̈ Θ ∈ H 1 ((ΠtempŸ× l(OK ) -multiple, we can and we shall assume that η̈ Θ = Im(sA(3) ) ∈ H 1 (Dx , ∆Θ ⊗Z/lZ),, ∆Θ ⊗ Z/lZ), which is well-defined up to an
and we obtain an element η̈ Θ ∈ H 1 (ΠtempŸ× l×(OK ) -multiple (not an OK -multiple), i.e., by the choice of X, the indeterminacy on
the ratio of sl and τl in the definition of η̈ Θ disappeared. In the above construction,
an element Sect(Dx  GK ) can be considered as “modulo l tangential basepoint” at
the cusp x, the theta function Θ̈ has a simple zero at the cusps (i.e., it is a uniformiser
at the cusps), and we made choices in such a way that η̈ Θ = Im(sA(3) ) holds. Hencethe covering X  X can be regarded as a covering of “taking a l-th root of the theta
function”.
Note that we have the following diagram
H 1 (sA(3) (GK ), ∆Θ ⊗ Z/lZ)O
H 1 (Dx , ∆Θ ⊗ Z/lZ)
O0/ H 1 (Dx /sA(3) (GK ), ∆Θ ⊗ Z/lZ)
O0,/ H 1 (Πtemp , ∆Θ ⊗ Z/lZ)Ÿ/ H 1 (Πtemp , ∆Θ ⊗ Z/lZ)
Ÿ

152

Go Yamashita

where the horizontal sequence and the vertical sequence are exact. Now, the image ofη̈ Θ = Im(sA(3) ) ∈ H 1 (Dx , ∆Θ ⊗Z/lZ) in H 1 (sA(3) (GK ), ∆Θ ⊗Z/lZ) vanishes by the calculation sA(3) (GK ) 3 sA(3) (g) 7→ sA(3) (g)(sA(3) (sA(3) (g)))−1 = sA(3) (g)(sA(3) (g))−1 = 1
and the above vertical sequence. Thus, η̈ Θ = Im(sA(3) ) comes from an element of
H 1 (Dx /sA(3) (GK ), ∆Θ ⊗ Z/lZ). Therefore, the image of η̈ Θ ∈ H 1 (Πtemp, ∆Θ ⊗ Z/lZ) in
Ÿ
temp
1
1
H (ΠŸ , ∆Θ ⊗Z/lZ) vanishes since it arises from the element of H (Dx /sA(3) (GK ), ∆Θ ⊗Z/lZ) and the above horizontal sequence. As a conclusion, the image of η̈ Θ ∈ H 1 (Πtemp, ∆Θ )
Ÿ
temp
temp
Θ
1
1
in H (ΠŸ , ∆Θ ) arises from an element η̈ ∈ H (ΠŸ , l∆Θ ), which is well-defined
×up to OK
. In some sense, η̈ Θ can be considered as an “l-th root of the étale theta
function”. We write η̈ Θ,lZ×µ2 for the Πtemp /Πtemp ∼= (lZ × µ2 )-orbits of η̈ Θ .
X

Definition 7.14.

Ÿ([EtTh, Definition 2.7]) We shall refer to η̈ Θ,lZ×µ2 as of stan-

dard type, if η̈ Θ,Z×µ2 is of standard type.
By combining Proposition 7.9 Lemma 7.12, and definitions, we obtain the following:
Corollary 7.15.
(Constant Multiple Rigidity of l-th Roots of the Étale Theta
Function, cf. [EtTh, Corollary 2.8]) Let X (resp. † X) be a smooth log-curve of type
(1, (Z/lZ)Θ ) over a finite extension K (resp. † K) of Qp . We use the notation † (−) for∼→ Πtempbe an isomorphism of topologicalthe associated objects with † X. Let γ : Πtemp†X
X
groups.
(1) The isomorphism γ preserves the property that η̈ Θ,lZ×µ2 is of standard type. Moreover, this property determines this collection of classes up to a µl -multiple.
(2) Assume that the cusps of X are rational over K, the residue characteristic of K
is prime to l, and that µl ⊂ K. Then the {±1}-structure of Proposition 7.9 (3)determinesa µ2l -structure (cf. Definition 7.3) at the decomposition groups of thecusps of X. Moreover, this µ2l -structure is compatible with the canonical integral
structure (cf. Definition 7.3) at the decomposition groups of the cusps of X, and ispreserved by γ.
Remark 7.15.1.
The statements in Corollary 7.15 are bi-anabelian ones (cf. Remark 3.4.4). However, we can reconstruct the set η̈ Θ,lZ×µ2 in Corollary 7.15 (1) in a
mono-anabelian manner, by a similar way as Remark 7.6.3 and Remark 7.9.1.
Lemma 7.16.
([EtTh, Corollary 2.9]) Assume that µl ⊂ K. We make a labelling on the cusps of X, which is induced by the labelling of the irreducible components
of Y by Z. Then this determines a bijection{}
Cusps of X /AutK (X) ∼= |Fl |

a proof of the abc conjecture after Mochizuki

153∼(cf. Section 0.2 for |Fl |), and this bijection is preserved by any isomorphism γ : Πtemp
→
X
Πtemp
of topological groups.
X
Proof. The first claim is trivial (cf. also Remark 7.12.1). The second claim follows
from Remark 6.12.1.
§ 7.4.

Three Fundamental Rigidities of Mono-theta Environments.

In this subsection, we introduce the notion of mono-theta environment, and show
important three rigidities of mono-theta environment, that is, the constant multiple
rigidity, the cyclotomic rigidity, and the discrete rigidity.
Definition 7.17.

For an integer N ≥ 1, we put
ΠµN ,K := µN o GK .

For a topological group Π with a surjective continuous homomorphism Π  GK , weput
Π[µN ] := Π ×GK ΠµN ,K , ∆[µN ] := ker (Π[µN ]  GK ) = ∆ × µN ,where ∆ := ker(Π  GK ), and we shall refer to Π[µN ] as cyclotomic envelope ofΠ  GK . We also writeµN (Π[µN ]) := ker(Π[µN ]  Π).
and we shall refer to µN (Π[µN ]) as the (mod N ) cyclotome of the cyclotomic
envelope Π[µN ]. Note that we have a tautological section GK → ΠµN ,K of ΠµN ,K 
GK , and that it determines a sectionsalgΠ : Π → Π[µN ],and we shall refer to it as a mod N tautological section. For any object with
Π[µN ]-conjugate action, we shall refer to a µN -orbit as a µN -conjugacy class.
Here, the µN in Π[µN ] plays a roll of “µN ” which comes from line bundles.
Lemma 7.18.
([EtTh, Proposition 2.11]) Let Π  GK (resp. † Π  G† K ) bean open subgroup of the tempered or profinite fundamental group of hyperbolic orbicurve
over a finite extension K (resp. † K) of Qp , and write ∆ := ker(Π  GK ) (resp.
†∆ := ker(† Π  G† K )).
(1) The kernel of the natural surjection ∆[µN ]  ∆ (resp. † ∆[µN ]  † ∆) is equal to the∼center of ∆[µN ] (resp. † ∆[µN ]). In particular, any isomorphism ∆[µN ] → † ∆[µN ]is compatible with the surjections ∆[µN ]  ∆, † ∆[µN ]  † ∆.

154

Go Yamashita(2) The kernel of the natural surjection Π[µN ]  Π (resp. † Π[µN ]  † Π) is equal to theunion of the center of the open subgroups of Π[µN ] (resp. † Π[µN ]). In particular,∼
any isomorphism Π[µN ] → † Π[µN ] is compatible with the surjections Π[µN ]  Π,†Π[µN ]  † Π.
Proof. Lemma follows from the temp-slimness (Lemma 6.4 (5)) or the slimness(Proposition 2.7 (2a), (2b)) of ∆, † ∆, Π, † Π.
Proposition 7.19.

([EtTh, Proposition 2.12])(1) We have an inclusion()[]
temp elltemp Θ
temp Θ
Θker (∆temp
)

(∆
)
=
l∆
⊂
(∆
)
,
(∆
)
.
Θ
X
X
X
X
(2) We have an equality[(∆temp)Θ [µN ], (∆temp)Θ [µN ]
X
X

]∩

(
)
alg
temp Θ
(l∆Θ )[µN ] = Im s(∆temp )Θ l∆Θ : l∆Θ → (∆X ) [µN ]X)(temp Θ
⊂ (l∆Θ )[µN ] ⊂ (∆X ) [µN ] ,where we write salg
for the restriction of the mod N tautological section(∆temp )Θ l∆ΘX: (∆temp)Θ → (∆temp)Θ [µN ] to l∆Θ (⊂ (∆temp
)Θ ).
salg
X
X
X
(∆temp)Θ
X
Proof. The inclusion of (1) follows from the structure of the theta group (=Heisen)Θ . The equality of (2) follows from (1).
berg group) (∆temp
X
Remark 7.19.1.
((cf. [EtTh,) Remark2.12.1]) As a conclusion of Proposition 7.19

the subgroup Im salg
, – i.e., the splitting l∆Θ × µN –, can be group(∆temp )Θ l∆ΘXtheoretically reconstructed, and the cyclotomic rigidity of mono-theta environment (cf.
Theorem 7.23 (1)), which plays an important role in inter-universal Teichmüller theory,
comes from this fact. Note that
7.19[(1) does not hold if we( the inclusion of Proposition)]
temp Θ
temp elltemp ΘΘ= ∆Θ 6⊂ (∆temp
)
,
(∆
)
.
use X instead of X, i.e., ker (∆X )  (∆X )X
X
We write salg
for the composite
Ÿ
salg
temp
Π
Ÿ

salg
: Πtemp−→ Πtemp[µN ] ,→ Πtemp[µN ],YŸŸŸ→ l∆Θ ⊗
and we shall refer to it as a mod N algebraic section. Let η : Πtemp
Ÿ
Z/N Z ∼= µN denote the composite of the reduction modulo N of any element (i.e., a

155a proof of the abc conjecture after Mochizuki

1-cocycle) of the collection of classes η̈ Θ,lZ×µ2 ⊂ H 1 (Πtemp, l∆Θ ), and the isomorphismŸl∆Θ ⊗ Z/N Z ∼= µN , which comes from a scheme theory (cf. Remark 3.15.1). We putsΘ:= η −1 · salg
: Πtemp→ Πtemp[µN ].
YŸŸŸWe shall refer to sΘ
as a mod N theta section. Note that sΘ
is a homomorphism,
Ÿ
Ÿ
since
sΘ
(gh) = η(gh)−1 salg(gh) = (g(η(h))η(g))−1 salg(g)salg
(h)
Ÿ
Ÿ
Ÿ
Ÿ
= (salg(g)η(h)salg(g)−1 η(g))−1 salg(g)salg
(h)
Ÿ
Ÿ
Ÿ
Ÿ
= η(g)−1 salg(g)η(h)−1 salg(h) = sΘ
(g)sΘ
(g).
Ÿ
Ÿ
Ÿ
Ÿ
Note also that the natural outer action
temp
temp ∼ temp
temp
temp
Gal(Y /X) ∼= ΠX /ΠY= ΠX [µN ]/ΠY [µN ] ,→ Out(ΠY [µN ])of Gal(Y /X) on Πtemp[µN ] fixes Im(salg: Πtemp→ Πtemp[µN ]) up to a conjugate by µN ,
Y
Y
Ÿ
Ÿ
since the mod N algebraic section salg
extends to a mod N tautological section salg
:
Πtemp
Ÿ
X

Πtemp
→ Πtemp[µN ]. Hence sΘup to Πtemp[µN ]-conjugates is independent of the choiceXXXŸ
of an element of η̈ Θ,lZ×µ2 ⊂ H 1 (Πtemp, l∆Θ ) (Recall that Πtemp
 Gal(Ÿ /X) ∼= lZ ×XŸµ2 ). Note also that conjugates by µN corresponds to modifying a 1-cocycle by 1coboundaries.
Note that we have a natural outer action∼

K ×  K × /(K × )N → H 1 (GK , µN ) ,→ H 1 (Πtemp, µN ) → Out(Πtemp[µN ]),
Y
Y
where the isomorphism is the Kummer map, and the last homomorphism is given bysending a 1-cocycle s to an outer homomorphism salg(g)a 7→ s(g)salg(g)a (g ∈
Πtemp
Πtemp
Y

Y

Πtemp
, a ∈ µN ) (Note that the last homomorphism is well-defined, since salg(g)asalg(g 0 )a0 (=
Y
Πtemp
Πtemp
Y

Y

0 −1
salg
(gg 0 )salg(a)a0 ) for g, g 0 ∈ Πtemp, a, a0 ∈ µN is sent totemp (g )
Y
Πtemp
Π
Y
Y

s(gg 0 )salg(gg 0 )salg(g 0 )−1 (a)a0 = g(s(g 0 ))s(g)salg(gg 0 )salg(g 0 )−1 asalg(g 0 )a0
Πtemp
Πtemp
Πtemp
Πtemp
Πtemp
Y

Y

Y

Y

Y

alg
alg
0 0
0
0 0
= s(g)g(s(g 0 ))salg(g)asalg
temp (g )a = s(g)s temp (g)s(g )as temp (g )a
Πtemp
Π
Π
ΠY
Y
Y
Y

by s, and since for a 1-coboundary s(g) = b−1 g(b) (b ∈ µN ) is sent tosalg(g)a 7→ s(g)salg(g)a = b−1 g(b)salg(g)a = b−1 salg(g)bsalg(g)−1 salg
(g)a
Πtemp
Πtemp
Πtemp
Πtemp
Πtemp
Πtemp
Y

Y

= b−1 salg(g)ba = b−1 salg(g)ab,
Πtemp
Πtemp
Y
Y

Y

Y

Y

Y

156

Go Yamashita

which is an inner automorphism). Note also any element Im(K × ) := Im(K × →Out(Πtemp[µN ])) lifts to an element of Aut(Πtemp[µN ]) which induces the identity auY
Ytomorphisms of both the quotient Πtemp[µN ]  Πtemp
and the kernel of this quotient.
Y
Y×In this natural outer action of K × , an OK-multiple on η̈ Θ,lZ×µ2 corresponds to an
×OK-conjugate of sΘ
.
Ÿ

Definition 7.20.

(Mono-theta Environment, [EtTh, Definition 2.13]) We write

DY := hIm(K × ), Gal(Y /X)i ⊂ Out(Πtemp[µN ])Y[µN ]) generated by Im(K × ) and Gal(Y /X) (∼for the subgroup of Out(Πtemp= lZ).
Y(1) We shall refer to the following collection of data as a mod N model mono-theta
environment:• the topological group Πtemp[µN ],Y• the subgroup DY (⊂ Out(Πtemp[µN ])), andY• the µN -conjugacy class of subgroups in Πtemp[µN ] determined by the image ofY
the theta section sΘ
.
Ÿ(2) We shall refer to any collection M = (Π, DΠ , sΘΠ ) of the following data as a mod N
mono-theta environment:• a topological group Π,• a subgroup DΠ (⊂ Out(Π)), and• a collection of subgroups sΘΠ of Π,∼such that there exists an isomorphism Π → Πtemp[µN ] of topological groups whichYmaps DΠ ⊂ Out(Π) to DY , and sΘΠ to the µN -conjugacy class of subgroups intempΠY [µN ] determined by the image of the theta section sΘ
.
Ÿ
†
†
Θ
(3) For two mod N mono-theta environments M = (Π, DΠ , sΘΠ ), M = ( Π, D† Π , s† Π ),∼ †we define an isomorphism of mod N mono-theta environments M → M∼
to be an isomorphism of topological groups Π → † Π which maps DΠ to D† Π ,Θ
and sΘ
Π to s† Π . For a mod N mono-theta environment M and a mod M monotheta environment † M with M | N , we define a homomorphism of mono-theta∼
environments M → † M to be an isomorphism MM → † M, where we write MM
for the mod M mono-theta environment induced by M.

157

a proof of the abc conjecture after Mochizuki

Remark 7.20.1. We can also consider a mod N bi-theta environment B =algΘ(Π, DΠ , sΘΠ , sΠ ), which is a mod N mono-theta environment (Π, DΠ , sΠ ) with a datumsalg
Π corresponding to the µN -conjugacy class of the image of mod N algebraic sectionalgsŸ (cf. [EtTh, Definition 2.13 (iii)]). As shown below in Theorem 7.23, three important rigidities (the cyclotomic reigidity, the discrete rigidity, and the constant multiple
rigidity) hold for mono-theta environments. On the other hand, the cyclotomic rigidity,
and the constant multiple rigidity trivially holds for bi-theta environments; however, thediscrete rigidity does not hold for them (cf. also Remark 7.23.1). We omit the detailsof bi-theta environments, since we will not use bi-theta environments in inter-universal
Teichmüller theory.
Lemma 7.21.

([EtTh, Proposition 2.14])(1) We have the following group-theoretic chracterisation of the image of the tautologicalsection of (l∆Θ )[µN ]  l∆Θ as the following subgroup of (∆temp)Θ [µN ]:Ÿ(l∆Θ )[µN ]∩{)Θ [µN ]γ(a)a−1 ∈ (∆temp
Y

}
temp
Θ
)
[µ
])
such
that
(∗)
,
a ∈ (∆temp
[µ
],
γ
∈
Aut(Π
N
N
Y
Y

where
(∗) : the image of γ in Out(Πtemp[µN ]) belongs to DY ,Yand γ induces the identity on the quotient Πtemp[µN ]  Πtemp
 GK .
Y
Y(2) Let tΘ: Πtemp→ Πtemp[µN ] be a section obtained as a conjugate of sΘ
relative to the
Y
Ÿ
Ÿ
Ÿ
actions of K × and lZ. Write δ := (sΘ)−1 tΘ
, which is a 1-cocycle of Πtemp
valued inŸŸŸµN . We write α̈δ ∈ Aut(Πtemp[µN ]) for the automorphism given by salg(g)a 7→
Πtemp
Ÿ
Ÿ

δ(g)salg(g)a (g ∈ Πtemp, a ∈ µN ), which induces the identity homomorphisms
Πtemp
Ÿ
Ÿ

on both the quotient Πtemp[µN ]  Πtemp
and the kernel of this quotient. ThenŸŸα̈δ extends to an automorphism αδ ∈ Aut(Πtemp[µN ]), which induces the identityYhomomorphisms on both the quotient Πtemp[µN ]  Πtemp
and the kernel of thisYYquotient. The conjugate by αδ maps sΘ
to tΘ, and preserves the subgroup DY ⊂ŸŸOut(Πtemp[µN ]).
Y(3) Let M = (Πtemp[µN ], DY , sΘ
) be the mod N model mono-theta environment. ThenY
Ÿevery automorphism of M induces an automorphism of Πtemp
by Lemma 7.18 (2),Youthence an automorphism of Πtemp= Aut(Πtemp) o Im(DY → Out(Πtemp
)) =
X
Y
Y

158

Go Yamashita

Aut(Πtemp) ×Out(Πtemp ) Im(DY → Out(Πtemp)). It also induces an automorphismYYYof the set of cusps of Y . Relative to the labelling by Z on these cusps, this induces
an automorphism of Z given by (lZ) o {±1}. This assignment gives us a surjective
homomorphism
Aut(M)  (lZ) o {±1}.
Proof. (1): Let γ ∈ Aut((Πtemp)[µN ]) be a lift of an element in Im(K × ) ⊂ DY (⊂Y
Out((Πtemp)[µN ])) such that γ satisfies (*). Then γ can be written as γ = γ1 γ2 , whereY
γ1 ∈ Inn(Πtemp[µN ]), γ2 ∈ Aut(Πtemp[µN ]), the image of γ2 in Out(Πtemp[µN ]) is inYYY
Im{K × → H 1 (GK , µN ) → H 1 (Πtemp, µN ) → Out(Πtemp[µN ])}, and the automorphismYY
induced by γ2 of the quotient Πtemp[µN ]  Πtemp
and the automorphism of its kernel (=YYµN ) are trivial. Since the composite H 1 (GK , µN ) → H 1 (Πtemp, µN ) → H 1 (∆temp, µN )YY
is trivial, the composite H 1 (GK , µN ) → H 1 (Πtemp, µN ) → H 1 (∆temp, µN ) → Out(∆temp[µN ])YY
Y
is trivial as well. Hence the automorphism induced by γ2 of ∆temp[µN ] is an inner auY
tomorphism. On the other hand, the automorphism induced by γ1 of GK is trivial
since the automorphism induced by γ2 of GK is trivial, and the condition (*). Thenthe center-freeness of GK (cf. Proposition 2.7 (1c)) implies that γ1 ∈ Inn(Πtemp[µN ])Y[µN ]). Hence the automorphism induced by γ = γ1 γ2 of ∆temp[µN ] isis in Inn(∆temp
YY
temp Θ
b
also an inner automorphism. Since (∆) [µN ](∼
× µN ) is abelian, the= lZ × Z(1)Y

inner automorphism induced by γ of (∆temp)Θ [µN ] is trivial. Then (1) follows fromY
Proposition 7.19 (2).
(2): By definition, the conjugate by α̈δ maps sΘto tΘ. Since the outer action of
Ÿ
Ÿ
temp
alg
Gal(Y /X) ∼= lZ on ∆Y [µN ] fixes sŸ up to µN -conjugacy, the cohomology class ofδ in H 1 (Πtemp, µN ) is in the submodule generated by the Kummer classes of K × andŸ(1/l)2l log(Ü ) = 2 log(Ü ) by the first displayed formula of Lemma 7.5 (2) (cf. Lemma 7.5(1) for the cohomology class log(Ü )). Here, note that the cohomology class of δ is in
Fil1 since both (salg)−1 · sΘand salg· tΘmaps to 1 in Fil0 /Fil1 = Hom(l∆Θ , l∆Θ ) by
Ÿ
Ÿ
Ÿ
Ÿ
Lemma 7.5 (2). Note also that “1/l” comes from that we are working with l-th rootsof the theta functions η̈ Θ,lZ×µ2 (cf. the proof of Lemma 7.5 (2)), and that “l” comesfrom lZ. Thus, δ descends to a 1-cocycle of Πtemp
valued in µN since the coordinate
YÜ 2 descends to Y . Hence α̈δ extends to an automorphism αδ ∈ Aut(Πtemp[µN ]), whichYinduces identity automorphisms on both the quotient Πtemp[µN ] → Πtemp
and the kernelYYof this quotient. The conjugate by αδ preserves DY ⊂ Out(Πtemp[µN ]), since the action
Yof Gal(Y /X) maps 2 log(Ü ) to a K × -multiple of 2 log(Ü ).
(3) comes from (2).

a proof of the abc conjecture after Mochizuki

159

Corollary 7.22.
(Group-Theoretic Reconstruction of Mono-theta Environment,[EtTh, Corollary 2.18]) Let N ≥ 1 be an integer, l a prime number and X a smoothlog-curve of type (1, (Z/lZ)Θ ) over a finite extension K of Qp . We assume that l and p
are odd, and K = K̈. Let MN be the resulting mod N model mono-theta environment,
which is independent of the choice of a member of η̈ Θ,lZ×µ2 , up to isomorphism over theidentity of Πtemp
by Lemma 7.21 (2).
Y
(1) Let † Πtemp
be a topological group which is isomorphic to Πtemp. Then there exists
XX
a group-theoretic algorithm for constructing• subquotients†Πtemp, † Πtemp, † GK , † (l∆Θ ), † (∆temp)Θ , † (Πtemp)Θ , † (∆temp)Θ , † (Πtemp
)Θ
Y
X
X
Y
Y
Ÿ

of † Πtemp, andX• a collection of subgroups of † Πtemp
for each element of (Z/lZ)/{±1},X∼such that any isomorphism † Πtemp→ Πtempmaps
XX• the above subquotients to the subquotientsΠtemp, Πtemp, GK , l∆Θ , (∆temp)Θ , (Πtemp)Θ , (∆temp)Θ , (Πtemp
)Θ
Y
X
X
Y
Y
Ÿ
of Πtemp
respectively, andX• the above collection of subgroups to the collection of cuspidal decompositiongroups of Πtemp
determined by the label in (Z/lZ)/{±1},Xin a functorial manner with respect to isomorphisms of topological groups (and noneed of any reference isomorphism to Πtemp).
X(2) “(Π 7→ M)”:
There exists a group-theoretic algorithm for constructing a mod N mono-theta environment † M = († Π, D† Π , sΘ† Π ), where†()Π := † Πtemp
×† GK († (l∆Θ ) ⊗ Z/N Z) o † GKY

up to isomorphism in a functorial manner with respect to isomorphisms of topolog). (cf. also [EtTh,
ical groups (and no need of any reference isomorphism to Πtemp
X
Corollary 2.18 (ii)] for a stronger form).
(3) “(M 7→ Π)”:Let † M = († Π, D† Π , sΘ† Π ) be a mod N mono-theta environment which is isomorphic

160

Go Yamashita

to MN . Then there exists a group-theoretic algorithm for constructing a quotient∼†Π  † Πtemp, such that any isomorphism † M → MN maps this quotient to theYquotient Πtemp[µN ]  Πtemp
in a functorial manner with respect to isomorphismsYYof mono-theta environments (and no need of any reference isomorphism to MN ).
∼Furthermore, any isomorphism † M → MN induces an isomorphism from†Πtemp:= Aut(† Πtemp) ×Out(† Πtemp ) Im(D† Π → Out(† Πtemp
))
X
Y
Y
Y

to Πtemp, where we set the topology of † Πtemp
as the topology determined by taking
X
X
†

∼

Πtemp
→ Aut(† Πtemp) ×Out(† Πtemp ) {1} ⊂ † Πtemp
YYXY

to be an open subgroup. Finally, if we apply the algorithm of (2) to † Πtemp, thenY
the resulting mono-theta environment is isomorphic to the original † M, via an isomorphism which induces the identity on † Πtemp.
Y‡
M = (‡ Π, D‡ Π , sΘ(4) Let † M = († Π, D† Π , sΘ‡ Π ) be mod N mono-theta environ† Π ), and† temp‡ tempments. Let ΠXand ΠXbe the topological groups constructed in (3) from†M and ‡ M respectively. Then the functoriality of the algorithm in (3) gives us anatural map
IsomµN -conj († M, ‡ M) → Isom(† Πtemp, ‡ Πtemp),
X
X

which is surjective with fibers of cardinality 1 (resp. 2) if N is odd (resp. even),
where we write IsomµN -conj for the set of µN -conjugacy classes of isomorphisms.
In particular, for any positive integer M with M | N , we have a natural homomorphism AutµN -conj († M) → AutµM -conj († MM ), where we write † MM for themod M mono-theta environment induced by † M such that the kernel and cokernel have the same cardinality (≤ 2) as the kernel and cokernel of the homomorphism Hom(Z/2Z, Z/N Z) → Hom(Z/2Z, Z/M Z) induced by the natural surjection
Z/N Z  Z/M Z, respectively.
 † GK by
Proof. (1): We can group-theoretically reconstruct a quotient † Πtemp
X
Lemma 6.2, other subquotients by Lemma 7.8, Lemma 7.12 and the definitions, and thelabels of cuspidal decomposition groups by Lemma 7.16.
(2) follows from the definitions (Note that we can reconstruct the set † η̈ Θ,lZ×µ2 oftheta classes by Remark 7.15.1, thus, the theta section sΘ† Π as well (cf. the construction
Θof the theta section sŸ before Definition 7.20)).
(3): We can group-theoretically reconstruct a quotient † Π  † Πtemp
by Lemma 7.18
Y(2). The reconstruction of † Πtemp
comes from the definitions and the temp-slimness ofXa proof of the abc conjecture after Mochizuki

161

†

Πtemp
(Lemma 6.4 (5)). The last claim of (3) follows from the definitions and theXdescription of the algorithm in (2).
(4): The surjectivity of the map comes from the last claim of (3). The fiber ofthis map is a ker(AutµN -conj († M) → Aut(† Πtemp))-torsor. By Theorem 7.23 (1) below
X∼(Note that there is no circular argument), the natural isomorphism † (l∆Θ ) ⊗ Z/N Z →µN († (l∆Θ [µN ])) is preserved by automorphisms of † M. Note that ker(AutµN -conj († M) →Aut(† Πtemp)) consists of automorphisms acting as the identity on † Πtemp, hence onXYker(† Π → † Πtemp) by the above natural isomorphism. Thus, we have
Y
temp
temp
temp
)) ∼
ker(AutµN -conj († M) → Aut(† Πtemp= Hom(† ΠY /† ΠŸ , ker(† Π → † ΠY )),X
temp∼
where † Πtemp/† Πtemp= µ2 and ker(† Π → † ΠY )) ∼= µN . The cardinality of this groupYŸ

is 1 (resp. 2) is N is odd (resp. even). The last claim follows from this description.
Theorem 7.23.
(Three Rigidities of Mono-theta Environments, [EtTh, Corollary 2.19]) Let N ≥ 1 be an integer, l a prime number and X a smooth log-curve oftype (1, (Z/lZ)Θ ) over a finite extension K of Qp . We assume that l and p are odd,
and K = K̈. Let MN be the resulting mod N model mono-theta environment (which is
independent of the choice of a member of η̈ Θ,lZ×µ2 , up to isomorphism over the identityof Πtemp
by Lemma 7.21 (2)).
Y(1) (Cyclotomic Rigidity) Let † M = († Π, D† Π , sΘ† Π ) be a mod N mono-theta environment which is isomorphic to MN . We write † Πtemp
for the topological groupX
obtained by applying Corollary 7.22 (3). Then there exists a group-theoretic algorithm for constructing subquotients†)Θ [µN ]) ⊂ † ((Πtemp)Θ [µN ])(l∆Θ [µN ]) ⊂ † ((∆tempYY∼of † Π such that any isomorphism † M → MN maps these subquotients to the subquotientsl∆Θ [µN ] ⊂ (∆temp)Θ [µN ] ⊂ (Πtemp)Θ [µN ]YYof Πtemp[µN ], in a functorial manner with respect to isomorphisms of mono-thetaYenvironments (no need of any reference isomorphism to MN ). Moreover, there exists
a group-theoretic algorithm for constructing two splittings of the natural surjection†(l∆Θ [µN ])  † (l∆Θ )∼such that any isomorphism † M → MN maps these two splittings to the two splittingsof the surjectionl∆Θ [µN ]  l∆Θ162

Go Yamashita

determined by the mod N algebraic section salg
and the mod N theta section sΘ
. in
Ÿ
Ÿ
a functorial manner with respect to isomorphisms of mono-theta environments (noneed of any reference isomorphism to M). Hence in particular, by taking the diﬀerence of these two splittings, there exists a group-theoretic algorithm for constructing
an isomorphism of cyclotomes†(Cyc. Rig. Mono-th.)∼(l∆Θ ) ⊗ Z/N Z → µN († (l∆Θ [µN ]))∼such that any isomorphism † M → MN maps this isomorphism of the cyclotomes tothe natural isomorphism of cyclotomes∼l∆Θ ⊗ Z/N Z → µN (l∆Θ [µN ])in a functorial manner with respect to isomorphisms of mono-theta environments(no need of any reference isomorphism to MN ).
(2) (Discrete Rigidity) Any projective system († MN )N ≥1 of mono-theta environments is isomorphic to the natural projective system of the model mono-theta environments (MN )N ≥1 .
(3) (Constant Multiple Rigidity) Assume that η̈ Θ,lZ×µ2 is of standard type. Let(† MN )N ≥1 be a projective system of mono-theta environments. Then there exists agroup-theoretic algorithm for constructing a collection of classes of H 1 († Πtemp, † (l∆Θ ))Ÿ∼such that any isomorphism († MN )N ≥1 → (MN )N ≥1 to the projective systems of themodel mono-theta environments maps the above collection of classes to the collection of classes of H 1 (Πtemp, l∆Θ ) given by some multiple of the collection of classesŸη̈ Θ,lZ×µ2 by an element of µl in a functorial manner with respect to isomorphismsof projective systems of mono-theta environments (no need of any reference isomorphism to (MN )N ≥1 ).
We shall refer to † (l∆Θ ) ⊗ Z/N Z as the (mod N ) internal cyclotome of themono-theta environment † M, and µN († (l∆Θ [µN ])) the (mod N ) external cyclotome of the mono-theta environment † M. We shall refer to the above isomorphism (Cyc. Rig. Mono-th.) as the cyclotomic rigidity of mono-theta environment.
Proof. (1): Firstly, note that the restrictions of the algebraic section salgandŸ
the theta section sΘto ker{Πtemp
 (Πtemp)Θ } coincide by Remark 7.2.1 (1). HenceYYŸ
we can reconstruct ker{† (Πtemp[µN ])  † ((Πtemp)Θ [µN ])} as the subset of (any µN YY
temp†
conjugacy class of) sΘ)  † ((Πtemp)Θ )}, via† Π whose elements project to ker{ (ΠYY

a proof of the abc conjecture after Mochizuki

163

the projection † (Πtemp[µN ])  † (Πtemp), where † (Πtemp[µN ])  † (Πtemp), † (Πtemp
),
Y
Y
Y
Y
Y
and † (Πtemp)  † ((Πtemp)Θ are reconstructed by Lemma 7.18 (2), Corollary 7.22
YY(3) and Corollary 7.22 (1) respectively. We can also reconstruct the subquotients†(l∆Θ [µN ]) ⊂ † ((∆temp)Θ [µN ]) ⊂ † ((Πtemp)Θ [µN ]) as the inverse images of † (l∆Θ ) ⊂YY†)Θ ) ⊂ † ((Πtemp)Θ ), which are reconstructed by Corollary 7.22 (1) (3), via((∆tempYY

the quotient † ((Πtemp)Θ [µN ])  † ((Πtemp)Θ ). We can reconstruct the splitting ofYY
the natural surjection † (l∆Θ [µN ])  † (l∆Θ ) given by the theta section directly assΘ
On the other hand, we can reconstruct the splitting of the natural surjection†Π.
†(l∆Θ [µN ])  † (l∆Θ ) given by the algebraic section by the algorithm of Lemma 7.21
(1).
(2) follows from Corollary 7.22 (4), since R1 limN Hom(Z/2Z, Z/N Z) = 0 and←−R1 limN µN = 0. cf. also Remark 7.23.1 (2).
←−(3) follows from Lemma 7.21 (3), Corollary 7.15, the cyclotomic rigidity (1), andthe discrete rigidity (2).
Remark 7.23.1. In this remark, we compare rigidity properties of mono-theta
environments and bi-theta environments (cf. Remark 7.20.1 for bi-theta environments).
(1) (Cyclotomic Rigidity) The proof of the cyclotimic rigidity for mono-theta environments comes from the reconstruction of the image of the algebraic section, andthis reconstruction comes from the quadratic structure of theta group (=Heisenberg
group) (cf. Remark 7.19.1). On the other hand, for a bi-theta environment, theimage of the algebraic section is included as a datum of a bi-theta environment,
hence the cyclotomic rigidity trivially holds for bi-theta environment.

(2) (Constant Multiple Rigidity) The proof of the constant multiple rigidity for monotheta environments comes from the elliptic cuspidalisation (cf. Proposition 7.9).
On the other hand, for a bi-theta environment, the image of the algebraic section is
included as a datum of a bi-theta environment. This means that the ratio (i.e., theétale theta class) determined by the given data of theta section and algebraic section
is independent of the simultaneous constant multiplications on theta section andalgebraic section, hence the constant multiple rigidity trivially holds for bi-theta
environment.

(3) (Discrete Rigidity) A mono-theta environment does not include a datum of algebraic
section, it includes only a datum of theta section. By this reason, a mono-theta
environment has “shifting automorphisms” α̈δ in Lemma 7.21 (2) (which comesfrom the “less-than-or-equal-to-quadratic” structure of theta group (=Heisenberg

164

Go Yamashitagroup)). This means that there is no “basepoint” relative to the lZ action on Y ,i.e., no distinguished irreducible component of the special fiber. If we work with aprojective system of mono-theta environments, then by the compatibility of mod
N theta sections, where N runs through the positive integers, the mod N theta
classes determine a single “discrete” lZ-torsor in the projective limit. The “shifting automorphisms” gives us a lZ-indeterminacy, which is independent of N (cf.
Lemma 7.21 (3)), and to find a common basepoint for the lZ/N lZ-torsor in the
projective system is the same thing to trivialise a limN lZ/lZ(= 0)-torsor, which←−
remains discrete. This is the reason that the discrete rigidity holds for mono-theta
environments. On the other hand, a bi-theta environment includes a datum of algebraic section as well. The basepoint indeterminacy is roughly N lZ-indeterminacy(i.e., the surjectivity of Lemma 7.21 (3) does not hold for bi-theta environments. forthe precise statement, see [EtTh, Proposition 2.14 (iii)]), which depends on N , andto find a common basepoint for the lZ/N lZ-torsor in the projective system is thebsame thing to trivialise a limN lZ/N lZ(= lZ)-torsor,
which does not remain discrete←−(it is profinite). Hence the discrete rigidity does not hold for bi-theta environments.
Note also that a short exact sequence of the projective systems
0 → N lZ → lZ → lZ/N lZ → 0(resp. 0 → lZ → lZ → lZ/lZ → 0 )with respect to N ≥ 1, which corresponds to bi-theta environments (resp. monotheta environments), induces an exact sequenceb → R1 lim N lZ(= lZ/lZ)b
0 → lim N lZ (= 0) → lZ → lZ
→0
←−
←−
N

N

(resp. 0 → lZ → lZ → 0 → R1 lim lZ (= 0) ),←−Nband that R1 limN N lZ = lZ/lZ(resp. R1 limN lZ = 0) exactly corresponds to the←−←−non-discreteness (resp. discreteness) phenomenon of bi-theta environment (resp.
mono-theta environment). cf. also [EtTh, Remark 2.16.1].
The following diagram is a summary of this remark (cf. also [EtTh, Introduction]):mono-theta env.

cycl. rig.

disc. rig.

const. mult. rig.

delicately OK

OK

delicately OK(structure of theta group)bi-theta env.

trivially OK(elliptic cuspidalisation)
Fails

trivially OK

a proof of the abc conjecture after Mochizuki

165

Remark 7.23.2. If we consider N -th power Θ̈N (N > 1) of the theta function Θ̈
instead of the first power Θ̈1 = Θ̈, then the cyclotomic rigidity of Theorem 7.23 (1) does
not hold since it comes from the quadratic structure of the theta group (=Heisenberg
group) (cf. Remark 7.19.1). The cyclotomic rigidity of the mono-theta environmentis one of the most important tools in inter-universal Teichmüller theory, hence if weuse Θ̈N (N > 1) instead of Θ̈, then inter-universal Teichmüller theory does not work.
If it worked, then it would give us a sharper Diophantine inequality, which would be
a contradiction with the results in analytic number theory (cf. [Mass2]). cf. also Remark 11.10.1 (the principle of Galois evaluation) and Remark 13.13.3 (2) (N -th power
does not work).
b× ∼bRemark 7.23.3.
The cyclotomic rigidity rigidifies the Z= Aut(Z(1))-indeterminacybbb× ∼of an object which is isomorphic to “Z(1)”,hence rigidifies the induced Z= Aut(Z(1))bindeterminacy of H 1 (−, “Z(1)”).
As for the cohomology class log(Θ̈) of the theta func×b log(Θ̈). The constant multiple rigidity rigidifies log(Θ̈) + Z.
btion Θ̈, it ridigifies Z
Hence the cyclotomic rigidity and the constant multiple rigidity rigidify the indetermib × log(Θ̈) + Z
b of the aﬃne transformation type. The discrete rigidity rigidifies
nacy Zb∼bbb
Z“Z(1)”).
Here the second “Z(1)”is a coeﬃcient cyclotome, and it is= Hom(“Z(1)”,× ∼bbsubject to Z = Aut(Z(1))-indeterminacy which is rigidified by the cyclotomic rigidity.
b
The first “Z(1)”is a cyclotome which arises as a subquotient of a (tempered) fundamental group. Hence three rigidities of mono-theta environments in Theorem 7.23
correspond to the structure of the theta group (=Heisenberg group) (∆temp
)Θ :
X


cyclotomic rigidity constant multiple rigidity



.


0
discrete rigidity
cf. also the filtration of Lemma 7.5 (1).
§ 7.5.

Some Analogous Objects at Good Places.

In inter-unversal Teichüller theory, X is the main actor for places in Vbad . In thissubsection, for the later use, we introduce a counterpart −X of X for places in Vgood and→related objects (However, the theory for the places in Vbad is more important than the
one for the places in Vgood ).
Let X be a hyperbolic curve of type (1, 1) over a field K of characteristic 0, Ca hyperbolic orbicurve of type (1, l-tors)± (cf. Definition 7.10) whose K-core C is also

166

Go Yamashita

the K-core of X. Then C determines a hyperbolic orbicurve X := C ×C X of type
(1, l-tors). Let ιX be the nontrivial element in Gal(X/C)(∼= Z/2Z). We write GK forthe absolute Galois group of K for an algebraic closure K. Let l ≥ 5 be a prime number.
Assumption We assume that GK acts trivially on ∆abX ⊗ (Z/lZ).
(In inter-universal Teichmüller theory, we will use for K = Fmod (EFmod [l]) later.) We
write 0 for the unique zero-cusp of X. We choose a non-zero cusp  and let 0 and00 be the cusps of X over , and let ∆X  ∆abX ⊗ (Z/lZ)  ∆ be the quotient ofab∆X ⊗ (Z/lZ) by the images of the inertia subgroups of all non-zero cusps except 0 and00 of X. Then we have the natural exact sequence
0 → I0 × I00 → ∆ → ∆E ⊗ (Z/lZ) → 0,with the natural actions of GK and Gal(X/C)(∼= Z/2Z), where E is the genus one
compactification of X, and I0 , I00 are the images in ∆ of the inertia subgroups ofthe cusps 0 , 0 respectively (we have non-canonically I0 ∼= I00 ∼= Z/lZ). Note that ιX∼
induces an isomorphism I0 = I00 , and that ιX acts on ∆E ⊗(Z/lZ) via the multiplication
by −1. Since l is odd, the action of ιX on ∆ induces a decomposition∼−∆ → ∆+
 × ∆ ,−where ιX acts on ∆+ and ∆ by +1 and −1 respectively. Note that the natural+composites I0 ,→ ∆  ∆+
 and I00 ,→ ∆  ∆ are isomorphisms. We define(ΠX )JX by pushing the short exact sequences 1 → ∆X → ΠX → GK → 1 and by∆X  ∆  ∆+
 :
/ ∆X
/ ΠX
/ GK
/1
1

1


/ ∆+




/ JX

=


/ GK

/ 1.

Next, we consider the cusps “20 ” and “200 ” of X corresponding to the points ofE obtained by multiplying 0 and 00 by 2 respectively, relative to the group law of
the elliptic curve determined by the pair (X, 0 ). These cusps are not over the cusp
 in C, since 2 6≡ ±1 (mod l) by l ≥ 5. Hence the decomposition groups of “20 ”
and “200 ” give us sections σ : GK → JX of the natural surjection JX  GK . The
element ιX ∈ Gal(X/C), which interchange I0 and I00 , acts trivially on ∆+ (Note also∼∼I0 −→ ∆ ←− I00 ), hence these two sections to JX coincides. This section is only
determined by “20 ” (or “200 ”) up to an inner automorphism of JX given by an element+∆+
 ; however, since the natural outer action of GK on ∆ is trivial by Assumption, it
follows that the section completely determined by “20 ” (or “200 ”) and the image of thea proof of the abc conjecture after Mochizuki

167

section is normal in JX . By taking the quotient by this image, we obtain a surjection(ΠX )JX  ∆+
 . Let
X →X
−
→
∼ + ∼
be the corresponding covering with Gal(X−→/X) = ∆ (= Z/lZ).

Definition 7.24.
([IUTchI, Definition 1.1]) An orbicurve over K is called oftype (1, l-tors) if it is isomorphic to →X over K for some l and .
−−−→
The arrow → in the notation →X indicates a direction or an order on the {±1}-orbits−(i.e., the cusps of C) of Q (in Assumption (1) before Definition 7.10) is determined by (Remark [IUTchI, Remark 1.1.1]). We omit the construction of “−C ” (cf. [IUTchI, §1]),→since we do not use it. This −X is the main actor for places in Vgood in inter-universal→
Teichmüller theory :main actor

local Vbad

local Vgood

global 

global 

Xv

X
→
−
v

XK

CK

Lemma 7.25.
([IUTchI, Corollary 1.2]) We assume that K is an NF or an
MLF. Then from Π X , there exists a group-theoretic algorithm to reconstruct ΠX and→−ΠC (as subgroups of Aut(X−→)) together with the conjugacy classes of the decompositiongroup(s) determined by the set(s) of cusps {0 , 00 } and {} respectively, in a functorial
manner with respect to isomorphisms of topological groups.
cf. also Lemma 7.8, Lemma 7.12 ([EtTh, Proposition 1.8, Proposition 2.4]).
Proof. First, since Π X , ΠX and ΠC are slim by Proposition 2.7 (2b), these are→−naturally embedded into Aut(Π X ) by conjugate actions. By the K-coricity of C, we→−
can also group-theoretically reconstruct (Π X ⊂) ΠC (⊂ Aut(Π X )). By Proposition 2.2→−→−
or Corollary 2.4, we can group-theoretically reconstruct the subgroups ∆ C ⊂ Π C and→−→−∆ X ⊂ Π X (In particular, we can reconstruct l by the formula [∆C : ∆ X ] = 2l2 ).
→
−
→
−
→
−
We can reconstruct ∆X as a unique torsion-free subgroup of ∆C of index 2. Then we
can reconstruct ΠX (⊂ ΠC ) as ΠX = H · Π X , where H := ker(∆X  ∆abX ⊗ (Z/lZ)).
→
−
0
0
00
The conjugacy classes of the decomposition groups of  ,  , and  in ΠX can bereconstructed as the decomposition groups of cusps (Corollary 2.9 and Remark 2.9.2)
whose image in ΠX /Π X is nontrivial. Then we can reconstruct the subgroup ΠC ⊂ ΠC→−
by constructing a splitting of the natural surjection ΠC /ΠX  ΠC /ΠX determined byΠC /ΠX , where the splitting is characterised (since l - 3) as the unique splitting (whose
image ⊂ ΠC /ΠX ) stabilising (via the outer action on ΠX ) the collection of conjugacy

168

Go Yamashitaclasses of the decomposition groups in ΠX of 0 , 0 , and 00 (Note that if an ivolutionof X fixed 0 and interchanged 0 and 00 , then we would have 2 ≡ −1 (mod l), i.e.,l | 3). Finally, the decomposition groups of 0 and 00 in ΠX can be reconstructed as thedecomposition group of cusps (Corollary 2.9 and Remark 2.9.2) whose image in ΠX /Π X→−
is nontrivial, and is not fixed, up to conjugacy, by the outer action of ΠC /ΠX (∼= Z/2Z)on ΠX .
Remark 7.25.1.

([IUTchI, Remark 1.2.1]) By Lemma 7.25, we have∼
AutK (X−→) = Gal(X−→/C) (= Z/2lZ)(cf. Remark 7.12.1).

§ 8.

Frobenioids.

Roughly speaking, we have the following proportional formula:Anabelioid (=Galois category) : Frobenioid = coverings : line bundles over coverings,

that is, the theory of Galois categories is a categorical formulation of coverings (i.e., it
is formulated in terms of category, and geometric terms never appear), and the theory
of Frobenioids is a categorical formulation of line bundles over coverings (i.e., it is formulated in terms of category, and geometric terms never appear). In [FrdI] and [FrdII],
Mochizuki developed a general theory of Frobenioids; however, in this survey, we mainly
forcus on model Frobenioids, which mainly used in inter-universal Teichmüller theory.
The main theorems of the theory of Frobenioids are category-theoretic reconstruction
algorithms of related objects (e.g., the base categories, the divisor monoids, and so on)under certain conditions; however, we avoid these theorems by including the objects,
which we want to reconstruct, as input data, as suggested in [IUTchI, Remark 3.2.1(ii)].
§ 8.1.

Elementary Frobenioids and Model Frobenioids.

For a category D, we shall refer to a contravariant functor Φ : D → Mon to thecategory of commutative monoids Mon as a monoid on D (In [FrdI, Definition 1.1], we
write some conditions on Φ. However, this has no problem for our objects used in interuniversal Teichmüller theory.) If any element in Φ(A) is invertible for any A ∈ Ob(D),then we shall refer to Φ as group-like.

a proof of the abc conjecture after Mochizuki

169

Definition 8.1.
(Elementary Frobenioid, [FrdI, Definition 1.1 (iii)]) Let Φ be a
monoid on a category D. We consider the following category FΦ :(1) Ob(FΦ ) = Ob(D).
(2) For A, B ∈ Ob(D), we put
HomFΦ (A, B) := {φ = (Base(φ), Div(φ), degFr (φ)) ∈ HomD (A, B) × Φ(A) × N≥1 } .
We define the composition of φ = (Base(φ), Div(φ), degFr (φ)) : A → B and ψ =(Base(ψ), Div(ψ), degFr (n)) : B → C asψ◦φ := (Base(ψ)◦Base(φ), Φ(Base(φ))(Div(ψ))+degFr (ψ)Div(φ), degFr (ψ)degFr (φ)) : A → C.
We shall refer to FΦ as an elementary Frobenioid associated to Φ. Note that
we have a natural functor FΦ → D, which sends A ∈ Ob(FΦ ) to A ∈ Ob(D), and
φ = (Base(φ), Div(φ), degFr (φ)) to Base(φ). We shall refer to D as the base categoryof FΦ .
For a category C and an elementary Frobenioid FΦ , we shall refer to a covariant
functor C → FΦ as a pre-Frobenioid structure on C (In [FrdI, Definition 1.1 (iv)],
we need conditions on Φ, D, and C for the general theory of Frobenioids). We shallrefer to a category C with a pre-Frobenioid structure as a pre-Frobenioid. For a
pre-Frobenioid C, we have a natural functor C → D by the composing with FΦ → D.
In a similar way, we obtain operations Base(−), Div(−), degFr (−) on C from the oneson FΦ by composing with FΦ → D. We often use the same notation on C as well, by
abuse of notation. We also shall refer to Φ and D as the divisor monoid and the basecategory of the pre-Frobenioid C respectively. We write
O× (A) := {φ ∈ AutC (A) | Base(φ) = id, degFr (φ) = 1} ⊂ AutC (A),andO (A) := {φ ∈ EndC (A) | Base(φ) = id, degFr (φ) = 1} ⊂ EndC (A)for A ∈ Ob(C). We also write µN (A) := {a ∈ O× (A) | aN = 1} for N ≥ 1.
Definition 8.2.
([IUTchI, Example 3.2 (v)]) When we are given a splitting spl :×O /O ,→ O (resp. a µN -orbit of a splitting spl : O /O× ,→ O for fixed N ) ofO  O /O× , i.e., functorial splittings (resp. functorial µN -orbit of splittings) ofO (A)  O (A)/O× (A) with respect to A ∈ Ob(C) and morphisms with degFr = 1,
then we shall refer to the pair (C, spl) as a split pre-Frobenioid (resp. a µN -split
pre-Frobenioid).
170

Go Yamashita

If a pre-Frobenioid satisfies certain technical conditions, then we call it a Frobenioid (cf. [FrdI, Definition 1.3]). (Elementary Frobenioids are, in fact, Frobenioids([FrdI, Proposition 1.5]).) In this survey, we do not recall the definition nor use thegeneral theory of Frobenioids, and we mainly focus on model Frobenioids.
Definition 8.3.
(Model Frobenioid, [FrdI, Theorem 5.2]) Let Φ : D → Mon
be a monoid on a category D. Let B : D → Mon be a group-like monoid on D, and
DivB : B → Φgp a homomorphism. We write Φbirat := Im(DivB ) ⊂ Φgp . We consider
the following category C:(1) The objects of C are pairs A = (AD , α), where AD ∈ Ob(D), and α ∈ Φ(AD )gp . We
write Base(A) := AD , Φ(A) := Φ(AD ), and B(A) := B(AD ).
(2) For A = (AD , α), B = (BD , β) ∈ Ob(C), we put{
HomC (A, B) :=φ = (Base(φ), Div(φ), degFr (φ), uφ ) ∈ HomD (AD , BD ) × Φ(A) × N≥1 × B(A)such that degFr (φ)α + Div(φ) = Φ(Base(φ))(β) + DivB (uφ )We define the composition of φ = (Base(φ), Div(φ), degFr (φ), uφ ) : A → B andψ = (Base(ψ), Div(ψ), degFr (ψ), uψ ) : B → C as(
ψ ◦ φ :=

Base(ψ) ◦ Base(φ), Φ(Base(φ))(Div(ψ)) + degFr (ψ)Div(φ),
degFr (ψ)degFr (φ), B(Base(φ))(uψ ) + degFr (ψ)uφ).

We equip C with a pre-Frobenioid structure C → FΦ by sending (AD , α) ∈ Ob(C) to
AD ∈ Ob(FΦ ) and (Base(φ), Div(φ), degFr (φ), uφ ) to (Base(φ), Div(φ), degFr (φ)). We
shall refer to the category C as the model Frobenioid defined by the divisor monoid
Φ and the rational function monoid B (Under some conditions, the model Frobenioid
is in fact a Frobenioid).
The main theorems of the theory of Frobenioids are category-theoretic reconstruction algorithms of related objects (e.g., the base categories, the divisor monoids, andso on), under certain conditions. However, in this survey, we consider isomorphisms
between pre-Frobenioids not to be just category equivalences, but to be category equivalences including pre-Frobenioid structures, i.e., for pre-Frobenioids F, F 0 with preFrobenioid structures F → FΦ , F 0 → FΦ0 , where FΦ , FΦ0 are defined by D → Φ,
D0 → Φ0 respectively, an isomorphism of pre-Frobenioids from F to F 0 consists of∼∼
isomorphism classes (cf. also Definition 6.1 (5)) of equivalences F 0 → F , D0 → D of
categories, and a natural transformation Φ0 → Φ|D0 (where Φ|D0 is the restriction of Φ∼∼
via D0 → D), such that it gives rise to an equivalence FΦ0 → FΦ of categories, and the}.

a proof of the abc conjecture after Mochizuki

171

diagram
F0

FΦ 0

∼

/F

∼


/ FΦ

is 1-commutative (i.e., one way of the composite of functors is isomorphic to the otherway of the composite of functors) (cf. also [IUTchI, Remark 3.2.1 (ii)]).
Definition 8.4.
(1) (Trivial Line Bundle) For a model Frobenioid F with base category D, we write
OA for the trivial line bundle over A ∈ Ob(D), i.e., the object determine by(A, 0) ∈ Ob(D) × Φ(A)gp (These objects are called “Frobenius-trivial objects” inthe terminology of [FrdI], which can category-theoretically be reconstructed only
from F under some conditions).
(2) (Birationalisation, “Z≥0Z”) Let C be a model Frebenioid. Let C birat be thecategory whose objects are the same as in C, and whose morphisms are given by
HomC birat (A, B) :=lim−→

HomC (A0 , B).

φ:A0 →A, Base(φ) : isom, degFr (φ)=1(For general Frobenioids, the definition of the birationalisation is a little more complicated. cf. [FrdI, Proposition 4.4]). We shall refer to C birat as the birationalisation of the model Frobenioid C. We have a natural functor C → C birat .
(3) (Realification, “Z≥0R≥0 ”) Let C be a model Frobenioid whose divisor monoid isΦ and whose rational function monoid is B. Then let C R be the model Frobenioid
obtained by replacing the divisor monoid Φ by ΦR := Φ ⊗Z≥0 R≥0 , and the rational
function monoid B by BR := R · Im(B → Φgp ) ⊂ (ΦR )gp (We need some conditions
on C, if we want to include more model Frobenioids which we do not treat in thissurvey. cf. [FrdI, Definition 2.4 (i), Proposition 5.2]). We shall refer to C R as therealification of the model Frobenioid C. We have a natural functor C → C R .
Definition 8.5.
(×-, ×µ-Kummer structure on pre-Frobenioid, [IUTchII, Example 1.8 (iv), Definition 4.9 (i)])(1) Let G be a toplogical group isomorphic to the absolute Galois group of an MLF.
Then we can group-theoretically reconstruct an ind-topological monoid G y O (G)with G-action, by Proposition 5.2 (Step 1). Write O× (G) := (O (G))× , Oµ (G) :=(O (G))tors and O×µ (G) := O× (G)/Oµ (G) (We use the notation O×µ (−), not172

Go Yamashita

O× (−)/Oµ (−), because we want to consider the object O× (−)/Oµ (−) as an abstract ind-topological module, i.e., without being equipped with the quotient structure O× /Oµ ). Write{∼
Isomet(G) = G-equivariant isomorphism O×µ (G) → O×µ (G) preserving}
the integral str. Im(O× (G)H → O×µ (G)H ) for any open H ⊂ G .
We shall refer to the compact topological group Isomet(G) as the group of Gisometries of O ×µ (G). If there is no confusion, we write just Isomet for Isomet(G).
(2) Let C be a pre-Frobenioid with base category D. We assume that D is equivalent
to the category of connected finite étale coverings of the spectrum of an MLF or a
CAF. Let A∞ be a universal covering pro-object of D. Write G := Aut(A∞ ), hence
G is isomorphic to the absolute Galois group of an MLF or a CAF. Then we havea natural action G y O (A∞ ). For N ≥ 1, we putµN (A∞ ) := {a ∈ O (A∞ ) | aN = 1} ⊂ Oµ (A∞ ) := O (A∞ )tors ⊂ O (A∞ ),and
O× (A∞ )  O×µN (A∞ ) := O× (A∞ )/µN (A∞ )  O×µ (A∞ ) := O× (A∞ )/Oµ (A∞ ).
These are equipped with natural G-actions. We assume that G is nontrivial (i.e.,
arising from an MLF). A ×-Kummer structure (resp. ×µ-Kummer strucb × -orbit (resp. an Isomet-orbit)ture) on C is a Z
×

κ

×

poly
∼

×

×µ

: O (G) → O (A∞ ) (resp. κ

: O

×µ

poly
∼

(G) → O×µ (A∞ ) )of isomorphisms of ind-topological G-modules. Note that the definition of a ×(resp. ×µ-) Kummer structure is independent of the choice of A∞ . Note also thatany ×-Kummer structure on C is unique, since ker(Aut(G y O× (G))  Aut(G)) =b × (= Aut(O× (G))) (cf. [IUTchII, Remark 1.11.1 (i) (b)]). We shall refer to a preZ
Frobenioid equipped with a ×-Kummer structure (resp. ×µ-Kummer structure)
as a ×-Kummer pre-Frobenioid (resp. ×µ-Kummer pre-Frobenioid). We
shall refer to a split pre-Frobenioid equipped with a ×-Kummer structure (resp.
×µ-Kummer structure) as a split-×-Kummer pre-Frobenioid (resp. split-×µKummer pre-Frobenioid).
Remark 8.5.1.
([IUTchII, Remark 1.8.1]) In the situation of Definition 8.5 (1),no automorphism of O×µ (G) induced by an element of Aut(G) is equal to an automorphism of O×µ (G) induced by an element of Isomet(G) which has nontrivial image in Z×p (Here p is the residual characteristic of the MLF under consideration),a proof of the abc conjecture after Mochizuki

173

since the composite with the p-adic logarithm of the cyclotomic character of G (which
can be group-theoretically reconstructed by Proposition 2.1 (6)) determines a natural
Aut(G) × Isomet(G)-equivariant surjection O×µ (G)  Qp , where Aut(G) trivially actsb ×  Z× .
on Qp and Isomet(G) acts on Qp via the natural surjection Zp§ 8.2.

Examples.

Example 8.6.
(Geometric Frobenioid, [FrdI, Example 6.1]) Let V be a proper
normal geometrically integral variety over a field k, k(V ) the function field of V , andk(V )∼ a (possibly inifinite) Galois extension. Write G := Gal(k(V )∼ /k(V )), and let
Dk(V ) be a set of Q-Cartier prime divisors on V . The connected objects Ob(B(G)0 )(cf. Section 0.2) of the Galois category (or connected anabelioid) B(G) can be thoughtof as schemes Spec L, where L ⊂ k(V )∼ is a finite extension of k(V ). We write VL
for the normalisation of V in L, and we write DL for the set of prime divisors of VL
which maps into (possibly subvarieties of codimension ≥ 1 of) prime divisors of Dk(V )We assume that any prime divisor of DL is Q-Cartier for any Spec L ∈ Ob(B(G)0 ).
We write Φ(L) ⊂ Z≥0 [DL ] for the monoid of eﬀective Cartier divisors D on VL suchthat every prime divisor in the support of D is in DL , and B(L) ⊂ L× for the group ofrational functions f on VL such that every prime divisor, at which f has a zero or a pole,
is in DL . Note that we have a natural homomorphism B(L) → Φ(L)gp which sends f to(f )0 − (f )∞ (Here, we write (f )0 and (f )∞ for the zero-divisor and the pole-divisor off respectively). This is functrial with respect to L. The data (B(G)0 , Φ(−), B(−), B →Φgp ) determines a model Frobenioid CV,k(V )∼ ,DK .
An object of CV,k(V )∼ ,DK , which is sent to Spec L ∈ Ob(B(G)0 ), can be thought ofas a line bundle L on VL , which is representable by a Cartier divisor D with supportin DL . For such line bundles L on Spec L and M on Spec M (L, M ⊂ k(V )∼ are finite
extensions of k(V )), a morphism L → M in CV,k(V )∼ ,DK can be thought of as consistingof a morphism Spec L → Spec M over Spec k(V ), an element d ∈ N≥1 , and a morphismof line bundles L⊗d → M|VL on VL whose zero locus is a Cartier divisor supported inDL .
Example 8.7.
(p-adic Frobenioid, [FrdII, Example 1.1], [IUTchI, Example 3.3])
Let Kv be a finite extension of Qpv (In inter-universal Teichmüller theory, we use v ∈Vgood ∩ Vnon ). Write0
Dv := B(X
−
→ ) ,
v

and Dv` := B(Kv )0 ,where −
X is a hyperbolic curve of type (1, l-tors→−−→) (cf. Definition 7.24). By pulling backv

finite étale coverings via the structure morphism −X → Spec Kv , we regard Dv` as a→v

full subcategory of Dv . We also have a left-adjoint Dv → Dv` to this functor, which is174

Go Yamashitaobtained by sending a Π X -set E to the GKv -set E/ker(Π X → GKv ) := ker(Π X →
→
−v

→
−v

→
−v

GKv )-orbits of E ([FrdII, Definition 1.3 (ii)]). Then
× pf
 pf
ΦCv : Spec L 7→ ord(OL)) := (OL /OL(cf. Section 0.2 for the perfection (−)pf ) gives us a monoid on Dv` . By composing theabove Dv → Dv` , it gives us a monoid ΦCv on Dv . Also, pf
ΦCv` : Spec L 7→ ord(Zpv ) (⊂ ord(OL ) )(cf. Section 0.2 for the perfection (−)pf ) gives us a submonoid ΦCv` ⊂ ΦCv on Dv` .
These monoids ΦCv on Dv and ΦCv` on Dv` determine pre-Frobenioids (In fact, these are
Frobenioid)Cv` ⊂ Cv
whose base categories are Dv` and Dv respectively. These are called pv -adic Frobenioids. These pre-Frobenioid can be regarded as model Frobenioids whose rational
function monoids B are given by Ob(Dv` ) 3 Spec L 7→ L× ∈ Mon, and L× 3 f 7→(f )0 − (f )∞ := image of f ∈ ΦCv` (L) ⊂ ΦCv (L) ([FrdII, Example 1.1]). Note that the`×element pv ∈ Z
pv gives us a splitting splv : O /O ,→ O , hence a split pre-Frobenioid

Fv` := (Cv` , spl`v ).
We also write
F v := Cv
for later use.
Example 8.8.
(Tempered Frobenioid, [EtTh, Definition 3.3, Example 3.9, thebeginning of §5], [IUTchI, Example 3.2]) Let X v := X K → X v := X Kv be a hyperbolicv

curve of type (1, l-torsΘ ) and a hyperbolic curve of type (1, Z/lZ) respectively (Definition 7.13, Definition 7.11) over a finite extension Kv of Qpv (As before, we always write
the log-structure associated to the cusps, and consider the log-fundamental groups).
Write
Dv := B temp (X v )0 , Dv` := B(Kv )0 ,temp
and D0 := Btemp (X v )0 (cf. Section 0.2 for (−)0 . Note also that we have π1 (Dv ) ∼= ΠX ,v
and π1 (D` ) ∼G(cf.
Definition
6.1(4))).
We
haveanaturalfunctor
D
→
D
,
which
= K
v
0
v

v

sends Y → X v to the composite Y → X v → X v .
For a tempered covering Z → X v and its stable formal model Z over OL , where L
is a finite extension of Kv , let Z∞ → Z be the universal combinatorial covering (i.e., thecovering determined by the universal covering of the dual graph of the special fiber ofZ), and Z∞ the Raynaud generic fiber of Z∞ .

175a proof of the abc conjecture after Mochizuki

Definition 8.9.
([EtTh, Definition 3.1], [IUTchI, Remark 3.2.4]) We write Div+ (Z∞ )for the monoid of the eﬀective Cartier divisors whose support lie in the union of thespecial fiber and the cusps of Z∞ . We shall refer to such a divisor as an eﬀective
Cartier log-divisor on Z∞ . Also, we write Mero(Z∞ ) for the group of meromorphic
functions f on Z∞ such that, for any N ≥ 1,f admits an N -th root over some tempered
covering of Z. We shall refer to such a function as a log-meromorphic function on
Z∞ .
Definition 8.10.
[IUTchI, Example 3.2])([EtTh, Definition 3.3, Example 3.9, the beginning of §5],(1) Let ∆ be a tempered group (Definition 6.1). We shall refer to a filtration {∆i }i∈I ,(where I is countable) of ∆ by characteristic open subgroups of finite index as atempred filter, if the following conditions are satisfied:(a) We have∩i∈I ∆i = ∆.

∞(b) Every ∆i admits an open characteristic subgroup ∆∞i such that ∆i /∆i isfree, and, for any open normal subgroup H ⊂ ∆i with free ∆i /H, we have∆∞i ⊂ H.
∞(c) For each open subgroup H ⊂ ∆, there exists unique ∆∞iH ⊂ H, and, ∆i ⊂ H∞
implies ∆∞i ⊂ ∆iH for every i ∈ I.

(2) Let {∆i }i∈I be a tempered filter of ∆temp
X v . Assume that, for any i ∈ I, the covering
detemined by ∆i has a stable model Zi over a ring of integers of a finite extension of
Kv , and all of the nodes and the irreducible components of the special fiber of Zi arerational (we say that Zi has split stable reduction). For any connected tempered
covering Y → X v , which corresponds to an open subgroup H ⊂ ∆temp, we putXv

Φ0 (Y ) := lim Div+ (Z∞ )Gal(Z∞ /Y ) ,
−→
∞
∆i ⊂H

B0 (Y ) := lim Mero(Z∞ )Gal(Z∞ /Y ) .
−→
∞
∆i ⊂H

These determine functors Φ0 : D0 → Mon, B0 : D0 → Mon. We also have a naturalconst⊂ B0 for thefunctor B0 → Φgp
0 , by taking f 7→ (f )0 − (f )∞ . We write B0
subfunctor defined by the constant log-meromorphic functions, and Φconst
⊂ Φgp
0
0
gp
for the image of Bconst
in
Φ
.
0
0
(3) We write D0ell ⊂ D0 for the full subcategory of tempered coverings which are unramified over the cusps of X v (i.e., tempered coverings of the underlying elliptic
curve E v of X v ). We have a left adjoint D0 → D0ell , which is obtained by sendinga ΠX v -set E to the ΠE v -set E/ker(ΠX v → ΠE v ) := ker(ΠX v → ΠE v )-orbits of E

176

Go Yamashita([FrdII, Definition 1.3 (ii)]). For Y ∈ Ob(Dv ), we write Y ell for the image of Y by
the composite Dv → D0 → D0ell . We put, for Y ∈ Ob(Dv ),)pf(
Φ(Y ) :=

lim Div+ (Z∞ )Gal(Z∞ /Y
−→

ell

)

⊂ Φ0 (the image of Y in D0 )pf ,Z∞

where Z∞ range over the connected tempered covering Z∞ → Y ell in D0ell suchthat the composite Z∞ → Y ell → X v arises as the generic fiber of the universal
combinatorial covering Z∞ of the stable model Z of some finite étale covering Z →X v in D0ell with split stable reduction over the ring of integers of a finite extensionof Kv (We use this Φ, not Φ0 , to consider only divisors related with the theta
function). We write (−)|Dv for the restriction, via Dv → D0 , of a functor whoseRdomain is D0 . We also write ΦR
0 := Φ0 ⊗Z≥0 R≥0 and Φ := Φ ⊗Z≥0 R≥0 . Write
B := B0 |Dv ×(ΦR )gp Φgp ,Φconst := (R · Φconst)|Dv ×(ΦR )gp Φ ⊂ ΦR ,0

and)|Dv ×(ΦR )gp Φgp ⊂ (ΦR )gp .
|Dv ×(ΦR )gp Φgp → (Φconst )gp = (R · Φconst
Bconst := Bconst
00
The data (Dv , Φ, B, B → Φgp ) and (Dv , Φconst , Bconst , Bconst → (Φconst )gp ) determine model Frobenioids
F v,

and Cv (= F base-field)v

respectively (In fact, these are Frobenioids). We have a natural inclusion Cv ⊂ F v .
We shall refer to F v a tempered Frobenioid and Cv as its base-field-theoretic
hull. Note that Cv is also a pv -adic Frobenioid.
(4) We write Θv ∈ O× (OŸbirat ) for the reciprocal (i.e., 1/(−)) of the l-th root of thevnormalised theta function, which is well-defined up to µ2l and the action of thegroup of automorphisms lZ ⊂ Aut(OŸ ) (Note that we use the notation Θ̈ inv

Section 8.3. This is not the reciprocal (i.e., not 1/(−)) one). We also write qv forthe q-parameter of the elliptic curve Ev over Kv . We consider qv as an elementqv ∈ O (OX ) (∼). We assume that any 2l-torsion point of Ev is rational over
= OKvv
∼ O ). Then we have
Kv . Then qv admits a 2l-root in O (OX ) (=v

Kv√
Θv ( −qv ) = q := qv1/2l ∈ O (OX ),vv√ −2 √√(which is well-defined up to µ2l ), since Θ̈( −q) = −q −1/2 −1 Θ̈( −1) = q −1/2
(in the notation of Lemma 7.4) by the formula Θ̈(q 1/2 Ü ) = −q −1/2 Ü −2 Θ̈(Ü ) in

a proof of the abc conjecture after Mochizuki

177

Lemma 7.4. The image of q determines a constant section, for which we writevlogΦ (q ) of the monoid ΦCv of Cv . The submonoidv

ΦCv` := N logΦ (q )|Dv` ⊂ ΦCv |Dv`v

gives us a pv -adic Frobenioid
Cv` (⊂ Cv = (F v )base-field ⊂ F v )whose base category is Dv` . The element q ∈ Kv determines a µ2l (−)-orbit spl`v ofv

the splittings of O  O /O× on Cv` . Hence
Fv` := (Cv` , spl`v )is a µ2l -split pre-Frobenioid.
Remark 8.10.1. We can category-theoretically reconstruct the base-field-theoretic
hull Cv from F v ([EtTh, Corollary 3.8]). However, in this survey, we include the basefield-theoretic hull in the deta of the tempered Frobenioid, i.e., we shall refer to a pair
F v = (F v , Cv ) as a tempered Frobenioid, by abuse of language/notation, in this survey.
Example 8.11.
(Archimedean Frobenioid, [FrdII, Example 3.3], [IUTchI, Example 3.4]) This example is not a model Frobenioid (In fact, it is not of isotropic type,which any model Frobenioids should be). Let Kv be a complex Archimedean local field(In inter-universal Teichmüller theory, we use v ∈ Varc ). We define a categoryCv
as follows: The objects of Cv are pairs (V, A) of a one-dimensional Kv -vector space×
V , and a subset A = B × C ⊂ V ∼× ord(Kv× ) (Here we write ord(Kv× ) :=
= OK
v
×
× ∼ 1
Kv× /OK. cf. Section 0.2 for OKv ), where B ⊂ OK(= S ) is a connected open subset,vv
× ∼and C ⊂ ord(Kv ) = R>0 is an interval of the form (0, λ] with λ ∈ R>0 (We shallrefer to A as an angular region). The morphisms φ from (V, A) to (V 0 , A0 ) in Cv∼
consist of an element degFr (φ) ∈ N≥1 and an isomorphism V ⊗degFr (φ) → V 0 of Kv vector spaces which sends A⊗degFr (φ) into A0 . We write Div(φ) := log(α) ∈ R≥0 for thelargest α ∈ R>0 such that α · Im(A⊗degFr (φ) ) ⊂ A0 . Let {Spec Kv } be the category of
connected finite étale coverings of Spec Kv (Thus, there is only one object, and only
one morphism), and Φ : {Spec Kv } → Mon the functor defined by sending Spec Kv (the− logunique object) to ord(OK) ∼= (0, 1] ∼= R≥0 . Write also Base(V, A) := Spec Kv forv(V, A) ∈ Ob(Cv ). Then the triple (Base(−), Φ(−), degFr (−)) gives us a pre-Frobenioid
structure Cv → FΦ on Cv (In fact, this is a Frobenioid). We shall refer to Cv as an

178

Go Yamashita

Archimedean Frobenioid (cf. the Archimedean portion of arithmetic line bundles).

Note also that we have a natural isomorphism O (Cv ) ∼of topological monoids= OKv(We can regarad Cv as a Frobenioid-theoretic representation of the topological monoid

OK
).
v
Let −
X be a hyperbolic curve of type (1, l-tors→−−→) (cf. Definition 7.24) over Kv , andv

we write −X for the Aut-holomorphic space (cf. Section 4) determined by −X , and put
→
→
v

v

Dv := −
X .
→v

Note also that we have a natural isomorphism∼

Kv → ADvof topological fields (cf. (Step 9) in Proposition 4.5), which determines an inclusion
κv : O (Cv ) ,→ ADv
of topological monoids. This gives us a Kummer structure (cf. Definition 4.6) on Dv .
Write
F v := (Cv , Dv , κv ),∼just as a triple. We define an isomorphism F v,1 → F v,2 of triples in an obvious manner.
Next, we consider the mono-analyticisation. Write
Cv` := Cv .
Note also that ADv naturally determines a split monoid (cf. Definition 4.7) by trans∼
porting the natural splitting of Kv via the isomorphism Kv → ADv of topological fields.
This gives us a splitting spl`v on Cv` , hence a split-Frobenioid (Cv` , spl`v ), as well as a split
monoidDv` := (O (Cv` ), spl`v ).
We putFv` := (Cv` , Dv` , spl`v ),∼``just as a triple. We define an isomorphism Fv,1
→ Fv,2of triples in an obvious manner.

Example 8.12.
(Global Realified Frobenioid, [FrdI, Example 6.3], [IUTchI, Example 3.5]) Let Fmod be a number field. Let {Spec Fmod } be the category of the trivial
connected finite étale covering of Spec Fmod . (Thus, there is only one object, and only
one morphism.) Write⊕⊕ΦCmod (Fmod ) :=ord(Ov ) ⊗Z≥0 R≥0 ⊕ord(Ov ),v∈V(Fmod )nonv∈V(Fmod )arc

a proof of the abc conjecture after Mochizuki

179

where ord(Ov ) := Ov /Ov× (cf. Section 0.2 for Ov and Ov , v ∈ V(Fmod )arc ). We shallrefer to an element of Φ(Fmod ) (resp. Φ(Fmod )gp ) as an eﬀective arithmetic divisor(resp. an arithmetic divisor). Note that ord(Ov ) ∼= Z≥0 for v ∈ V(Fmod )non , andord(Ov ) ∼= R≥0 for v ∈ V(Fmod )arc . We have a natural homomorphism×
B(Fmod ) := Fmod
→ Φ(Fmod )gp .

Then the data ({Spec Fmod }, ΦCmod , B) determines a model Frobenioid
Cmod .
(In fact, it is a Frobenioid.) We shall refer to it as a global realified Frobenioid.
We have a natural bijection∼

Prime(Cmod ) → Vmod(by abuse of notation, we write Prime(Cmod ) := Prime(ΦCmod (Spec Fmod ))), where
Prime(−) is defined as follows:
Definition 8.13.
Let M be a commutative monoid such that 0 is the only invertible element in M , the natural homomorphism M → M gp is injective, and anya ∈ M gp with na ∈ M for some n ∈ N≥1 is in the image of M ,→ M gp . We define theset Prime(M ) of primes of M as follows ([FrdI, §0]):(1) For a, b ∈ M , we write a ≤ b, if there is c ∈ M such that a + c = b.
(2) For a, b ∈ M , we write a 4 b, if there is n ∈ N≥1 such that a ≤ nb.
(3) For 0 6= a ∈ M , we say that a is primary, if a 4 b holds for any M 3 b 4 a, b 6= 0.
(4) The relation a 4 b is an equivalence relation among the set of primary elements in
M , and we shall refer to an equivalence class as a prime of M (this definition is
diﬀerent from a usual definition of primes of a monoid). We write Prime(M ) for
the set of primes of M .
Note that pv determines an elementlog`mod (pv ) ∈ ΦCmod ,v
for v ∈ Vmod ∼= Prime(Cmod ), where we write ΦCmod ,v (∼= R≥0 ) for the v-portion ofΦCmod .
§ 8.3.

From Tempered Frobenioids to Mono-theta Environments.

Let F v be the tempered Frobenioid constructed in Example 8.8. Recall that it has
temp
a base category Dv with π1 (Dv ) ∼(=: Πv ). We write O for the object in F
=Π
X

v

Ÿ

v

180

Go Yamashita

corresponding to the trivial line bundle on Ÿ (i.e., OŸ = (Ÿ , 0) ∈ Ob(Dv ) × Φ(Ÿ ). cf.
Definition 8.4 (1)). Let ŸlN , ZlN , Z̈lN , LlN , and L̈lN as in Section 7.1. We can interpretthe pull-backs to Z̈lN of(1) the algebraic section slN ∈ Γ (ZlN , LlN |ZlN ) of Lemma 7.1, and()(2) the theta trivialisation τlN ∈ Γ ŸlN , L̈lN after Lemma 7.1.
as morphismstsu
N , sN : OZ̈lN → L̈lN |Z̈lN

in F v respectively. For A ∈ Ob(F v ), we write Abirat for the image of A in the birationalisation F v → (F v )birat (Definition 8.4 (2)). Then by definition, we have
t −1
su
= Θ̈
N ◦ (sN )1/N)(∈ O× OZ̈biratlN

for an N -th root of Θ̈, where Θ̈ := Θ̈1/l is a l-th root of the theta function Θ̈ ([EtTh,
Proposition 5.2 (i)]), as in Section 7.1 (cf. also the claim (7.2)). We write H(Z̈lN ) (⊂AutDv (Z̈lN )) for the image of Πtemp
under the surjective outer homomorphism Πtemp

X
Ÿ
v

×

AutDv (Z̈lN ). We also write H(OZ̈lN ) (⊂ AutF (OZ̈lN )/O (OZ̈lN )) (resp. H(L̈lN |Z̈lN ) (⊂v

×

AutF (L̈lN |Z̈lN )/O (L̈lN |Z̈lN )) ) for the inverse image of H(Z̈lN ) of the natural injecv

tion AutF (OZ̈lN )/O× (OZ̈lN ) ,→ AutDv (Z̈lN ) (resp. AutF (L̈lN |Z̈lN )/O× (L̈lN |Z̈lN ) ,→v

v

AutDv (Z̈lN )):
Πtemp
X
Ov

/ / AutDv (Z̈lN ) o
O

?

?
/ / H(Z̈lN ) o

Πtemp
Ÿ

? _ AutF (Ov=Z̈lN )/O×(OZ̈lN ) (resp. AutF (L̈lN |Z̈lN )/O× (L̈lN |Z̈lN ) )v
O?
Z̈lN ) (resp. H(L̈lN |Z̈lN ) ).

? _ H(O∼ H(Z̈lN ) =∼ H(L̈lN | ). Choose a
Note that we have natural isomorphisms H(OZ̈lN ) =Z̈lN
section of AutF (OZ̈lN )  AutDv (Z̈lN ), which gives us a homomorphismv

striv
N : H(OZ̈lN ) → AutF (OZ̈lN ).
v

temp
t
Then by taking the group actions of H(L̈lN |Z̈lN ) on su
N , and sN (cf. the actions of ΠŸ
on sN and τN in Section 7.1), we have unique groups homomorphisms

su-gp, st-gp: H(L̈lN |Z̈lN ) → AutF (L̈lN |Z̈lN ),
N
N
v

181

a proof of the abc conjecture after Mochizuki

which make diagrams
OZ̈lN

su
N

(striv
N |L̈lN )(h)


OZ̈lN

su
N

/ L̈lN |/ L̈lN |

OZ̈lN

Z̈lN
su-gp
(h)
N

st
N

(striv
N |L̈lN )(h)


OZ̈lN

Z̈lN ,stN/ L̈lN |/ L̈lN |

Z̈lN
st-gp
(h)
N

Z̈lN ,

triv
commutative for any h ∈ H(L̈lN |Z̈lN ), where striv
with
N |L̈lN is the composite of sN
u-gp
t-gp −1∼
the natural isomorphism H(L̈lN |Z̈lN ) = H(OZ̈lN ). Then the diﬀerence sN ◦ (sN )gives us a 1-cocycle H(L̈lN |Z̈lN ) → µN (L̈lN |Z̈lN ), whose cohomology class in

H 1 (H(L̈lN |Z̈lN ), µN (L̈lN |Z̈lN )) (⊂ H 1 (Πtemp, µN (L̈lN |Z̈lN )))Ÿ
is, by construction, equal to the (mod N ) Kummer class of an l-th root Θ̈ of thetheta function, and also equal to the η̈ Θ modulo N constructed before Definition 7.14
under the natural isomorphisms l∆Θ ⊗(Z/N Z) ∼= lµlN (L̈lN | ) ∼= µN (L̈lN | ) ([EtTh,
Z̈lN

Z̈lN

Proposition 5.2 (iii)]). (cf. also Remark 7.2.1.)
 (Πtemp)Θ , l∆Θ ⊂ (Πtemp)Θ in Section 7.1
Note that the subquotients Πtemp
X
X
X
Θdetermine subquotients AutDv (S)  AutΘ
Dv (S), (l∆Θ )S ⊂ AutDv (S) for S ∈ Ob(Dv ).
As in Remark 7.6.3, Remark 7.9.1, and Remark 7.15.1, by considering the zero-divisor
and the pole-divisor (as seen in this subsection too) of the normalised theta function√Θ̈( −1)−1 Θ̈, we can category-theoretically reconstruct the lZ × µ2 -orbit of the theta
classes of standard type with µN (−)-coeﬃcient ([EtTh, Theorem 5.7]). As in the case of
the cyclotomic rigidity on mono-theta environment (Theorem 7.23 (1)), by considering
the diﬀerence of two splittings of the surjection (l∆Θ )S [µN (S)]  (l∆Θ )S , we cancategory-theoretically reconstruct the cyclotomic rigidity isomorphism(Cyc. Rig. Frd)∼(l∆Θ )S ⊗ Z/N Z → µN (S) (= lµlN (S))

for an object S of F v such that µlN (S) ∼= Z/lN Z, and (l∆Θ )S ⊗ Z/N Z ∼= Z/N Z
as abstract groups ([EtTh, Theorem 5.6]). We shall refer to this isomorphism as thecyclotomic rigidity in tempered Frobenioid.
) (⊂ AutDv (Z̈lN )) to be the image of Πtemp(Note thatWrite (H(Z¨lN ) ⊂) Im(Πtemp
YY
temp
we used Πin the definition of H(Z¨lN )) under the natural surjective outer homoŸ

morphism Πtemp
 AutDv (Z̈lN ), andXv

EN := su-gp)) · µN (L̈lN |Z̈lN ) ⊂ AutF (L̈lN |Z̈lN ).
(Im(Πtemp
NYv

Write also
temp
,
EΠ
N := EN ×Im(Πtemp ) ΠYY

182

Go Yamashita

where the homomorphism Πtemp
 Im(Πtemp) is well-defined up to Πtemp-conjugate.
Y
Y
X
Then the natural inclusions µN (L̈lN |Z̈lN ) ,→ EN and Im(Πtemp) ,→ EN induce anYisomorphism of topological groups
∼

temp
EΠ
[µN ].
N → ΠYWe write (Kv× )1/N ⊂ O× ((L̈lN |Z̈lN )birat ) for the subgroup of elements whose N th power is in the image of the natural inclusion Kv× ,→ O× ((L̈lN |Z̈lN )birat ), and we
× 1/N
write (OK):= (Kv× )1/N ∩ O× (L̈lN |Z̈lN ). Then the set of elements of O× (L̈lN |Z̈lN )v
which normalise the subgroup EN ⊂ AutF (L̈lN |Z̈lN ) is equal to the set of elements onv

which Πtemp
acts by multiplication by an element of µN (L̈lN |Z̈lN ), and it is equal to
Y
∼

× 1/N
× 1/N
×
)
. Hence we have a natural outer action of (OK)/µN (L̈lN |Z̈lN ) → OK
on
(OK
v
v
v
∼

EN , and it extends to an outer action of (Kv× )1/N /µN (L̈lN |Z̈lN ) → Kv× on EN ([EtTh,
Lemma 5.8]). On the other hand, by composing the natural outer homomorphism∼Πtemp AutDv (Z̈lN ) with su-gp, we obtain a natural outer action lZ → Πtemp/Πtemp→X
NX
Yv

Out(EN ). We write DF := hIm(Kv× ), lZi ⊂ Out(EΠN ) for the subgroup generated by×these outer actions of Kv and lZ.
We also note that st-gp: H(L̈lN |Z̈lN ) → AutF (L̈lN |Z̈lN ) factors through EN , and
N
v

→ EΠ
we write st-Π: Πtemp
N for the homomorphism induced by by taking (−)×Im(Πtemp
N
)
Ÿ
Y
Πtemp
to the homomorphism H(L̈lN |Z̈lN ) → EN . We write sΘF for the µN (L̈lN |Z̈lN ) Y
conjugacy classes of the subgroup given by the image of the homomorphism st-ΠN .
Then the tripleΘM(F v ) := (EΠN , DF , sF )reconstructs a (mod N ) mono-theta environment (We omitted the details here to verify
that this is indeed a “category-theoretic” reconstruction algorithms. In fact, in interuniversal Teichmüller theory, for holomorphic Frobenioid-theoretic objects, we can use“copies” of the model object (category), instead of categories which are equivalent to
the model object (category), and we can avoid “category-theoretic reconstruction algorithms” cf. also [IUTchI, Remark 3.2.1 (ii)]). Hence we obtain:
Theorem 8.14.
(“F 7→ M”, [EtTh, Theorem 5.10], [IUTchII, Proposition 1.2(ii)]) We have a category-theoretic algorithm to reconstruct a (mod N) mono-theta environment M(F v ) from a tempered Frobenioid F v .
Corollary 7.22 (2) reconstructs a mono-theta environment from a topological group(“Π 7→ M”) and Theorem 8.14 reconstructs a mono-theta environment from a tempered
Frobenioid (“F 7→ M”). We relate group-theoretic constructions (étale-like objects)
and Frobenioid-theoretic constructions (Frobenius-like objects) by transforming thema proof of the abc conjecture after Mochizuki

183

into mono-theta environments (and by using Kummer theory, which is available by thecyclotomic rigidity of mono-theta environment), in inter-universal Teichmüller theory,
especially, in the construction of Hodge-Arakelov-theoretic evaluation maps:†

Πv 7−→ † M →−7 † F v .

cf. Section 11.2.
§ 9.

Preliminaries on the NF Counterpart of Theta Evaluation.

§ 9.1.
Definition 9.1.

Pseudo-Monoids of κ-Coric Functions.
([IUTchI, §0])

(1) A topological space P with a continuous map P ×P ⊃ S → P is called a topologicalpseudo-monoid if there exists a topological abelian group M (we write its group
operation multiplicatively) and an embedding ι : P ,→ M of topological spaces suchthat S = {(a, b) ∈ P × P | ι(a) · ι(b) ∈ ι(P ) ⊂ M } and the restriction of the group
operation M × M → M to S gives us the given map S → P .
(2) If M is equipped with the discrete topology, we shall refer to P simply as a pseudomonoid.
(3) A pseudo-monoid is called divisible if there exist M and ι as above such that, for
any n ≥ 1 and a ∈ M , there exists b ∈ M with bn = a, and if, for any n ≥ 1 anda ∈ M , a ∈ ι(P ) if and only if an ∈ ι(P ).
(4) A pseudo-monoid is called cyclotomic if there exist M and ι as above such that,the subgroup µM ⊂ M of torsion elements of M is isomorphic to Q/Z, and ifµM ⊂ ι(P ), µM · ι(P ) ⊂ ι(P ) hold.
(5) For a cyclotomic pseudo-monoid P , write µZb (P ) := Hom(Q/Z, P ) and shall refer
to it as the cyclotome of a cycltomic pseudo-monoid P .
Definition 9.2.
([IUTchI, Remark 3.1.7]) Let Fmod be a number field, andCFmod = (EFmod \ {O})//{±1} a semi-elliptic orbicurve (cf. Section 3.1) over Fmod
which is an Fmod -core (Here, the model EFmod over Fmod is not unique in general). Let
L be Fmod or (Fmod )v for some place v of Fmod , and write CL := CFmod ×Fmod L and wewrite |CL | for the coarse scheme of the algebraic stack CL (which is isomorphic to theaﬃne line over L), and |CL | the canonical smooth compactification of |CL |. We write

184

Go Yamashita

LC for the function field of CL and take an algebraic closure LC of LC . Let L be thealgebraic closure of L in LC . We putF
if L = Fmod or L = (Fmod )v for v : non-Archimedean,
mod
•
L :=
(Fmod )v if L = (Fmod )v for v : Archimedean,
and
UL :=


L×

if L = Fmod ,O ×

if L = (Fmod )v .

L

(1) A closed point of the proper smooth curve determined by some finite subextension
of LC ⊂ LC is called a critical point if it maps to a closed point of |CL | whicharises from one of the 2-torsion points of EFmod .
(2) A critical point is called a strictly critical point if it does not map to the closed
point of |CL | which arises from the unique cusp of CL .
(3) A rational function f ∈ LC on LC is called κ-coric (κ stands for “Kummer”) if thefollowing conditions hold:(a) If f 6∈ L, then f has precisely one pole (of any order) and at least two distinct
zeroes over L.
(b) The divisor (f )0 of zeroes and the divisor (f )∞ of poles are defined over afinite extension of L• and avoid the critical points.
(c) The values of f at any strictly critical point of |CL | are roots of unity.
(4) A rational function f ∈ LC is called ∞ κ-coric, if there is a positive integer n ≥ 1such that f n is κ-coric.
(5) A rational function f ∈ LC is called ∞ κ×-coric, if there is an element c ∈ UL suchthat c · f is ∞ κ-coric.
Remark 9.2.1.
(1) A rational function f ∈ LC is κ-coric if and only if f is ∞ κ-coric(2) An ∞ κ×-coric function f ∈ LC is ∞ κ-coric if and only if the value at some strictly
critical point of the proper smooth curve determined by some finite subextension
of LC ⊂ LC containing f is a root of unity.
(3) The set of κ-coric functions (⊂ LC ) forms a pseudo-monoid. The set of ∞ κ-coricfunctions (⊂ LC ) and the set of ∞ κ×-coric functions (⊂ LC ) form divisible cyclotomic pseudo-monoids.

185

a proof of the abc conjecture after Mochizuki§ 9.2.

Cyclotomic Rigidity via κ-Coric Functions.

Let F be a number field, l ≥ 5 a prime number, XF = EF \ {O} a once-punctured
elliptic curve, and Fmod (⊂ F ) the field of moduli of XF . Write CF := XF //{±1},and K := F (EF [l]). Let C K be a smooth log-orbicurve of type (1, l-tors)± (cf. Definition 7.10) with K-core given by CK := CF ×F K. Note that CF admits a unique(up to unique isomorphism) model CFmod over Fmod , by the definition of Fmod andK-coricity of CK . Note that C K determines an orbicurve X K of type (1, l-tors) (cf.
Definition 7.10).
Let † D} be a category, which is equivalent to D} := B(C K )0 . We have an isomorphism † Π} := π1 († D} ) ∼= ΠC K (cf. Definition 6.1 (4) for π1 ((−)0 )), well-defined up to
inner automorphism.
Lemma 9.3.
([IUTchI, Remark 3.1.2] (i)) From † D} , we can group-theoreticallyreconstruct a profinite group † Π}± (⊂ † Π} ) corresponding to ΠX K .
Proof. First, we can group-theoretically reconstruct an isomorph † ∆} of ∆C K
from † Π} , by Proposition 2.2 (1). Next, we can group-theoretically reconstruct anisomorph † ∆}± of ∆X K from † ∆} as the unique torsion-free subgroup of † ∆} of index 2.
Thirdly, we can group-theoretically reconstruct the decomposition subgroups of the nonzero cusps in † ∆}± by Remark 2.9.2 (Here, non-zero cusps can be group-theoreticallygrasped as the cusps whose inertia subgroups are contained in † ∆}± ). Finally, we
can group-theoretically reconstruct an isomorph † Π}± of ΠX K as the subgroup of † Π}
generated by any of these decomposition groups and † ∆}± .
Definition 9.4.
([IUTchI, Remark 3.1.2] (ii)) From † Π} (= π1 († D} )), insteadof reconstructing an isomorph of the function field of C K directly from † Π} by Theorem 3.17, we apply Theorem 3.17 to † Π}± via Lemma 9.3 to reconstruct an isomorphof the function field of X K with † Π} /† Π}± -action. We shall refer to this procedure as† }the Θ-approach. We also write µΘb ( Π ) to be the cyclotome defined in Definition 3.13
Z
which we think of as being applied via Θ-approach.
Later, we may also use Θ-approach not only to ΠC K , but also ΠC v , ΠX , and Π Xv→−v(cf. Section 10.1 for these objects). We will always apply Theorem 3.17 to these objects
via Θ-approach (As for ΠX (resp. Π X ), see also Lemma 7.12 (resp. Lemma 7.25)).
v

Remark 9.4.1.

→−v([IUTchI, Remark 3.1.2] (iii)) The extension
ell
1 → ∆Θ → ∆Θ
X → ∆X → 1

in Section 7.1 gives us an extension class in∼ 2 ell b∼
H 2 (∆ellb (ΠX ), ∆Θ ),X , ∆Θ ) = H (∆X , Z) ⊗ ∆Θ = Hom(µZ186

Go Yamashita

which determines an tautological isomorphism∼µZb (ΠX ) → ∆Θ .
This also gives us∼µZb (ΠX ) → l∆Θ .

(Cyc. Rig. Ori. &Theta)

As already seen in Section 7, the cyclotome l∆Θ plays a central role in the theory of theétale theta function. In inter-universal Teichmüller theory, we need to use the above
tautological isormophism in the construction of Hodge-Arakelov-theoretic evaluationmap (cf. Section 11).
By applying Theorem 3.17 to † Π} (= π1 († D} )), via the Θ-approach (Definition 9.4),
we can group-theoretically reconstruct an isomorph~

M († D} )~of the field F with † Π} -action. We also write M~ († D} ) := M († D} )× , which is×an isomorph of F . We can also group-theoretically reconstruct a profinite group† ~Π (⊃ † Π} ) corresponding to ΠCFmod , by a similar way (“Loc”) as in (Step 2) of
the proof of Theorem 3.7 (We considered “Π’s over G’s” in (Step 2) of the proof of
Theorem 3.7; however, in this case, we consider “Π’s without surjections to G’s”).
Hence we obtain a morphism†

D} → † D~ := B(† Π~ )0 ,~which corresponding to C K → CFmod . Then the action of † Π} on M († Π} ) naturally
extends to an action of † Π~ . In a similar way, by using Theorem 3.17 (especially Belyi
cuspidalisations), we can group-theoretically reconstruct from † Π} an isomorph(† Π~ )rat ( † Π~ )of the absolute Galois group of the function field of CFmod in a functorial manner. By
using elliptic cuspidalisations as well, we can also group-theoretically reconstruct from† }
Π isomorphs
~
† }
~ † }
† }
M~
κ ( D ), M∞ κ ( D ), M∞ κ× ( D )of the pseudo-monoids of κ-, ∞ κ-, and ∞ κ×- coric rational functions associated with
CFmod with natural († Π~ )rat -actions (Note that we can group-theoretically reconstructevaluations at strictly critical points).

a proof of the abc conjecture after Mochizuki

187

Example 9.5.
(Global non-Realified Frobenioid, [IUTchI, Example 5.1 (i), (iii)])~
By using the field structure on M († D} ), we can group-theoretically reconstruct theset
V(† D} )~of valuations on M († D} ) with † Π~ -action, which corresponds to V(F ). Note also thatthe set†Vmod := V(† D} )/† Π~ , (resp. V(† D} ) := V(† D} )/† Π} )of † Π~ -orbits (resp. † Π} -orbits) of V(† D} ) reconstructs Vmod (resp. V(K)), and that
we have a natural bijection∼~
Prime(† Fmod
) → † Vmod

(cf. Definition 8.13 for Prime(−)). Thus, we can also reconstruct the monoid
Φ~ († D~ )(−)
on † D~ , which associates to A ∈ Ob(† D~ ) the monoid Φ~ († D~ )(A) of stack-theoretic~~arithmetic divisors on M († D} )A (⊂ M († D} )) (i.e., we are considering the coverings
over the stack-theoretic quotient (Spec OK )//Gal(K/Fmod )(∼= Spec OFmod )) with the~ † } A~ † ~gpnatural homomorphism M ( D ) → Φ ( D )(A) of monoids. Then these data~(† D~ , Φ~ († D~ ), M († D} )(−) → Φ~ († D~ )(−)gp ) determine a model Frobenioid
F ~ († D} )whose base category is † D~ . We shall refer to this as a global non-realified Frobenioid.
Let † F ~ be a pre-Frobenioid, which is isomorphic to F ~ († D} ). Suppose that weare given a morphism † D} → Base(† F ~ ) which is abstractly equivalent (cf. Section 0.2)
to the natural morphism † D} → † D~ . We identify Base(† F ~ ) with † D~ (Note that thisidentification is uniquely determined by the Fmod -coricity of CFmod and Theorem 3.17).
We write† }F := † F ~ |† D} (→ † F ~ )for the restriction of † F ~ to † D} via the natural † D} → † D~ . We also shall refer tothis as a global non-realified Frobenioid. We also write
†

~
Fmod
:= † F ~ |terminal object in † D~(⊂ † F ~ )for the restriction of † F ~ to the full subcategory consisting of the terminal object in† ~D (which corresponds to CFmod ). We also shall refer to this as a global non-realified
Frobenioid. Note that the base category of † Fmod has only one object and only one

188

Go Yamashita~
morphism. We can regard † Fmod
as the Frobenioid of (stack-theoretic) arithmetic line
bundles over (Spec OK )//Gal(K/Fmod ) (∼= Spec Fmod ). In inter-universal Teichmüller
theory, we use the global non-realified Frobenioid for converting -line bundles into-line bundles and vice versa (cf. Section 9.3 and Corollary 13.13).

Definition 9.6.
b × = {1})via Q>0 ∩ Z(∞ κ-Coric and ∞ κ×-Coric Structures, and Cyclotomic Rigidity(1) (Global case, [IUTchI, Example 5.1 (ii), (iv), (v)]) We consider O× (OA ) (which
is isomorphic to the multiplicative group of non-zero elements of a finite Galois
extension of Fmod ), varying Galois objects A ∈ Ob(† D~ ) (Here OA is a trivial line
bundle on A. cf. Definition 8.4 (1)). Then we obtain a pair†e ~×Π~ y † O

well-defined up to inner automorphisms of the pair arising from conjugation by † Π~ .
For each p ∈ Prime(Φ† F ~ (OA )), where we write Φ† F ~ for the divisor monoid of† ~F , we obtain a submonoid†

biratOp ⊂ † O× (OA),

by taking the inverse image of p ∪ {0} ⊂ Φ† F ~ (OA ) via the natural homomorphism
birat
birat
O× (OA) → Φ† F ~ (OA )gp (i.e., the submonoid of integral elements of O× (OA
)
×
birat
with respect to p). Note that the natural action of Aut† F ~ (OA ) on O (OA )permutes the Op ’s. For each p0 ∈ Prime(Φ† F ~ (OA0 )), where A0 ∈ Ob(† D~ ) is theterminal object, we obtain a closed subgroup†Πp0 ⊂ † Π~

(well-defined up to conjugation) by varying Galois objects A ∈ Ob(† D~ ), and byconsidering the elements of Aut† F ~ (OA ) which fix the submonoid † Op for systemof p’s lying over p0 (i.e., a decomposition group for some v ∈ V(Fmod )). Note thatp0 is non-Archimedean if and only if the p-cohomological dimension of † Πp0 is equal
to 2 + 1 = 3 for inifinitely many prime numbers p (Here, 2 comes from the absolute
Galois group of a local field, and 1 comes from “∆-portion (or geometric portion)” of† ~Π ). By taking the completion of † Op with respect to the corresponding valuation,
varying Galois objects A ∈ Ob(† D~ ), and considering a system of p’s lying over p0 ,we also obtain a pair†eΠp y † O0bp0

of a toplogical group acting on an ind-topological monoid, which is well-defined up
to the inner automorphisms of the pair arising from conjugation by † Πp0 (since † Πp0
is commensurably terminal in † Π~ (Proposition 2.7)).

a proof of the abc conjecture after Mochizuki

189

We write(† Π~ )rat y † M~
e ~× . Suppose that we are given isomorphs
for the above pair († Π~ )rat y † O(† Π~ )rat y † M~
∞ κ×(† Π~ )rat y † M~,
∞κ(Note that these are Frobenius-like object) of(† Π~ )rat y M~(† D} )∞ κ×(† Π~ )rat y M~(† D } )∞κ

respectively (Note that these are étale-like object) as cyclotomic pseudo-monoids
with a continuous action of († Π~ )rat . We shall refer to such a pair as an ∞ κ-coricstructure, and an ∞ κ×-coric structure on † F ~ respectively.
(† D} ), and M~
We recall that the étale-like objects M~(† D} ) are constructed
∞κ
∞ κ×
† }
† }
as subsets of ∞ H 1 ((† Π~ )rat , µΘ
H 1 (H, µΘb ( Π )) := limb ( Π )):Z
Z−→H⊂(† Π~ )rat : open
† }M~(† D} ) (resp. M~(† D} ) ) ⊂ ∞ H 1 ((† Π~ )rat , µΘb ( Π )).
∞κ
∞ κ×
Z

On the other hand, by taking Kummer classes, we also have natural injections†

M~⊂ ∞ H 1 ((† Π~ )rat , µZb († M~
)),
∞κ
∞κ

†

)),
⊂ ∞ H 1 ((† Π~ )rat , µZb († M~
M~
∞ κ×
∞ κ×

where ∞ H 1 ((† Π~ )rat , −) := limH⊂(† Π~ )rat : open H 1 (H, −). (The injectivity follows−→(† D} ) respectively.)from the corresponding injectivity for M~(† D} ) and M~∞κ∞ κ×b × -torsor, and that
Recall that the isomorphisms between two cyclotomes form a Z
κ-coric functions distinguish zeroes and poles (since it has precisely one pole (of anyb ⊃)Q>0 ∩ Z
b × = {1}, there exist
order) and at least two zeroes). Hence by (Q ⊗ Z
unique isomorphisms
∼

∼

† }
† ~
Θ † }
† ~
µΘ
b ( M∞ κ ), µZ
b ( M∞ κ× )b ( Π ) → µZ
b ( Π ) → µZ
Z

(Cyc. Rig. NF1)

characterised as the ones which induce Kummer isomorphisms
†

Kum
∼

M~
−→ M~
(† D} ),
∞κ
∞κ

†

Kum
∼

(† D } )
−→ M~
M~
∞ κ×
∞ κ×

e ~× ,
respectively. In a similar manner, for the isomorph † Π} y † M~ of † Π} y O
there exists a unique isomorphism
∼

† }
† ~
µΘ
b( M )b ( Π ) → µZ
Z

(Cyc. Rig. NF2)

characterised as the one which induces a Kummer isomorphism
†

Kum
∼

M~ −→ M~ († D} )

190

Go Yamashita

between the direct limits of cohomology modules described in (Step 4) of Theorem 3.17, in a fashion which is compatible with the integral submonoids “Op ”. We
shall refer to the isomorphism (Cyc. Rig. NF2) as the cyclotmoic rigidity via
b × = {1} (cf. [IUTchI, Example 5.1 (v)]). By the above discussions, it
Q>0 ∩ Z
follows that † F ~ always admits an ∞ κ-coric and an ∞ κ×-coric structures, which
are unique up to uniquely determined isomorphisms of pseudo-monoids with continuous actions of († Π~ )rat respectively. Thus, we regard † F ~ as being equippedwith these uniquely determined ∞ κ-coric and ∞ κ×-coric structures without notice.
We also put
†

~ rat

†

† ~ ( Π )M~,
mod := ( M )

†

~ rat

†

† ~ ( Π )M~,
κ := ( M∞ κ )† }~ † } ( Π )M~,mod ( D ) := (M ( D ))† }~ † } ( Π )M~,
κ ( D ) := (M∞ κ ( D ))
†

~ rat

where we write (−)( Π )

†

~ rat

†

~ rat

for the († Π~ )rat -invariant part.

(2) (Local non-Archimedean case, [IUTchI, Definition 5.2 (v), (vi)]) For v ∈ Vnon , let†0
Dv be a category equivalent to B temp (X v )0 (resp. B(X−→ ) ) over a finite extensionv

Kv of Qpv , where X v (resp. →X ) is a hyperbolic orbicurve of type (1, (Z/lZ)Θ )−v(Definition 7.13) (resp. of type (1, l-tors−−→) (Definition 7.24)) such that the field of
moduli of the hyperbolic curve “X” of type (1, 1) in the start of the definition of
hyperbolic orbicurve of type (1, (Z/lZ)Θ ) (resp. of type (1, l-tors−−→)) is a number field
Fmod . By Corollary 3.19, we can group-theoretically reconstruct an isomorph†Πv y Mv († Dv )of Πtemp
y OK(resp. Π X y OK
) from † Πv := π1 († Dv ).
X
v

→
−v

v

v

Let v ∈ Vmod = V(Fmod ) be the valuation lying under v. From † Πv , we cangroup-theoretically reconstruct a profinite group † Πv corresponding to C(Fmod )v by
a similar way (“Loc”) as in (Step 2) of the proof of Theorem 3.7. We write†

Dv

for B(† Πv )0 . We have a natural morphism † Dv → † Dv (This corresponds to
X v → C(Fmod )v (resp. −X → C(Fmod )v )). In a similar way, by using Theorem 3.17→v

(especially Belyi cuspidalisations), we can group-theoretically reconstruct from † Πv
an isomorph(† Πv )rat ( † Πv )a proof of the abc conjecture after Mochizuki

191of the absolute Galois group of the function field of C(Fmod )v in a functorial manner.
By using elliptic cuspidalisations as well, we can also group-theoretically reconstruct, from † Πv , isomorphsMκv († Dv ), M∞ κv († Dv ), M∞ κ×v († Dv )of the pseudo-monoids of κ-, ∞ κ-, and ∞ κ×- coric rational functions associatedwith C(Fmod )v with natural († Πv )rat -actions (Note that we can group-theoreticallyreconstruct evaluations at strictly critical points).
Let † Fv be a pre-Frobenioid isomorphic to the pv -adic Frobenioid Cv = (F v )base-fieldin Example 8.8 (resp. to the pv -adic Frobenioid Cv in Example 8.7) whose basecategory is equal to † Dv . We write(† Πv )rat y † Mv
for an isomorph of († Πv )rat y Mv († Dv ) determined by † Fv . Suppose that we are
given isomorphs(† Πv )rat y † M∞ κv , († Πv )rat y † M∞ κ×v(Note that these are Frobenius-like object) of(† Πv )rat y M∞ κv († Dv ), († Πv )rat y M∞ κ×v († Dv )(Note that these are étale-like objects) as cyclotomic pseudo-monoids with a continuous action of († Πv )rat . We shall refer to such pairs as an ∞ κ-coric structure,
and an ∞ κ×-coric structure on † Fv respectively.
We recall that the étale-like objects M∞ κv († Dv ), M∞ κ×v († Dv ) is constructed as††
subsets of ∞ H 1 ((† Πv )rat , µΘ
H 1 (H, µΘb ( Πv )) := limb ( Πv )):Z
Z−→H⊂(† Πv )rat : open†
M∞ κv († Dv ) (resp. M∞ κ×v († Dv ) ) ⊂ ∞ H 1 ((† Πv )rat , µΘb ( Πv )).
Z

On the other hand, by taking Kummer classes, we also have natural injections†

M∞ κv ⊂ ∞ H 1 ((† Πv )rat , µZb († M∞ κv )),†⊂ ∞ H 1 ((† Πv )rat , µZb († M∞ κ×v )).
M~∞ κ×(The injectivity follows from the corresponding injectivity for M∞ κv († Dv ) and M∞ κ×v († Dv )b×respectively.) Recall that the isomorphisms between two cyclotomes form a Z
torsor, and that κ-coric functions distinguish zeroes and poles (since it has preciselyb ⊃)Q>0 ∩ Z
b × = {1},one pole (of any order) and at least two zeroes). Hence by (Q⊗ Z
there exist unique isomorphisms(Cyc. Rig. NF3)

∼

∼

†
†
Θ †
†
µΘ
b ( M∞ κv ), µZ
b ( M∞ κ×v )b ( Πv ) → µZ
b ( Πv ) → µZ
Z

192

Go Yamashitacharacterised as the ones which induce Kummer isomorphisms
†

Kum
∼

M∞ κv −→ M∞ κv († Dv ),

†

Kum
∼

M∞ κ×v −→ M∞ κ×v († Dv )respectively. In a similar manner, for the isomorph † Πv y † Mv of † Πv y Mv († Dv ),there exists a unique isomorphism
∼

†
†
µΘ
b ( Mv )b ( Πv ) → µZ
Z

(Cyc. Rig. NF4)

characterised as the one which induces a Kummer isomorphism
†

Kum
∼

Mv −→ Mv († Dv )between the direct limits of cohomology modules described in (Step 4) of Theorem 3.17. We also shall refer to the isomorphism (Cyc. Rig. NF4) as the cyclotmoic
b × = {1} (cf. [IUTchI, Definition 5.2 (vi)]). By the above disrigidity via Q>0 ∩ Z
cussions, it follows that † Fv always admits an ∞ κ-coric and ∞ κ×-coric structures,
which are unique up to uniquely determined isomorphisms of pseudo-monoids with
continuous actions of († Πv )rat respectively. Thus, we regard † Fv as being equippedwith these uniquely determined ∞ κ-coric and ∞ κ×-coric structures without notice.
We also put†Mκv († Dv ) := (M∞ κv († Dv ))( Πv ) ,†where we write (−)( Πv )

rat

rat

†

†

Mκv := († M∞ κv )( Πv ) ,rat

for the († Πv )rat -invariant part.

(3) (Local Archimedean case, [IUTchI, Definition 5.2 (vii), (viii)]) For v ∈ Varc , let † Dv
be an Aut-holomorphic orbispace isomorphic to the Aut-holomorphic orbispace →X−v

associated to →X , where −X is a hyperbolic orbicurve of type (1, l-tors−→−−→) (Definivv

tion 7.24) such that the field of moduli of the hyperbolic curve “X” of type (1, 1)
in the start of the definition of hyperbolic orbicurve of type (1, l-tors−−→) is a number
field Fmod .
Let v ∈ Vmod = V(Fmod ) be the valuation lying under v. By Proposition 4.5, we
can algorithmically reconstruct an isomorph†Dvof the Aut-holomorphic orbispace Cv associated with C(Fmod )v from † Dv . We havea natural morphism † Dv → † Dv (This corresponds to −X → C(Fmod )v . Note that
→
†

∼

v

we have a natural isomorphism Aut( Dv ) → Gal(Kv /(Fmod )v ) (⊂ Z/2Z), since CK
is a K-core. Write† rat
Dv := lim(† Dv \ Σ) (→ † Dv ),←−a proof of the abc conjecture after Mochizuki

193

where we choose a projective system of († Dv \ Σ)’s which arise as universal covering
spaces of † Dv with Σ ⊃ {strictly critical points}, #Σ < ∞ (cf. Definition 9.2 forstrictly critical points). Note that † Dvrat is well-defined up to deck transformations
over † Dv . We write†
Mv († Dv ) ⊂ A Dv
for the topological submonoid of non-zero elements with norm ≤ 1 (which is an††
isomorph of OC ) in the topological field A Dv (cf. Proposition 4.5 for A Dv ). By
using elliptic cuspidalisations, we can also algorithmically reconstruct, from † Dv ,isomorphs
Mκv († Dv ), M∞ κv († Dv ), M∞ κ×v († Dv ) (⊂ Homco-hol († Dvrat , Mv († Dv )gp ) )of the pseudo-monoids of κ-, ∞ κ-, and ∞ κ×- coric rational functions associatedwith C(Fmod )v as sets of morphisms of Aut-holomorphic orbispaces from † Dvrat to†Mv († Dv )gp (= A Dv ) which are compatible with the tautological co-holomorphicisation†(Recall that A Dv has a natural Aut-holomorphic structure and a tautological coholomorphicisation (cf. Definition 4.1 (5) for co-holomorphicisation)).
†Let † Fv = († Cv , † Dv , † κv : O († Cv ) ,→ A Dv ) be a triple isomorphic to the triple(Cv , Dv , κv ) in Example 8.11, where the second data is equal to the above † Dv .
Write†
Mv := O († Cv ).
Then the Kummer structure † κv gives us an isomorphism
†

†

Kum
∼

κv : Mv → Mv († Dv )of topological monoids, to which we shall refer as a Kummer isomorphism. We
can algorithmically reconstruct the pseudo-monoids†

M∞ κv ,†

M∞ κ×vof ∞ κ-coric and ∞ κ×-coric rational functions associated to C(Fmod )v as the sets of
maps
a
† rat
† gp
Dv −→ Mv († Dv )gp
Mv (disjoint union)†
gp
which send strictly critical points to † Mgp
v , otherwise to Mv ( Dv ) , such that the` †`id (( κv )gp )−1
composite † Dvrat → Mv († Dv )gp † Mgp−−−−−−−−−−−→ Mv († Dv )gp is an elementvof M∞ κv († Dv ), M∞ κ×v († Dv ) respectively. We shall refer to them as an ∞ κ-coricstructure, and an ∞ κ×-coric structure on † Fv respectively. Note also that†Mκv (⊂ † M∞ κv ) can be reconstructed as the subset of the maps which descend to194

Go Yamashita

some † Dv \ Σ in the projective limit of † Dvrat , and are equivariant with the uniqueembedding Aut(† Dv ) ,→ Aut(A Dv ). Hence the Kummer structure † κv in † Fv
determines tautologically isomorphisms
†

†

Kum
∼

†

Mκv −→ Mκv ( Dv ),

†

Kum
∼

†

M∞ κv −→ M∞ κv ( Dv ),

†

Kum
∼

M∞ κ×v −→ M∞ κ×v († Dv )of pseudo-monoids, to which we also shall refer as Kummer isomorphisms.
Remark 9.6.1.
(Mono-anabelian Transport) The technique of mono-anabeliantransport is one of the main tools of reconstructing an alien ring structure in a scheme
theory from another (after admitting mild indeterminacies). In this occasion, we explainit.
Let † Π, ‡ Π be profinite groups isomorphic to ΠX , where X is a hyperbolic orbicurveof strictly Belyi type over non-Archimedean local field k (resp. isomorphic to ΠC K as
in this section). Then by Corollary 3.19 (resp. by Theorem 3.17 as mentioned in thissubsection), we can group-theoretically construct isomorphs O († Π), O (‡ Π) (resp.
M~ († Π), M~ (‡ Π)) of Ok (resp. F ) with † Π-, ‡ Π-action from the abstract topologicalgroups † Π, ‡ Π respectively (These are étale-like objects). Suppose that we are given
isomorphs † O , ‡ O (resp. † M~ , ‡ M~ ) of O († Π), O (‡ Π) (resp. M~ († Π), M~ (‡ Π))
respectively (This is a Frobenius-like object), and that an isomorphism † Π ∼= ‡ Π of topological groups. The topological monoids † O and ‡ O (resp. the multiplicative groups† ~M and ‡ M~ of fields) are a priori have no relation to each other, since an “isomorph”
only means an isomorphic object, and an isomorphism is not specified. However, we cancanonically relate them, by using the Kummer theory (cf. the Kummer isomorphism inRemark 3.19.2), which is available by relating two kinds of cyclotomes (i.e., cyclotomes
arisen from Frobenius-like object and étale-like object) via the cyclotomic rigidity via
b × = {1}):LCFT (resp. via Q>0 ∩ Z††Kummer∼( Π y O ) −→Frobenius-like† †( Π y O ( Π))induced by

∼
=

† Π∼
=‡ Π

‡

 ‡

Kummer∼( Π y O ( Π)) ←−étale-likeétale-like(‡ Π y † O )Frobenius-like(resp.
Kummer∼(† Π y † M~ ) −→Frobenius-like(† Π y M~ († Π))

Kummer∼induced byétale-like

∼
=

† Π∼
=‡ Π

(‡ Π y M~ (‡ Π)) ←−étale-likeIn short,†Π∼= ‡ Π,(† Π y † M~ )no relation←→a priori( ‡ Π y ‡ M~ )(‡ Π y † M~ )Frobenius-like).

195

a proof of the abc conjecture after Mochizuki

mono-anabelian⇒transport

cyclotomic rigidity

makes available⇒(† Π y † M~ )Kummer theory

canonicallyapplied⇒∼=(‡ Π y ‡ M~ ),mono-anabelian transport.

This technique is called the mono-anabelian transport.
Remark 9.6.2. (diﬀerences between three cyclotomic rigidities) We already met
three kinds of cyclotomic rigidities: the cyclotomic rigidity via LCFT (Cyc. Rig. LCFT2)
in Remark 3.19.2, of mono-theta environment (Cyc. Rig. Mono-th.) in Theorem 7.23 (1),b × = {1} (Cyc. Rig. NF2) in Definition 9.6:
and via Q>0 ∩ Z∼µZb (Gk ) → µZb (M ),†∼(l∆Θ ) ⊗ Z/N Z → µN († (l∆Θ [µN ])),

∼

† }
† ~
µΘ
b ( M ).
b ( Π ) → µZ
Z

In inter-universal Teichmüller theory, we use these three kinds of cyclotomic rigidities
to three kinds of Kummer theory respectively, and they correspond to three portions of
Θ-links, i.e.,(U) we use the cyclotomic rigidity via LCFT (Cyc. Rig. LCFT2) for the constant monoids
at local places in Vgood ∩ Vnon , which is related with the local unit (modulo torsion)
portion of the Θ-links,(V) we use the cyclotomic rigidity of mono-theta environment (Cyc. Rig. Mono-th.) for
the theta functions and their evaluations at local places in Vbad , which is relatedwith the local value group portion of the Θ-links, andb × = {1} (Cyc. Rig. NF2) for the non(G) we use the cyclotomic rigidity of via Q>0 ∩ Z
realified global Frobenioids, which is related with the global realified portion of theΘ-links.
We explain more.
(1) In Remark 9.6.1, we used † O (∼= Ok ) and as examples to explain the techniqueof mono-anabelian transport. However, in inter-universal Teichmüller theory, themono-anabelian transport using the cyclotomic rigidity via LCFT is useless in theimportant situation i.e., at local places in Vbad (However, we use it in the lessimportant situation i.e., at local places in Vgood ∩ Vnon ), because the cyclotomic
rigidity via LCFT uses essentially the value group portion in the construction, and,at places in Vbad in inter-universal Teichmüller theory, we deform the value group
portion in Θ-links! Since the value group portion is not shared under Θ-links, if

196

Go Yamashita

we use the cyclotomic rigidity via LCFT for the Kummer theory for theta functions/theta values at places in Vbad in a Hodge theatre, then the algorithm is only
valid with in the same Hodge theatre, and we cannot see it from another Hodge
theatre (i.e., the algorithm is uniradial. (cf. Remark 11.4.1, Proposition 11.15(2), and Remark 11.17.2 (2)). Therefore, the cyclotomic rigidity via LCFT is notsuitable at local places in Vbad , which deforms the value group portion.
(2) Instead, we use the cyclotomic rigidity via LCFT at local places in Vgood ∩ Vnon .
In this case too, only the unit portion is shared in Θ-links, and the value group
portion is not shared (even though the value group portion is not deformed inb × -indeterminacy to makethe case of Vgood ∩ Vnon ), thus, we ultimately admit Z
an algorithm multiradial (cf. Definition 11.1 (2), Example 11.2, and § A.4. cf.
also Remark 11.4.1, and Proposition 11.5). Mono-analytic containers, or local logb × -indeterminacy.
volumes in algorithms have no eﬀect by this Z(3) In Vbad , we use the cyclotomic rigidity of mono-theta environment for the Kummer
theory of theta functions (cf. Proposition 11.14, and Theorem 12.7). The cyclotomic
rigidity of mono-theta environment only uses µN -portion, and does not use the value
group portion! Hence the Kummer theory using the cyclotomic rigidity of monotheta environment in a Hodge theatre does not harm/aﬀect the ones in other Hodge
theatres. Therefore, these things make algorithms using the cyclotomic rigidity ofmono-theta environment multiradial (cf. also Remark 11.4.1).
×(4) In Remark 9.6.1, we used † M~ (∼= F ) and as examples to explain the techniqueof mono-anabelian transport. However, in inter-universal Teichmüller theory, we×cannot transport † M~ (∼= F ) by the technique of the mono-anabelian transport
by the following reason (cf. also [IUTchII, Remark 4.7.6]): In inter-universal Teichmüller theory, we consider ΠCF as an abstract topological group. This meansthat the subgroups ΠC K , ΠX K are only well-defined up to ΠCF -conjugacy, i.e., thesubgroups ΠC K , ΠX K are only well-defined up to automorphisms arising from theirnormalisers in ΠCF . Therefore, we need to consider these groups ΠC K , ΠX K as being subject to indeterminacies of F>l -poly-actions (cf. Definition 10.16). However,
×
×
>
† ~ ∼
Fl nontrivially acts on M (= F ). Therefore, † M~ (∼= F ) is inevitablyy subject
×
† ~ ∼
† ~
to F>
l -indeterminacies. Instead of M (= F ), we can transport the Π -invariant† ~×part † Mmod := († M~ ) Π (∼), since F>= Fmodl trivially poly-acts on it, and there is>
no Fl -indeterminacies (cf. also Remark 11.22.1).
(5) Another important diﬀerence is as follows: The cyclotomic rigidity via LCFT andof mono-theta environment are compatible with the profinite topology, i.e., it is the
projective limit of the “mod N ” levels. On the other hand, the cyclotomic rigidityb × = {1} is not compatible with the profinite topology, i.e., it has no
via Q>0 ∩ Z

a proof of the abc conjecture after Mochizuki

197× =) lim(k × ← k × ← · · · ), wesuch “mod N ” levels. In the Kummer tower (kc←−have the field structures on each finite levels k × (∪{0}); however, we have no field∑ xn
× . On the other hand, the logarithm “structure on the limit level kc
n n ” needs
field structure. Hence we need to work in “mod N ” levels to construct log-links,b × = {1} is notand the Kummer theory using the cyclotomic rigidity via Q>0 ∩ Z
compatible with the log-links. Therefore, we cannot transport global non-realified
Frobenioids under log-links. On the realified Frobenioids, we have the compatibilityof the log-volumes with log-links (i.e., the formulae (5.1) and (5.2) in Proposition 5.2
and Proposition 5.4 respectively). (Note that N -th power maps are not compatible
with addtions, hence we caanot work in a single scheme-theoretic basepoint overboth the domain and the codomain of Kummer N -th power map. This means that
we should work with diﬀerent scheme-theoretic basepoints over both the domain
and the codomain of Kummer N -th power map, hence the “isomorphism classcompatibility” i.e., the compatibility with the convention that various objects ofthe tempered Frobenioids are known only up to isomorphism, is crucial here (cf.
[IUTchII, Remark 3.6.4 (i)], [IUTchIII, Remark 2.1.1 (ii)]) (This is also related toRemark 13.13.3 (2b))).

Cyclotomic rigidity

via LCFTof mono-theta env.

b × = {1}via Q>0 ∩ Z

Related Component

units

value groupglobal realifiedof Θ-links

modulo torsion(theta values)

component

Radiality

uniradial or

multiradial

multiradialcompatible

incompatibleb × -indet.
multiradial up to Z
Compatibility withcompatible

profinite top.

§ 9.3.

-Line Bundles and -Line Bundles.

We continue to use the notation in the previous section. Moreover, we assume that
we are given a subset V ⊂ V(K) such that the natural surjection V(K)  V(Fmod )∼induces a bijection V → V(Fmod ) (Note that, as we will see in the following definitions,
we are regarding V as an “analytic section” of the morphism Spec OK  Spec OFmod ).
Write Vnon := V ∩ V(K)non and Varc := V ∩ V(K)arc .
Definition 9.7.

~([IUTchIII, Example 3.6]) We write Fmod
(i.e., without “† ”)

198

Go Yamashita

for the global non-realified Frobenioid which is constructed by the model D(C K )0 (i.e.,
without “† ”).
(1) (-line bundle) A -line bundle on (Spec OK )//Gal(K/Fmod ) is a data L =(T, {tv }v∈V ), where
×(a) T is an Fmod
-torsor, and×(b) tv is a trivialisation of the torsor Tv := T ⊗F × (Kv× /OK) for each v ∈ V,
v
mod

×
×
where Fmod
→ Kv× /OK
is the natural group homomorphism,v

satisfying the condition that there is an element t ∈ T such that tv is equal to thetrivialisation determined by t for all but finitely many v ∈ V. We can define atensor product (L )⊗n of a -line bundle L for n ∈ Z in an obvious manner.
(2) (morphism of -line bundles) Let L
1 = (T1 , {t1,v }v∈V ), L2 = (T2 , {t2,v }v∈V ) be-line bundles. An elementary morphism L
1 → L2 of -line bundles is∼
×
an isomorphism T1 → T2 of Fmod
-torsors which sends the trivialisation t1,v toan element of the OKv -orbit of t2,v (i.e., the morphism is integral at v) for eachv ∈ V. A morphism of -line bundles from L
1 to L2 is a pair of a positive⊗n→ L
integer n ∈ Z>0 and an elementary morphism (L
2 . We can define1)
a composite of morphisms in an obvious manner. Then the -line bundles on(Spec OK )//Gal(K/Fmod ) and the morphisms between them form a category (infact, a Frobenioid)
~
FMOD
.

We have a natural isomorphism
∼

~
~
Fmod
→ FMOD××of (pre-)Frobenioids, which induces the identity morphism Fmod
→ Fmodon Φ((−)birat ).
~Note that the category FMOD
is defined by using only the multiplicative () structure.

(3) (-line bundle) A -line bundle on (Spec OK )//Gal(K/Fmod ) is a data L ={Jv }v∈V , where Jv ⊂ Kv is a fractional ideal for each v ∈ V (i.e., a finitely generated
non-zero OKv -submodule of Kv for v ∈ Vnon , and a positive real multiple of OKv
for v ∈ Varc (cf. Section 0.2 for OKv )) such that Jv = OKv for finitely many v ∈ V.
We can define a tensor product (L )⊗n of a -line bundle L for n ∈ Z in an
obvious manner.
(4) (morphism of -line bundles) Let L
1 = {J1,v }v∈V , L2 = {J2,v }v∈V be -line bundles. An elementary morphism L
1 → L2 of -line bundles is an element

a proof of the abc conjecture after Mochizuki

199×f ∈ Fmodsuch that f · J1,v ⊂ J2,v (i.e., f is integral at v) for each v ∈ V. A morphism of -line bundles from L
1 to L2 is a pair of a positive integer n ∈ Z>0 and⊗nan elementary morphism (L→ L
1)2 . We can define a composite of morphisms
in an obvious manner. Then the -line bundles on (Spec OK )//Gal(K/Fmod ) andthe morphisms between them form a category (in fact, a Frobenioid)
~
Fmod
.

We have a natural isomorphism
∼

~
~
Fmod
→ Fmod××of (pre-)Frobenioids, which induces the identity morphism Fmod
→ Fmodon Φ((−)birat ).
~Note that the category Fmod
is defined by using both the multiplicative () and the
additive () structures.

Hence by combining the isomorphisms, we have a natural isomorphism
∼

~
~
Fmod
→ FMOD(Convert)××of (pre-)Frobenioids, which induces the identity morphism Fmod
→ Fmodon Φ((−)birat ).

§ 10.

Hodge Theatres.

In this section, we construct Hodge theatres after fixing an initial Θ-data (Section 10.1). More precisely, we construct Θ±ell NF-Hodge theatres (In this survey, we
shall refer to them as -Hodge theatres). We can consider Z/lZ as a finite approximation of Z for l >> 0 (Note also that we take l >> 0 approximately of order of a value ofo±height function. cf. Section ). Then we can consider F>
as a “multiplicative
l and Flfinite approximation” and an “additive finite approximation” of Z respectively. Moreover, it is important that two operations (multiplication and addition) are separated in“these finite approximations” (cf. Remark 10.29.2). Like Z/lZ is a finite approximationof Z (Recall that Z = Gal(Y/X)), a Hodge theatre, which consists of various data involved by X v , −X , C K and so on, can be seen as a finite approximation of upper half→v

plane.
Before preceeding to the detailed constructions, we briefly explain the structureof a Θ±ell NF-Hodge theatre (or -Hodge theatre). A Θ±ell NF-Hodge theatre (or a-Hodge theatre) will be obtained by “gluing” (Section 10.6)• a ΘNF-Hodge theatre, which has a F>l -symmetry, is related to a number field, ofarithmetic nature, and is used to Kummer theory for NF (In this survey, we shallrefer to it as a -Hodge theatre, Section 10.4) and200

Go Yamashita• a Θ±ell -Hodge theatre, which has a Fo±l -symmetry, is related to an elliptic curve,of geometric nature, and is used to Kummer theory for Θ (In this survey, we shallrefer to it as a -Hodge theatre, Section 10.5).

Separating the multiplicative () symmetry and the additive () symmetry is also
important (cf. [IUTchII, Remark 4.7.3, Remark 4.7.6]).

ΘNF-Hodge theatreF>l -symmetry ()arithmetic nature

Kummer theory for NF

Θ±ell -Hodge theatreFo±l -symmetry ()geometric nature

Kummer theory for Θ

As for the analogy with upper half plane, the multiplicative symmetry (resp. the additive symmetry) corresponds to supersingular points of the reduction modulo p of modular curves (resp. the cusps of the modular curves). cf. the following tables ([IUTchI,
Fig. 6.4]):
-symmetry

Basepoint

Functions(cf. Remark 10.29.1)(cf. Corollary 11.23)

upper half planez cos(t)+sin(t)z 7→ zz cos(t)−sin(t)sin(t)+cos(t) , z 7→ z sin(t)−cos(t)supersingular pts.

rat. fct. w = z−iz+iHodge theatreF>l -symm.

Bor
F>
l yV

elements of Fmod

-symmetry

Basepoint

Functions(cf. Remark 10.29.1)(cf. Corollary 11.21)

upper half planez 7→ z + a, z 7→ −z + a

cusp

trans. fct. q = e2πi

Hodge theatreFo±l -symm.

V±theta values {q j }1≤j≤l>2v

a proof of the abc conjecture after Mochizuki

201

Coric symmetry (cf. Proposition 10.34 (3))
upper half planez 7→ z, −z

Hodge theatre{±1}

These three kinds of Hodge theatres have base-Hodge theatres (like Frobenioids)
respectively, i.e., a Θ±ell NF-Hodge theatre (or a -Hodge theatre) has a base-Θ±ell NFHodge theatre (or D-Θ±ell NF-Hodge theatre, or D--Hodge theatre), which is obtained
by “gluing”• a base-ΘNF-Hodge theatre (or D-ΘNF-Hodge theatre, or D--Hodge theatre) and• a base-Θ±ell -Hodge theatre (or D-Θ±ell -Hodge theatre, or D--Hodge theatre).

A D-ΘNF-Hodge theatre (or D--Hodge theatre) consists• of three portions– (local object) a holomorphic base-(or D-)prime-strip † D> = {† D>,v }v∈V , wheregood†0∩ Vnon , or a category
D>,v is a category equivalent to B(X−→ ) for v ∈ V
vequivalent to B temp (X v )0 for v ∈ Vbad , or an Aut-holomorphic orbispace isomorphic to −X for v ∈ Varc (Section 10.3),→v– (local object) a capsule † DJ = {† Dj }j∈J of D-prime-strips indexed by J (∼=>Fl ) (cf. Section 0.2 for the term “capsule”), and– (global object) a category † D} equivalent to B(C K )0 ,• and of two base-bridges†– a base-(or D-)Θ-bridge † φΘ
> , which connects the capsule DJ of D-prime-strips
to the D-prime-strip † D> , and†– a base-(or D-)NF-bridge † φNF> , which connects the capsule DJ of D-primestrips to the global object † D} .

Here, for a holomorphic base-(or D-)prime-strip † D = {† Dv }v∈V , we can associate its
mono-analyticisation (cf. Section 3.5) † D` = {† Dv` }v∈V , which is a mono-analytic base(or D` -)prime-strip.
On the other hand, a D-Θ±ell -Hodge theatre (or D--Hodge theatre) similarly
consists

202

Go Yamashita• of three portions– (local object) a D-prime-strip † D = {† D,v }v∈V ,– (local object) a capsule † DT = {† Dt }t∈T of D-prime-strips indexed by T (∼=Fl ), and– (global object) a category † D}± equivalent to B(X K )0 ,• and of two base-bridges±†– a base-(or D-)Θ± -bridge † φΘ
± , which connects the capsule DT of D-primestrips to the D-prime-strip † D , and†– a base-(or D-)Θell -bridge † φΘ
± , which connects the capsule DT of D-primestrips to the global object † D}± .
ellHence the structure of a D-Θ±ell NF-Hodge theatre (or D-  -Hodge theatre) is as follows (For the torsor structures, Aut, and gluing see Proposition 10.20, Proposition 10.34,
Lemma 10.38, and Definition 10.39):
D-Θ±ell NF-HT(Aut = {±1}) D-Θ±ell -HT

±

φΘ
±

(t ∈ T (∼= Fl ))††ell

D-Θell -bridge

Geometric (X K

)

φΘ
±

†

gluing (>={0,})

D
O

†

D-Θ± -bridge

-Symm.

†/ † D>
O({±1}×{±1}V -torsor)DT(rigid)gluing (J=(T \{0})/{±1})†

D-ΘNF-HT (Aut = {1})φΘ>

D-Θ-bridge/ † DJ (j ∈ J (∼= F> ))

-Symm.

l(F±l -torsor)(F>l -torsor)


D}±

†

†

φNF
>


D}

D-NF-bridge(

C K ) Arithmetic

We can also draw a picture as follows (cf. [IUTchI, Fig. 6.5]):
D = / ±O>={0,}⇒

D> = />
O

±

φΘ
>

φΘ
±

± ± ±
±
{±1} y DT = /±−l> · · · /−1 /0 /1 · · · /l>

J=(T \{0})/{±1}⇒>>DJ = / >1 / · · · /l>

ell

φΘ
±



±→±

Fo±
y ↑ ↓ D}± = B(X K )0
l
±←±



φNF
>

>→>

}
0
F>
l y ↑ ↓ D = B(C K ) ,>←>a proof of the abc conjecture after Mochizuki

203

where /’s express prime-strips.
These are base Hodge theatres, and the structure of the total space of Hodge
theatres is as follows: A ΘNF-Hodge theatre (or -Hodge theatre) consists• of five portions– (local and global realified object) a Θ-Hodge theatre † HT Θ = ({† F v }v∈V , † Fmod ),which consists of∗ (local object) a pre-Frobenioid † F v isomorphic to the pv -adic Frobenioid
F v (Example 8.7) for v ∈ Vgood ∩ Vnon , or a pre-Frobenioid isomorphic to
the tempered Frobenioid F v for v ∈ Vbad (Example 8.8), or a triple † F v =(† Cv , † Dv , † κv ), isomorphic to the triple F v = (Cv , Dv , κv ) (Example 8.11)of the Archimedean Frobenioid Cv , the Aut-holomorphic orbispace Dv =X and its Kummer structure κv : O (Cv ) ,→ ADv for v ∈ Varc , and−→v∗ (global realified object with localisations) a quadruple∼†
Fmod = († Cmod , Prime(† Cmod ) → V, {† Fv` }v∈V , {† ρ`v }v∈V ) of a pre-Frobenioid
isomorphic to the global realified Frobenioid Cmod (Example 8.12), a bijec∼
tion Prime(† Cmod ) → V, a mono-analytic Frobenioid-(or F ` -)prime-strip{† Fv` }v∈V (cf. below), and global-to-local homomorphisms {† ρ`v }v∈V .
– (local object) a holomorphic Frobenioid-(or F-)prime-strip † F> = {† F>,v }v∈V ,where † F>,v is equalto the † Fv ’s in the above Θ-Hodge theatre † HT Θ .
– (local object) a capsule † FJ = {† Fj }j∈J of F-prime-strips indexed by J (∼= F> )
l(cf. Section 0.2 for the term “capsule”),– (global object) a pre-Frobenioid † F } isomorphic to the global non-realified
Frobenioid F } († D} ) (Example 9.5), and– (global object) a pre-Frobenioid † F ~ isomorphic to the global non-realified
Frobenioid F ~ († D} ) (Example 9.5).
• and of two bridgesΘ– a Θ-bridge † ψ>
, which connects the capsule † FJ of prime-strips to the primestrip † F> , and to the Θ-Hodge theatre † F> 99K † HT Θ , andNF– an NF-bridge † ψ>
, which connects the capsule † FJ of prime-strips to theglobal objects † F } 99K † F ~ .

and these objects are “lying over” the corresponding base objects.
Here, for a holomorphic Frobenioid-(or F-)prime-strip † F = {† Fv }v∈V , we canalgorithmically associate its mono-analyticisation (cf. Section 3.5) † F` = {† Fv` }v∈V ,which is a mono-analytic Frobenioid-(or F ` -)prime-strip.

204

Go Yamashita

On the other hand, a Θ±ell -Hodge theatre (or -Hodge theatre) similarly consists• of three portions– (local object) an F-prime-strip † F = {† F,v }v∈V ,– (local object) a capsule † FT = {† Ft }t∈T of F-prime-strips indexed by T (∼= Fl ),and– (global object) the same global object † D}± as in the D--Hodge theatre,• and of two bridges±Θ– a Θ± -bridge † ψ±
, which connects the capsule † FT of prime-strips to the primestrip † F , andΘ– a Θell -bridge † ψ±is equal to the D-Θell -bridge † φΘ
± ,

ell

ell

and these objects are “lying over” the corresponding base objects.
Hence the structure of a Θ±ell NF-Hodge theatre (or -Hodge theatre) is asfollows (For the torsor structures, Aut, and gluing, see Lemma 10.25, Lemma 10.37,
Lemma 10.38, and Definition 10.39):

Θ±ell NF-HT

†

HTO Θ

F -prime-strip(Aut = {±1}) Θ±ell -HT
Θ± -bridge

-Symm.

†: DT →† D }± ) Θell -bridge

Geometric†±({±1}×{±1}V -torsor)

Θ
ψ±

ell

Θ
ψ±

†

gluing (>={0,})F
O(t ∈ T (∼= Fl )) † FTell †(† φ Θ
±†/ † F>
O(rigid)gluing (J=(T \{0})/{±1})(F±l -torsor)

†

NF
ψ>


† }
F

†

Θ-bridge

-Symm.

NF-bridge

Arithmetic

D } →† D ~†Kummer for Θ

Definition 10.1.

Θψ>/ † FJ (j ∈ J (∼
= F>
l ))

(F>
l -torsor)

D}±§ 10.1.

†

ΘNF-HT (Aut = {1})F~Initial Θ-Data.

We shall refer to a collection of data
(F /F, XF , l, C K , V, Vbad
mod , )Kummer for NF

205

a proof of the abc conjecture after Mochizuki

as an initial Θ-data, if it satisfies the following conditions:√(1) F is a number field such that −1 ∈ F , and F is an algebraic closure of F . We
write GF := Gal(F /F ).
(2) XF is a once-punctured elliptic curve over F , which admits stable reduction overall v ∈ V(F )non . We write EF (⊃ XF ) for the elliptic curve over F obtaine by thesmooth compactification of XF . We also write CF := XF //{±1}, where we write“//” for the stack-theoretic quotient, and −1 is the F -involution determined by
the multiplication by −1 on EF . Let Fmod be the field of moduli (i.e., the field
generated by the j-invariant of EF over Q). We assume that F is Galois over Fmodof degree prime to l, and that 2 · 3-torsion points of EF are rational over F .
nonnon(3) Vbadmod ⊂ Vmod := V(Fmod ) is a non-empty subset of Vmod \ {v ∈ Vmod | v | 2}such that XF has bad (multiplicative in this case by the condition above) reduction
goodbad
at the places of V(F ) lying over Vbad
mod . Write Vmod := Vmod \ Vmod (Note thatbad
XF may have bad reduction at some places V(F ) lying over Vgood:=
mod ), V(F )
good
bad
good
Vmod ×Vmod V(F ), and V(F ):= Vmod ×Vmod V(F ). We also write ΠXF :=π1 (XF ) ⊂ ΠCF := π1 (CF ), and ∆XF := π1 (XF ×F F ) ⊂ ∆CF := π1 (CF ×F F ).

(4) l is a prime number ≥ 5 such that the image of the outer homomorphism GF →GL2 (Fl ) determined by the l-torsion points of EF contains the subgroup SL2 (Fl ) ⊂GL2 (Fl ). Write K := F (EF [l]), which corresponds to the kernel of the abovehomomorphism (Thus, since 3-torsion points of EF are rational, K is Galois over
Fmod by Lemma 1.7 (4). We also assume that l is not divisible by any place in
Vbad
mod , and that l does not divide the order (normalised as being 1 for a uniformiser)of the q-parameters of EF at places in V(F )bad .
(5) C F is a hyperbolic orbicurve of type (1, l-tors)± (cf. Definition 7.10) over K with
K-core given by CK := CF ×F K (Thus, C K is determined, up to K-isomorphism,
by CF by the above (4)). Let X K be a hyperbolic curve of type (1, l-tors) (cf.
Definition 7.10) over K determined, up to K-isomorphism, by C K . Recall that wehave uniquely determined open subgroup ∆X ⊂ ∆C corresponding to the hyperbolic
curve X F of type (1, l-torsΘ ) (cf. Definition 7.11), which is a finite étale covering of
C F := C F ×F F (cf. the argument after Assumption (2) in Section 7.3, where theelldecomposition ∆X ∼= ∆ × ∆Θ does not depend on the choice of ι ).
X

X(6) V ⊂ V(K) is a subset such that the composite V ⊂ V(K)  Vmod is a bijection,i.e., V is a section of the surjection V(K)  Vmod . Write Vnon := V ∩ V(K)non ,Varc := V ∩ V(K)arc , Vgood := V ∩ V(K)good , and Vbad := V ∩ V(K)bad . For a placev ∈ V, write (−)v := (−)F ×F Kv or (−)v := (−)K ×K Kv for the base change of206

Go Yamashitaa hyperbolic orbicurve over F and K respectively. For v ∈ Vbad , we assume thatthe hyperbolic orbicurve C v is of type (1, Z/lZ)± (cf. Definition 7.13) (Note that
we have “K = K̈” since 2-torsion points of EF are rational). For a place v ∈ V,
it follows that X F ×F F v admits a natural model X v over Kv , which is hyperbolic
curve of type (1, (Z/lZ)Θ ) (cf. Definition 7.13), where v is a place of F lying overv (Roughly speaking, X v is defined by taking “l-root of the theta function”). Forv ∈ Vbad , we write Πv := Πtemp.
Xv(7)  is a non-zero cusp of the hyperbolic orbicurve C K . For v ∈ V, we write v for thecusp of C v determined by . If v ∈ Vbad , we assume that v is the cusp, which arises
b via the surjection ΠX  Z
b determinedfrom the canonical generator (up to sign) of Z
temp
by the natural surjection ΠX  Z (cf. Section 7.1 and Definition 7.13). Thus, theX of type (1, l-tors
data (XK := XF ×F K, C K , ) determines a hyperbolic curve −→−−→)(cf. Definition 7.24). For v ∈ V

good

K

, we write Πv := Π X .
→−v

Note that C K and  can be regarded as “a global multiplicative subspace and acanonical generator up to {±1}”, which was one of main interests in Hodge-Arakelov
theory (cf. Appendix A). At first glance, they do not seem to be a global multiplicative
subspace and a canonical generator up to {±1}; however, by going outside the scheme
theory (Recall we cannot obtain (with finitely many exceptions) a global multiplicative
subspace within a scheme theory), and using mono-anabelian reconstruction algorithms,
they behave as though they are a global multiplicative subspace and a canonical generator up to {±1}.
From now on, we take an initial Θ-data (F /F, XF , l, C K , V, Vbad
mod , ), and fix it until
the end of Section 13.
§ 10.2.

Model Objects.

From now on, we often use the convention (cf. [IUTchI, §0]) that, for categories C, D,
we call any isomorphism class of equivalences C → D of categories an isomorphism
C → D (Note that this termniology diﬀers from the standard terminology of category
theory).
Definition 10.2.
(Local Model Objects, [IUTchI, Example 3.2, Example 3.3,
Example 3.4]) For the fixed initial Θ-data, we define model objects (i.e., without “† ”)
as follows:(1) (Dv : holomorphic, base) We write Dv for the category B temp (X v )0 of connected0
objects of the connected temperoid B temp (X v ) for v ∈ Vbad , the category B(X−→ )v

a proof of the abc conjecture after Mochizuki

207

good
of connected objects of the connected anabelioid B(X∩ Vnon , and−→ ) for v ∈ V
v

the Aut-holomorphic orbispace −X associated with −X for v ∈ Varc (cf. Section 4).
→
→
v

v

(2) (Dv` : mono-analytic, base) We write Dv` for the category B(Kv )0 of connected
objects of the connected anabelioid B(Kv ) for v ∈ Vnon , and the split monoid(O (Cv` ), spl`v ) in Example 8.11. We also write Gv := π1 (Dv` ) for v ∈ Vnon .
(3) (Cv : holomorphic, Frobenioid-theoretic) We write Cv for the base-field-theoretic hull
(F v )base-field (with base category Dv ) of the tempered Frobenioid F v in Example 8.8
for v ∈ Vbad , the pv -adic Frobenioid Cv (with base category Dv ) in Example 8.7 forv ∈ Vgood ∩ Vnon , and the Archimedean Frobenioid Cv (whose base category has
only one object Spec Kv and only one morphism) in Example 8.11 for v ∈ Varc .
(4) (F v : holomorphic, Frobenioid-theoretic) We write F v for the tempered Frobenioid
F v (with base category Dv ) in Example 8.8 for v ∈ Vbad , the pv -adic Frobenioid
Cv (with base category Dv ) in Example 8.7 for v ∈ Vgood ∩ Vnon , and the triple(Cv , Dv , κv ) of the Archimedean Frobenioid, the Aut-holomorphic orbispace, andthe Kummer structure κv : O (Cv ) ,→ ADv in Example 8.11 for v ∈ Varc .
(5) (Cv` : mono-analytic, Frobenioid-theoretic) We write Cv` for the pv -adic Frobenioid
Cv` (with base category Dv` ) in Example 8.8 for v ∈ Vbad , the pv -adic Frobenioid Cv`(with base category Dv` ) in Example 8.7 for v ∈ Vgood ∩ Vnon , and the Archimedean
Frobenioid Cv (whose base category has only one object Spec Kv and only one
morphism) in Example 8.11 for v ∈ Varc .
(6) (Fv` : mono-analytic, Frobenioid-theoretic) We write Fv` for the µ2l -split pre-Frobenioid(Cv` , spl`v ) (with base category Dv` ) in Example 8.8 for v ∈ Vbad , the split preFrobenioid (Cv` , spl`v ) (with base category Dv` ) in Example 8.7 for v ∈ Vgood ∩ Vnon ,and the triple (Cv` , Dv` , spl`v ), where (Cv` , spl`v ) is the split Archimedean Frobenioid, and Dv` = (O (Cv` ), spl`v ) is the split monoid (as above) in Example 8.11
for v ∈ Varc .
cf. the following table (We use Dv ’s (resp. Dv` ’s, resp. Fv` ’s) with v ∈ V for
D-prime-strips (resp. D` -prime-strips, F ` -prime-strips) later (cf. Definition 10.9 (1)(2)). However, we use Cv (not F v ) with v ∈ Vnon and F v with v ∈ Varc for F-primestrips (cf. Definition 10.9 (3)), and F v ’s with v ∈ V for Θ-Hodge theatres later (cf.
Definition 10.7)):

208

Go Yamashita

Vbad (Example 8.8)

Vgood ∩ Vnon (Example 8.7)

Varc (Example 8.11)

Dv

B temp (X v )0 (Πv )0B(X−→v ) (Πv )

X
−
→
v

Dv`

B(Kv )0 (Gv )B(Kv )0 (Gv )(O (Cv` ), spl`v )Cv(F v )base-field (Πv y (OF )pf )Fvv

temp. Fr’d F v (Θ-fct.)

Πv y (OF )pf
v

Arch. Fr’d Cv (ang. region)

equal to Cv(Cv , Dv , κv )Cv`

Gv y OF× · q N

Gv y OF× · pNv

equal to Cv

Fv`(Cv` , spl`v )(Cv` , spl`v )(Cv` , Dv` , spl`v )

v

v

v

We continue to define model objects.
Definition 10.3.
6.1 (v)]) We put(Model Global Objects, [IUTchI, Definition 4.1 (v), Definition
D} := B(C K )0 ,D}± := B(X K )0 .

Isomorphs of the global objects will be used in Proposition 10.19 and Proposition 10.33 to write “labels” on each local objects in a consistent manner (cf. also Remark 6.11.1). We will use D} for (D-)-Hodge theatre (Section 10.4), and D}± for(D-)-Hodge theatre (Section 10.5).
Definition 10.4.
(Model Global Realified Frobenioid with Localisations, [IUTchI,
Example 3.5]) Let Cmod be the global realified Frobenioid in Example 8.12. Note that∼
we have the natural bijection Prime(Cmod ) → Vmod , and an element log`mod (pv ) ∈ ΦCv ,vfor each v ∈ Vmod . For v ∈ Vmod , we write v ∈ V for the corresponding element under∼the bijection V → Vmod . For each v ∈ V, we also have the (pre-)Frobenioid Cv` (cf.
Definition 10.2 (5)). We write Cv`R for the realification of Cv` (Definition 8.4 (3)) forv ∈ Vnon , and Cv itself for v ∈ Varc . We write logΦ (pv ) ∈ ΦR
C ` for the element deterv`R
mined by pv , where we write ΦR
Cv` for the divisor monoid of Cv . We have the natural
restriction functor

Cmod → Cv`R

209

a proof of the abc conjecture after Mochizuki

for each v ∈ V. This is determined, up to isomorphism, by the isomorphism
ρv : ΦCmod ,v

gl. to loc.
∼−→

log`mod (pv ) 7→

ΦR
Cv`

1
logΦ (pv )[Kv : (Fmod )v ]of topological monoids (For the assignment, consider the volume interpretations of thearithmetic divisors, i.e., logpv #(O(Fmod )v /pv ) = [Kv :(F1mod )v ] logpv #(OKv /pv )). Recall
also the point of view of regarding V(⊂ V(K)) as an “analytic section” of Spec OK 
Spec OFmod (The left-hand side ΦCmod ,v is an object on (Fmod )v , and the right-hand sideΦRC ` is an object on Kv ). We write Fmod for the quadruplev∼

Fmod := (Cmod , Prime(Cmod ) → V, {Fv` }v∈V , {ρv }v∈V )of the global realified Frobenioid, the bijection of primes, the model objects Fv` ’s in
Definition 10.2 (6), and the localisation homomorphisms. We define an isomorphism∼
Fmod,1 → Fmod,2 of quadruples in an obvious manner.
Isomorphs of the global realified Frobenioids are used to consider log-volume functions.
Definition 10.5.
(Θ-version, [IUTchI, Example 3.2 (v), Example 3.3 (ii), Example 3.4 (iii), Example 3.5 (ii)])(1) (Vbad ) Let v ∈ Vbad . We write DvΘ (⊂ Dv ) for the category whose objects areAΘ := A × Ÿ v for A ∈ Ob(Dv` ), where × is the product in Dv , and morphisms aremorphisms over Ÿ v in Dv (Note also that Ÿ v ∈ Ob(Dv ) is defined over Kv ). Taking∼“(−) × Ÿ v ” induces an equivalenc Dv` → DvΘ of categories. The assignmentbirat
Ob(DvΘ ) 3 AΘ 7→ O× (OAΘ ) · (ΘN|) ⊂ O× (OA
Θ )
v OAΘ

determines a monoid OCΘ (−) on DvΘ (cf. Example 8.8 for Θv ∈ O× (OŸbirat ), andv∼v

O(−) for Definition 8.4 (1)). Under the above equivalence Dv` → DvΘ of categories,∼
we have natural isomorphism OC` (−) → OCΘ (−). These are compatible with thevv
assignmentq |OA 7→ Θv |OAΘv∼

and a natural isomorphism O× (OA ) → O× (OAΘ ) induced by the projection AΘ =A × Ÿ v → A (cf. Example 8.8 for q ∈ O (OX )). Hence the monoid OCΘ (−)
v

determines a pv -adic Frobenioid
CvΘ (⊂ F birat
)
v

v

v

210

Go Yamashita

whose base category is DvΘ . Note also Θv determines a µ2l (−)-orbit of splittings
∼

Θ
`
Θ
splΘ
v of Cv . We have a natural equivalence Cv → Cv of categories, which sends
spl`v to splΘv , hence we have an isomorphism∼

Fv` (= (Cv` , spl`v )) → FvΘ := (CvΘ , splΘv)of µ2l -split pre-Frobenioids.
(2) (Vgood ∩ Vnon ) Let v ∈ Vgood ∩ Vnon . Recall that the divisor monoid of Cv` is of
the form OC` (−) = OC×` (−) × N log(pv ), where we write log(pv ) for the element pv
v
v
considered additively. We putOCΘ (−) := OC×Θ (−) × N log(pv ) log(Θ),
vvwhere log(pv ) log(Θ) is just a formal symbol. We have a natural isomorphism∼OC` (−) → OCΘ (−). Then the monoid OCΘ (−) determines a pv -adic Frobenioid
v

v

v

CvΘ
whose base category is DvΘ := Dv` . Note also that log(pv ) log(Θ) determines aΘΘ` ∼splitting splΘv of Cv . We have a natural equivalence Cv → Cv of categories, which`Θ
sends splv to splv , hence we have an isomorphism∼

Fv` (= (Cv` , spl`v )) → FvΘ := (CvΘ , splΘv)of split pre-Frobenioids.
(3) (Varc ) Let v ∈ Varc . Recall that the image ΦCv` of spl`v of the split monoid (OC` , spl`v )v

is isomorphic to R≥0 . We write log(pv ) ∈ ΦCv` for the element pv considered additively (cf. Section 0.2 for pv with Archimedean v). We put
ΦCvΘ := R≥0 log(pv ) log(Θ),
where log(pv ) log(Θ) is just a formal symbol. We also write OC×` := (OC` )× , andOC×Θ := OC×` . Then we obtain a split pre-Frobenioid
v

v

v

v

(CvΘ , splΘv ),∼such that O (CvΘ ) = OC×Θ × ΦCvΘ . We have a natural equivalence Cv` → CvΘ of
v

∼

`
`
categories, which sends spl`v to splΘv , hence we have an isomorphism (Cv , splv ) →(CvΘ , splΘv ) of split pre-Frobenioids, and an isomorphism∼

Fv` (= (Cv` , Dv` , spl`v )) → FvΘ := (CvΘ , DvΘ , splΘv)of triples, where we write DvΘ := Dv` .

211

a proof of the abc conjecture after Mochizuki(4) (Global Realified with Localisations) Let Cmod be the global realified Frobenioid
considered in Definition 10.4. For each v ∈ Vmod , we write v for the corresponding∼
element under the bijection V → Vmod . WriteΦCtheta := ΦCmod · log(Θ),
where log(Θ) is just a formal symbol. This monoid ΦCtheta determines a global
realified Frobenioid
Ctheta∼with a natural equivalence Cmod → Ctheta of categories and a natural bijection∼
Prime(Ctheta ) → Vmod . For each v ∈ Vmod , the element log`mod (pv ) ∈ ΦCmod ,v ⊂ΦCmod determines an element log`mod (pv ) log(Θ) ∈ ΦCtheta ,v ⊂ ΦCtheta . As in the casewhere Cmod , We have the natural restriction functor
Ctheta → CvΘR

for each v ∈ V. This is determined, up to isomorphism, by the isomorphism
gl. to loc.
 [Kv :(F1mod )v ] logΦ (pv ) log(Θ) v ∈ Vgood ,∼`RΘlogΦ (Θ )ρv : ΦCtheta ,v −→ ΦCvΘ logmod (pv ) log(Θ) 7→
logΦ (pv )v
v ∈ Vbad [Kv :(Fmod )v ] log (q )Φvof topological monoids, where we write logΦ (pv ) log(Θ) ∈ ΦR
C Θ for the element dev

termined by logΦ (pv ) for v ∈ Vgood , and logΦ (Θv ), logΦ (pv ), and we write logΦ (q )v

for the element determined by Θv , pv , and q respectively for v ∈ V

badv(Note thatlogΦ (Θv ) is not a formal symbol). Note that for any v ∈ V, the localisation homo∼morphisms ρv and ρΘv are compatible with the natural equivalences Cmod → Ctheta ,` ∼Θand Cv → Cv :log`mod (pv ) _/ log`mod (pv ) log(Θ)_“mod→theta”

ρv



1
[Kv :(Fmod )v ] logΦ (pv )


/



ρΘ
v

“`→Θ”

1
[Kv :(Fmod )v ] logΦ (pv ) log(Θ)

for v ∈ Vgood , andlog`mod (pv ) _“mod→theta”

ρv



1
[Kv :(Fmod )v ] logΦ (pv )/ log`mod (pv ) log(Θ)
_

/



“`→Θ”

ρΘ
v

logΦ (Θ )logΦ (pv )v[Kv :(Fmod )v ] logΦ (q )v

212

Go Yamashita

for v ∈ Vbad . We write Ftheta for the quadruple∼

Ftheta := (Ctheta , Prime(Ctheta ) → V, {FvΘ }v∈V , {ρΘv }v∈V )of the global realified Frobenioid, the bijection of primes, the Θ-version of model
objects FvΘ ’s in (1), (2), and (3), and the localisation homomorphisms.
Note that we have group-theoretic or category-theoretic reconstruction algorithms
such as reconstructing Dv` from Dv . We summarise these as follows ([IUTchI, Example
3.2 (vi), Example 3.3 (iii)]):F F v

 except //>Cv Varc D_v y
~
~~
~
~~

~~

/ Cv` / Dv`Fv`

up to lZ-indet.
on Θ for v∈Vbad
v



FvΘ 


/ DvΘ .

/ CvΘ 

(Note also the remark given just before Theorem 8.14.)
Definition 10.6.
(iii)]) We write(D-version or “log-shell version”, [IUTchI, Example 3.5 (ii),
Dmod∼

for a copy of Cmod . Let ΦDmod , Prime(Dmod ) → Vmod , logD
mod (pv ) ∈ ΦDmod ,v ⊂ ΦDmod∼

be the corresponding objects under the tautological equivalence Cmod → Dmod . For each∼v ∈ Vmod , we write v for the corresponding element under the bijection V → Vmod .
For v ∈ Vnon , we can group-theoretically reconstruct from Dv`(R`≥0 )v := Rnon (Gv ) (∼= R≥0 )and Frobenius element F(Gv ) ∈ (R`≥0 )v by (Step 3) in Proposition 5.2 (Recall that
Gv = π1 (Dv` )). Write also`logD
Φ (pv ) := ev F(Gv ) ∈ (R≥0 )v ,where we write ev for the absolute ramification index of Kv .
For v ∈ Varc , we can also group-theoretically reconstruct from the split monoid
Dv` = (OC` , spl`v )v(R`≥0 )v := Rarc (Dv` ) (∼= R≥0 )

a proof of the abc conjecture after Mochizuki

213

and Frobenius element F(Dv` ) ∈ (R`≥0 )v by (Step 4) in Proposition 5.4. Write alsologD
Φ (pv ) :=F(Dv` )∈ (R`≥0 )v ,2π

where 2π ∈ R× is the length of the perimeter of the unit circle (Note that (R`≥0 )v hasa natural R× -module structure).
Hence for any v ∈ V, we obtain a uniquely determined isomorphism
ρDv : ΦDmod ,v

gl. to loc.
∼

−→

logD
mod (pv ) 7→

(R`≥0 )v

1logD
Φ (pv )
[Kv : (Fmod )v ]of topological monoids.
We write FD for the quadruple∼

FD := (Dmod , Prime(Dmod ) → V, {Dv` }v∈V , {ρDv }v∈V )of the global realified Frobenioid, the bijection of primes, the D` -version of model objects
Dv` ’s, and the localisation homomorphisms.
§ 10.3.
Definition 10.7.
atre is a collection

Θ-Hodge Theatres and Prime-strips.
(Θ-Hodge theatre, [IUTchI, Definition 3.6]) A Θ-Hodge the†HT Θ = ({† F v }v∈V , † Fmod ),where(1) (local object) † F v is a pre-Frobenioid (resp. a triple († Cv , † Dv , † κv )) isomorphic to
the model F v (resp. isomorphic to the model triple F v = (Cv , Dv , κv )) in Definition 10.2 (4) for v ∈ Vnon (resp. for v ∈ Varc ). We write † Dv , † Dv` , † DvΘ , † Fv` , † FvΘ
(resp. † Dv` , † DvΘ , † Fv` ,† FvΘ ) for the objects algorithmically reconstructed from † F v
corresponding to the model objects (i.e., the objects without † ).
(2) (global realified object with localisations) † Fmod is a quadruple∼`(† Cmod, Prime(† Cmod ) → V, {† Fv` }v∈V , {† ρv }v∈V ),∼``where † Cmod
is a category equivalent to the model Cmodin Definition 10.4, Prime(† Cmod ) →V is a bijection of sets, † Fv` is the reconstructed object from the above local data†

F v , and † ρv : Φ† Cv ,vgl. to loc.
∼

−→

ΦR
† C ` is an isomorphism of topological monoids (Herev† `Cv is the reconstructed object from the above local data † F v ), such that there ex∼
ists an isomorphism of quadruples † Fmod → Fmod . We write † Ftheta , † FD for thealgorithmically reconstructed object from † Fmod corresponding to the model objects†(i.e., the objects without ).

214

Go Yamashita

Definition 10.8.

(Θ-link, [IUTchI, Corollary 3.7 (i)]) Let † HT Θ = ({† F v }v∈V , † Fmod ),‡HT Θ = ({‡ F v }v∈V , ‡ Fmod ) be Θ-Hodge theatres (with respect to the fixed initial Θdata). We shall refer to the full poly-isomorphism (cf. Section 0.2)†

Fthetafull poly
∼
‡

−→

Fmod

as the Θ-link from † HT to ‡ HT (Note that the full poly-isomorphism is non-empty),
and we write it as†ΘHT Θ −→‡HT Θ ,

and we shall refer to this diagram as the Frobenius-picture of Θ-Hodge theatres([IUTchI, Corollary 3.8]). Note that the essential meaning of the above link is∼“ ΘN−→ q N ”vv

for v ∈ Vbad .
Remark 10.8.1.

([IUTchI, Corollary 3.7 (ii), (iii)])(1) (Preservation of D` ) For each v ∈ V, we have a natural composite full polyisomorphism†∼
Dv` → † DvΘ

full poly
∼
‡

−→

Dv` ,

where the first isomorphism is the natural one (Recall that it is tautological forv ∈ Vgood , and that it is induced by (−)× Ÿ v for v ∈ Vbad ), and the second full polyisomorphism is the full poly-isomorphism of the Θ-link. Hence the mono-analytic
base “Dv` ” is preserved (or “shared”) under the Θ-link (i.e., Dv` is horizontally coric).
Note that the holomorphic base “Dv ” is not shared under the Θ-link (i.e., Θ-link
shares the underlying mono-analytic base structures, but not the arithmeticallyholomorphic base structures).
(2) (Preservation of O× ) For each v ∈ V, we have a natural composite full polyisomorphism∼
O†×C ` → O†×C Θvvfull poly∼−→ O‡×C ` ,vwhere the first isomorphism is the natural one (Recall that it is tautological forv ∈ Vgood , and that it is induced by (−) × Ÿ v for v ∈ Vbad ), and the second fullpoly-isomorphism is induced by the full poly-isomorphism of the Θ-link. Hence“OC×` ” is preserved (or “shared”) under the Θ-link (i.e., OC×` is horizontally coric).
v
v
Note also that the value group portion is not shared under the Θ-link.

a proof of the abc conjecture after Mochizuki

215

We can visualise the “shared” and “non-shared” relation as follows:†

Dv − − >

(

†

) (
)
×
‡ `
Dv` y O†×C ` ∼DyO> − − ‡ Dv
=
‡ C`
v
v

v

We shall refer to this diagram as the étale-picture of Θ-Hodge theatres ([IUTchI,
Corollary 3.9]). Note that, there is the notion of the order in the Frobenius-picture (i.e.,†(−) is on the left, and ‡ (−) is on the right), on the other hand, there is no such anorder and it has a permutation symmetry in the étale-picture (cf. also the last table in
Section 4.3).
This Θ-link is the primitive one. We will update the Θ-link to Θ×µ -link, Θ×µgau -link×µ×µ(cf. Corollary 11.24), and ΘLGP -link (resp. Θlgp -link) (cf. Definition 13.9 (2)) in interuniversal Teichmüller theory :Θ-link“Hodge-Arakelov-theoretic eval.”“theta fct.7→theta values”
and O × 7→O × /µ

Definition 10.9.
(iii), (iv)])

Θ×µgau -link“ log -link”

×µΘ×µLGP -link (resp. Θlgp -link).

([IUTchI, Definition 4.1 (i), (iii), (iv) Definition 5.2 (i), (ii),(1) (D : holomorphic, base) A holomorphic base-prime-strip, or D-prime-strip isa collection†

D = {† Dv }v∈Vof data such that † Dv is a category equivalent to the model Dv in Definition 10.2
(1) for v ∈ Vnon , and † Dv is an Aut-holomorphic orbispace isomorphic to the model
Dv in Definition 10.2 (1). A morphism of D-prime-strips is a collection ofmorphisms indexed by V between each component.
(2) (D` : mono-analytic, base) A mono-analytic base-prime-strip, or D ` -primestrip is a collection†D` = {† Dv` }v∈Vof data such that † Dv` is a category equivalent to the model Dv` in Definition 10.2(2) for v ∈ Vnon , and † Dv` is a split monoid isomorphic to the model Dv` in Definition 10.2 (2). A morphism of D ` -prime-strips is a collection of morphisms
indexed by V between each component.

216

Go Yamashita(3) (F : holomorphic, Frobenioid-theoretic) A holomorphic Frobenioid-prime-strip,
or F -prime-strip is a collection†

F = {† Fv }v∈Vof data such that † Fv is a pre-Frobenioid isomorphic to the model Cv (not F v ) in

Definition 10.2 (3) for v ∈ Vnon , and † Fv = († Cv , † Dv , † κv ) is a triple of a category,an Aut-holomorphic orbispace, and a Kummer structure, which is isomorphic to
the model F v in Definition 10.2 (3). An isomorphism of F -prime-strips is acollection of isomorphisms indexed by V between each component.
(4) (F ` : mono-analytic, Frobenioid-theoretic) A mono-analytic Frobenioid-primestrip, or F ` -prime-strip is a collection† `F = {† Fv` }v∈Vof data such that † Fv` is a µ2l -split pre-Frobenioid (resp. split pre-Frobenioid)
isomorphic to the model Fv` in Definition 10.2 (6) for v ∈ Vbad (resp. v ∈ Vgood ∩Vnon ), and † Fv` = († Cv` , † Dv` , † spl`v ) is a triple of a category, a split monoid, and a
splitting of † Cv , which is isomorphic to the model Fv` in Definition 10.2 (6). An
isomorphism of F ` -prime-strips is a collection of isomorphisms indexed by V
between each component.
(5) (F : global realified with localisations) A global realified mono-analytic Frobenioidprime-strip, or F -prime-strip is a quadruple†

where † C∼

F = († C , Prime(† C ) → V, † F` , {† ρv }v∈V ),`is a pre-Frobenioid isomorphic to the model Cmodin Definition 10.4,∼

gl. to loc.
∼

Prime(† C ) → V is a bijection of sets, † F` is an F ` -prime-strip, and † ρv : Φ† C ,v −→† `ΦR
Cv is the object recon† C ` is an isomorphism of topological monoids (Here,v

structed from † Fv` ), such that the quadruple † F is isomorphic to the model Fmod
in Definition 10.4. An isomorphism of F -prime-strips is an isomorphism ofquadruples.

(6) Let AutD (−), IsomD (−, −) (resp. AutD` (−), IsomD` (−, −) resp. AutF (−), IsomF (−, −)resp. AutF ` (−), IsomF ` (−, −) resp. AutF (−), IsomF (−, −)) be the group of
automorphisms of a D-(resp. D` -, resp. F-, resp. F ` -, resp. F -)prime-strip,
and the set of isomorphisms between D-(resp. D` -, resp. F-, resp. F ` -, resp.
F -)prime-strips.

a proof of the abc conjecture after Mochizuki

217

Remark 10.9.1. We use global realified prime-strips with localisations for calculating (group-theoretically reconstructed) local log-volumes (cf. Section 5) with the global
product formula. Another necessity of global realified prime-strips with localisations is
as follows: If we were working only with the various local Frobenioids for v ∈ V (which
are directly related to computations of the log-volumes), then we could not distinguish,
nonfor example, pm, since the isomorphism ofv OKv from OKv with m ∈ Z for v ∈ V
these Frobenioids arising from (the updated version of) Θ-link preserves only the isomorphism classes of objects of these Frobenioids. By using global realified prime-strips
with localisations, we can distinguish them (cf. [IUTchIII, (xii) of the proof of Corollary3.12]).
Note that we can algorithmically associate D` -prime-strip † D` to any D-primestrip † D and so on. We summarise this as follows (cf. also [IUTchI, Remark 5.2.1 (i),(ii)]):
Θ 
† 
†
/ †D
HT
8/ _F
_
_
x
xx
x
x
xx
 {xx



†
† ` 
† `
/
/
F
F
D .
Lemma 10.10.

([IUTchI, Corollary 5.3, Corollary 5.6 (i)])(1) Let 1 F ~ , 2 F ~ (resp. 1 F } , 2 F } ) be pre-Frobenioids isomorphic to the global nonrealifed Frobenioid † F ~ (resp. † F } ) in Example 9.5 , then the natural map
Isom(1 F ~ , 2 F ~ ) → Isom(Base(1 F ~ ), Base(2 F ~ ))(resp. Isom(1 F } , 2 F } ) → Isom(Base(1 F } ), Base(2 F } )) )is bijective.
(2) For F-prime-strips 1 F, 2 F, whose associated D-prime-strips are 1 D, 2 D respectively,
the natural map
IsomF (1 F, 2 F) → IsomD (1 D, 2 D)
is bijective.
(3) For F ` -prime-strips 1 F` , 2 F` , whose associated D` -prime-strips are 1 D` , 2 D`
respectively, the natural map
IsomF ` (1 F` , 2 F` ) → IsomD` (1 D` , 2 D` )is bijective.

218

Go Yamashita(4) For v ∈ Vbad , let F v be the tempered Frobenioid in Example 8.8, whose base category
is Dv then the natural map
Aut(F v ) → Aut(Dv )is bijective.
(5) For T h-Hodge theatres 1 HT Θ , 2 HT Θ , whose associated D-prime-strips are 1 D> ,2
D> respectively, the natural map
Isom(1 HT Θ , 2 HT Θ ) → IsomD (1 D> , 2 D> )is bijective.
Proof. (1) follows from the category-theoretic construction of the isomorphism∼
M~ († D} ) → † M~ in Example 9.5. (2) follows from the mono-anabelian reconstruction
algorithms via Belyi cuspidalisation (Corollary 3.19), and the Kummer isomorphism
in Remak 3.19.2) for v ∈ Vnon , and the definition of the Kummer structure for Autholomorphis orbispaces (Definition 4.6) for v ∈ Varc . (3) follows from Proposition 5.2
and Proposition 5.4. We show (4). By Theorem 3.17, automorphisms of Dv arises fromautomorphisms of X v , thus, the surjectivity of (4) holds. To show the injectivity of(4), let α be in the kernel. Then it suﬃces to show that α induces the identity on therational functions and divisor monoids of F v . By the category-theoretic reconstructionof cyclotomic rigidity (cf. isomorphism (Cyc. Rig. Frd)) and the naturality of Kummer
map, (which is injective), it follows that α induces the identity on the rational functionsof F v . Since α preserves the base-field-theoretic hull, α also preserves the non-cuspidal
portion of the divisor of the Frobenioid-theoretic theta function and its conjugate (theseare preserved by α since we already show that α preserves the rational function monoidof F v ), hence α induces the identity on the non-cuspidal elements of the divisor monoidof F v . Similary, since any divisor of degree 0 on an elliptic curve supported on thetorsion points admits a positive multiple which is principal, it follows that α induces
the identityo on the cuspidal elements of the divisor monoid of F v as well. by considering
the cuspidal portions of divisor of a suitable rational functions (these are preserved byα since we already show that α preserves the rational function monoid of F v ). (Notethat we can simplify the proof by suitably adding F v more data, and considering theisomorphisms preserving these data. cf. also the remark given just before Theorem 8.14
and [IUTchI, Remark 3.2.1 (ii)]). (5) follows from (4).
Remark 10.10.1. ([IUTchI, Remark 5.3.1]) Let 1 F, 2 F be F-prime-strips, whose
associated D-prime-strips are 1 D, 2 D respectively. Let
φ : 1D → 2D

a proof of the abc conjecture after Mochizuki

219

be a morphism of D-prime-strips, which is not necessarily an isomorphism, such that allof the v(∈ Vgood )-components are isomorphisms, and the induced morphism φ` : 1 D` →2 `D on the associated D` -prime-strips is also an isomorphism. Then φ uniquely lifts toan “arrow”
ψ : 1 F → 2 F,
which we say that ψ is lying over φ, as follows: By pulling-back (or making categoricalfiber products) of the (pre-)Frobenioids in 2 F via the various v(∈ V)-components of φ,
we obtain the pulled-back F-prime-strip φ∗ (2 F) whose associated D-prime-strip is tautologically equal to 1 D. Then this tautological equality uniquely lifts to an isomorphism∼
1
F → φ∗ (2 F) by Lemma 10.10 (2):1∼

/ φ∗ (2 F)
FF
_
 FF
FF
FF
FF 
"
1
D

/ F
_

pull back 2φ/ 2 D.

Definition 10.11.
([IUTchI, Definition 4.1 (v), (vi), Definition 6.1 (vii)]) Let† }±D (resp. D ) is a category equivalent to the model global object D} (resp. D}± )in Definition 10.3.

†}(1) Recall that, from † D} (resp. † D}± ), we can group-theoretically reconstruct a set
V(† D} ) (resp. V(† D}± )) of valuations corresponding to V(K) by Example 9.5 (resp.
in a slimilar way as in Example 9.5, i.e., firstly group-theoretically reconstructing
an isomorph of the field F from π1 († D}± ) by Theorem 3.17 via the Θ-approach(Definition 9.4), secondly group-theoretically reconstructing an isomorph V(† D}± )of V(F ) with π1 († D}± )-action, by the valuations on the field, and finally consider
the set of π1 († D}± )-orbits of V(† D}± )).
For w ∈ V(† D} )arc (resp. w ∈ V(† D}± )arc ), by Proposition 4.8 and Lemma 4.9, we
can group-theoretically reconstruct, from † D} (resp. † D}± ), an Aut-holomorphic
orbispaceC(† D} , w) (resp. X(† D}± , w) )corresponding to C w (resp. X w ). For an Aut-holomorphic orbispace U, a morphism
U → † D} (resp. U → † D}± )is a morphism of Aut-holomorphic orbispaces U → C(† D} , w) (resp. U → X(† D}± , w))
for some w ∈ V(† D} )arc (resp. w ∈ V(† D}± )arc ).
(2) For a D-prime-strip † D = {† Dv }v∈V , a poly-morphism†poly

D −→ † D}

poly(resp. † D −→ † D}± )220

Go Yamashita
poly

poly

is a collection of poly-morphisms {† Dv −→ † D} }v∈V (resp. {† Dv −→ † D}± }v∈V )indexed by v ∈ V (cf. Definition 6.1 (5) for v ∈ Vnon , and the above definition in(1) for v ∈ Varc ).
(3) For a capsule E D = {e D}e∈E of D-prime-strips and a D-prime-strip † D, a polymorphismE

poly

D −→ † D}

poly(resp. E D −→ † D}± ,poly

resp. E D −→ † D )

poly

poly

is a collection of poly-morphisms {e D −→ † D} }e∈E (resp. {e D −→ † D}± }e∈E ,polyresp. {e D −→ † D}e∈E ).
Definition 10.12.
([IUTchII, Definition 4.9 (ii), (iii), (iv), (v), (vi), (vii), (viii)])‡ `Let F = { Fv }v∈V be an F ` -prime-strip with associated D` -prime-strip ‡ D` ={‡ Dv` }v∈V .
‡ `(1) Recall that ‡ Fv` is a µ2l -split pre-Frobenioid (resp. a split pre-Frobenioid, resp. a
triple (‡ Cv` , ‡ Dv` , ‡ spl`v )) for v ∈ Vbad (resp. v ∈ Vgood ∩ Vnon , resp. v ∈ Varc ). Let‡
A∞ be a universal covering pro-object of ‡ Dv` , and write ‡ G := Aut(‡ A∞ ) (hence‡
G is a profinite group isomorphic to Gv ). For v ∈ Vbad (resp. v ∈ Vgood ∩ Vnon ),we writeO⊥ (‡ A∞ ) (⊂ O (‡ A∞ ))for the submonoid generated by µ2l (‡ A∞ ) and the image of the splittings on ‡ Fv`(resp. the submonoid determined by the image of the splittings on ‡ Fv` ), and put
OI (‡ A∞ ) := O⊥ (‡ A∞ )/µ2l (‡ A∞ ) (resp. OI (‡ A∞ ) := O⊥ (‡ A∞ ) ),andOI×µ (‡ A∞ ) := OI (‡ A∞ )×O×µ (‡ A∞ ) (resp. OI×µ (‡ A∞ ) := OI (‡ A∞ )×O×µ (‡ A∞ ) ).
These are equipped with natural ‡ G-actions.
Next, for v ∈ Vnon , we can group-theoretically reconstruct, from ‡ G, ind-topologicalmodules ‡ G y O× (‡ G), ‡ G y O×µ (‡ G) with G-action, by Proposition 5.2 (Step 1)
b × -orbit(cf. Definition 8.5 (1)). Then by Definition 8.5 (2), there exists a unique Z
of isomorphisms
‡ `×
κv

× ‡

poly
∼

: O ( G) → O× (‡ A∞ )of ind-topological modules with ‡ G-actions. Moreover, ‡ κ`×v induces an Isomet-orbit
‡ `×µ
κv

poly
∼

: O×µ (‡ G) → O×µ (‡ A∞ )

a proof of the abc conjecture after Mochizuki

221

of isomorphisms.
For v ∈ Vnon , the rational function monoid determined by OI×µ (‡ A∞ )gp with ‡ Gaction and the divisor monoid of ‡ Fv` determine a model Frobenioid with a splitting.
The Isomet-orbit of isomorphisms ‡ κ`×µdetermines a ×µ-Kummer structure (Defvinition 8.5 (2)) on this model Frobenioid. For v ∈ Vnon (resp. v ∈ Varc ), we write‡

Fv`I×µfor the resulting split-×µ-Kummer pre-Frobenioid (resp. the collection of data obtained by replacing the split pre-Frobenioid ‡ Cv in ‡ Fv` = (‡ Cv` , ‡ Dv` , ‡ spl`v ) by the
inductive system, indexed by the multiplicative monoid N≥1 , of split pre-Frobenioids
obtained from ‡ Cv` by taking the quotients by the N -torsions for N ∈ N≥1 . Thus,
the units of the split pre-Frobenioids of this inductive system give rise to an inductive system · · ·  O×µN (A∞ )  · · ·  O×µN M (A∞ )  · · · , and a system ofcompatible surjections {(‡ Dv` )×  O×µN (A∞ )}N ∈N≥1 (which can be regard as akind of Kummer structure on ‡ Fv`I×µ ) for the split monoid ‡ Dv` ), and, by abuse ofnotation,‡ `Fv
for the split-×-Kummer pre-Frobenioid determined by the split pre-Frobenioid ‡ Fv`
with the ×-Kummer structure determined by ‡ κ`×v .
(2) Write‡ `I×µF:= {‡ Fv`I×µ }v∈V .

We also write‡ `×F= {‡ Fv`× }v∈V

(resp. ‡ F`×µ := {‡ Fv`×µ }v∈V )for the collection of data obtained by replacing the various split pre-Frobenioidsof ‡ F` (resp. ‡ F`I×µ ) by the split Frobenioid with trivial splittings obtainedby considering the subcategories determined by morphisms φ with Div(φ) = 0(i.e., the “units” for v ∈ Vnon ) in the pre-Frobenioid structure. Note that ‡ Fv`×(resp. ‡ Fv`×µ ) is a split-×-Kummer pre-Frobenioid (resp. a split-×µ-Kummer preFrobenioid).
(3) An F `× -prime-strip (resp. an F `×µ -prime-strip, resp. an F `I×µ -primestrip) is a collection∗ `×F= {∗ Fv`× }v∈V (resp. ∗ F`×µ = {∗ Fv`×µ }v∈V , resp. ∗ F`I×µ = {∗ Fv`I×µ }v∈V )of data such that ∗ Fv`× (resp. ∗ Fv`×µ , resp. ∗ Fv`I×µ ) is isomorphic to ‡ Fv`×(resp. ‡ Fv`×µ , resp. ‡ Fv`I×µ ) for each v ∈ V. An isomorphism of F `× -primestrips (resp. F `× -prime-strips, resp. F `× -prime-strips) is a collection of
isomorphisms indexed by V between each component.

222

Go Yamashita(4) An F I×µ -prime-strip is a quadruple∗∼

F I×µ = (∗ C , Prime(∗ C ) → V, ∗ F`I×µ , {∗ ρv }v∈V )`where ∗ C is a pre-Frobenioid isomorphic to the model Cmodin Definition 10.4,∼∗∗ `I×µ`I×µPrime( C ) → V is a bijection of sets, F
is an F
-prime-strip, and ∗ ρv :

Φ∗ C ,v

gl. to loc.
∼

−→

∗ `
ΦR
Cv is the∗ C ` is an isomorphism of topological monoids (Here,v

object reconstructed from ∗ Fv`I×µ ), such that the quadruple ∗ F is isomorphic to
the model Fmod in Definition 10.4. An isomorphism of F I×µ -prime-strips isa collection of isomorphisms indexed by V between each component.
(5) Let AutF `× (−), IsomF `× (−, −) (resp. AutF `×µ (−), IsomF `×µ (−, −) resp. AutF `I×µ (−),
IsomF `I×µ (−, −) resp. AutF I×µ (−), IsomF I×µ (−, −)) be the group of automorphisms of an F `× -(resp. F `×µ -, resp. F `I×µ -, resp. F I×µ -)prime-strip, andthe set of isomorphisms between F `× -(resp. F `×µ -, resp. F `I×µ -, resp. F I×µ )prime-strips.
Remark 10.12.1. In the definition of ‡ Fv`I×µ for v ∈ Varc in Definition 10.12, weconsider an inductive system. We use this as follows: For the crucial non-interference
property for v ∈ Vnon , we use the fact that the pv -adic logarithm kills the torsionµ(−) ⊂ O× (−). However, for v ∈ Varc , the Archimedean logarithm does not kill thetorsion. Instead, in the notation of Section 5.2, we replace a part of log-link by k ∼ (Ok )gp  (Ok )gp /µN (k) and consider k ∼ as being reconstructed from (Ok )gp /µN (k),not from (Ok )gp , and write weight N on the corrsponding log-volume. Then there is no
problem. cf. also Definition 12.1 (2), (4), Proposition 12.2 (2) (cf. [IUTchIII, Remark
1.2.1]), Proposition 13.7, and Proposition 13.11.
Definition 10.13.

([IUTchIII, Definition 2.4])(1) Let‡ `F = {‡ Fv` }v∈Vbe an F ` -prime-strip. Then by Definition 10.12 (1), for each w ∈ Vbad , the splittingsof the µ2l -split-Frobenioid ‡ Fw` determine submonoids O⊥ (−) ⊂ O (−) and quotient monoids O⊥ (−)  OI (−) = O⊥ (−)/Oµ (−). Similarly, for each w ∈ Vgood ,the splitting of the split Frobenioid ‡ Fw` determines a submonoid O⊥ (−) ⊂ O (−).
In this case, we write OI (−) := O⊥ (−). We write‡ `⊥F= {‡ Fv`⊥ }v∈V ,

‡ `I

F

= {‡ Fv`I }v∈Vfor the collection of data obtained by replacing the µ2l -split/split Frobenioid portion of each ‡ Fv` by the pre-Frobenioids determined by the subquotient monoidsO⊥ (−) ⊂ O (−) and OI (−), respectively.

a proof of the abc conjecture after Mochizuki

223(2) An F `⊥ -prime-strip (resp. an F `I -prime-strip) is a collection∗ `⊥F= {∗ Fv`⊥ }v∈V (resp. ∗ F`I = {∗ Fv`I }v∈V )of data such that ∗ Fv`⊥ (resp. ∗ Fv`I ) is isomorphic to ‡ Fv`⊥ (resp. ‡ Fv`I ) for eachv ∈ V. An isomorphism of F `⊥ -prime-strips (resp. F `I -prime-strips) is acollection of isomorphisms indexed by V between each component.
(3) An F ⊥ -prime-strip (resp. F I -prime-strip) is a quadruple∗∼

F ⊥ = (∗ C , Prime(∗ C ) → V, ∗ F`⊥ , {∗ ρv }v∈V )∼(resp. ∗ F I = (∗ C , Prime(∗ C ) → V, ∗ F`I , {∗ ρv }v∈V ) )`where ∗ C is a pre-Frobenioid isomorphic to the model Cmodin Definition 10.4,
∼
∗
∗ `⊥
∗ `I
Prime( C ) → V is a bijection of sets, F(resp. F ) is an F `⊥ -prime-strip(resp. F `I -prime-strip), and ∗ ρv : Φ∗ C ,v

gl. to loc.
∼

−→

ΦR
∗ C ` is an isomorphism of topov

logical monoids (Here, ∗ Cv` is the object reconstructed from ∗ Fv`⊥ (resp. ∗ Fv`I )),such that the quadruple ∗ F ⊥ (resp. ∗ F I ) is isomorphic to the model Fmod in Definition 10.4. An isomorphism of F ⊥ -prime-strips (resp. F I -prime-strips)
is a collection of isomorphisms indexed by V between each component.
§ 10.4.

The Multiplicative Symmetry  : ΘNF-Hodge Theatres and NF-,
Θ-Bridges.

We begin constructing the multiplicative portion of full Hodge theatres.
([IUTchI, Definition 4.1 (i), (ii), (v)]) Let † D = {† Dv }v∈V be

Definition 10.14.
a D-prime-strip.

(1) For v ∈ Vbad (resp. v ∈ Vgood ∩ Vnon ), we can group-theoretically reconstruct
in a functorial manner, from π1 († Dv ), a tempered group (resp. a profinite group)(⊃ π1 († Dv )) corresponding to C v by Lemma 7.12 (resp. by Lemma 7.25). We write†Dv

for its B(−)0 . We have a natural morphism † Dv → † Dv (This corresponds to
X v → C v (resp. →X → C v )). Similarly, for v ∈ Varc , we can algorithmically−v

reconstruct, in a functorial manner, from † Dv , an Aut-holomorphic orbispace † Dv
corresponding to C v by translating Lemma 7.25 into the theory of Aut-holomorphic
spaces (since −X admits a Kv -core) with a natural morphism † Dv → † Dv . Write→v†

D := {† Dv }v∈V .

224

Go Yamashita(2) Recall that we can algorithmically reconstruct the set of conjugacy classes of cuspidal decomposition groups of π1 († Dv ) or π1 († Dv ) by Corollary 6.12 for v ∈ Vbad , byCorollary 2.9 for v ∈ Vgood ∩ Vnon , and by considering π0 (−) of a cofinal collectionof the complements of compact subsets of the underlying topological space of † Dv
or † Dv for v ∈ Varc . We say them the set of cusps of † Dv or † D v .
For v ∈ V, a label class of cusps of † Dv is the set of cusps of † Dv lying overa single non-zero cusp of † Dv (Note that each label class of cusps consists of two
cusps). We write
LabCusp(† Dv )for the set of label classes of cusps of † Dv . Note that LabCusp(† Dv ) has a natural×F>
l -torsor structure (which comes from the action of Fl on Q in the definition of X
in Section 7.1). Note also that, for any v ∈ V, we can algorithmically reconstruct a
canonical element†

η v ∈ LabCusp(† Dv )corresponding to v in the initial Θ-data, by Lemma 7.16 for v ∈ Vbad , Lemma 7.25
for v ∈ Vgood ∩ Vnon , and a translation of Lemma 7.25 into the theory of Autholomorphic spaces for v ∈ Varc .
good(Note that, if we used † Dv (i.e., “C v ”) instead of † Dv (i.e., “X∩−→ ”) for v ∈ V
vVnon , then we could not reconstruct † η v . In fact, we could make the action of
the automorphism group of † Dv on LabCusp transitive for some v ∈ Vgood ∩ Vnon ,by using Chebotarev density theorem (i.e., by making a decomposition group in
Gal(K/F ) ,→ GL2 (Fl ) to be the subgroup of diagonal matrices with determinant1). cf. [IUTchI, Remark 4.2.1].)(3) Let † D} is a category equivalent to the model global object D} in Definition 10.3.
Then by Remark 2.9.2, similarly we can define the set of cusps of † D } and theset of label classes of cusps
LabCusp(† D} ),which has a natural F>
l -torsor structure.
From the definitions, we immediately obtain the following proposition:
Proposition 10.15.
([IUTchI, Proposition 4.2]) Let † D = {† Dv }v∈V be a Dprime-strip. Then for any v, w ∈ V, there exist unique bijections∼

LabCusp(† Dv ) → LabCusp(† Dw )

a proof of the abc conjecture after Mochizuki

225†
which are compatible with the F>
l -torsor structures and send the canonical element η v

to the canonical element † η w . By these identifications, we can write
LabCusp(† D)for them. Note that it has a canonical element which comes from † η v ’s. The F>
l -torsor
structure and the canonical element give us a natural bijection∼

LabCusp(† D) → F>l .
Definition 10.16.

(Model D-NF-Bridge, [IUTchI, Example 4.3]) We writeAut (C K ) ⊂ Aut(C K ) ∼= Out(ΠC K ) ∼= Aut(D} )for the subgroup of elements which fix the cusp  (The firs isomorphisms follows from
Theorem 3.17). By Theorem 3.7, we can group-theoretically reconstruct ∆X from ΠC K .
We obtain a natural homomorphism
Out(ΠC K ) → Aut(∆ab
X ⊗ Fl )/{±1},since inner automorphisms of ΠC K act by multiplication by ±1 on EF [l]. By choosing∼aba suitable basis of ∆ab
X ⊗ Fl , which induces an isomorphism Aut(∆X ⊗ Fl )/{±1} →GL2 (Fl )/{±1}, the images of Aut (C K ) and Aut(C K ) are identified with the following
subgroups{()} {( )}∗ ∗∗∗⊂⊂ Im(GFmod ) (⊃ SL2 (Fl )/{±1})0 ±10∗of GL2 (Fl )/{±1}, where Im(GFmod ) ⊂ GL2 (Fl )/{±1} is the image of the natural actionof GFmod := Gal(F /Fmod ) on EF [l]. Write also
V±un := Aut (C K ) · V ⊂ VBor := Aut(C K ) · V ⊂ V(K).
Hence we have a natural isomorphism∼

Aut(C K )/Aut (C K ) → F>l ,±un
. By the above discussions, from π1 (D} ), we canthus, VBor is the F>
l -orbit of V
group-theoretically reconstructAut (D} ) ⊂ Aut(D} )corresponding to Aut (C K ) ⊂ Aut(C K ) (cf. also Definition 10.11 (1), (2)).

226

Go Yamashita

For v ∈ Vbad (resp. v ∈ Vgood ∩ Vnon , resp. v ∈ Varc ), We write}
φNF•,v : Dv → D

for the natural morphism correponding to X v → C v → C K (resp. −X → Cv → CK ,
→
v

∼

}

resp. a tautological morphism Dv = −X → Cv → C(D , v)) (cf. Definition 10.11 (1)).
→
v
Write
poly
}
NF
}
φNF
v := Aut (D ) ◦ φ•,v ◦ Aut(Dv ) : Dv −→ D .

Let Dj = {Dvj }v∈V be a copy of the tautological D-prime-strip {Dv }v∈V for eachj ∈ F>l (Here, we write v j for the pair (j, v)). Write
poly

NF
}
φNF
1 := {φv }v∈V : D1 −→ D(cf. Definition 10.11 (2)). Since φNF
is stable under the action of Aut (D} ), we obtain1
a poly-morphismpoly
φNF:= (action of j) ◦ φNF: Dj −→ D} ,
j
1
}
}
}
∼
by post-composing a lift of j ∈ F>
l = Aut(D )/Aut (D ) to Aut(D ). Hence we obtaina poly-morphism
poly

NF
}
φNF
> := {φj }j∈F> : D> := {Dj }j∈F> −→ Dllfrom a capsule of D-prime-strip to the global object D} (cf. Definition 10.11 (3)). Thisis called the model base-(or D-)NF-bridge. Note that φNF> is equivariant with the>}natural poly-action (cf. Section 0.2) of Fl on D and the natural permutation polyaction of F>l (via capsule-full poly-automorphisms (cf. Section 0.2)) on the components}
NF
of the cupsule D> . In particular, we obtain a poly-action of F>
l on (D> , D , φ> ).
Definition 10.17.
(Model D-Θ-Bridge, [IUTchI, Example 4.4]) Let v ∈ Vbad .
Recall that we have a natural bijection between the set of cusps of C v and |Fl | byLemma 7.16. Thus, we can write labels (∈ |Fl |) on the collections of cusps of X v , X v
by considering fibers over C v . We writeµ− ∈ X v (Kv )for the unique torsion point of order 2 such that the closures of the cusp labelled 0 ∈ |Fl |and µ− in the stable model of X v over OKv intersect the same irreducible componentZof the special fiber (i.e., “−1” in Grigm /qX v ). We shall refer to the points obtained bytranslating the cusps labelled by j ∈ |Fl | by µ− with respect to the group scheme
structure of E v (⊃ X v ) (Recall that the origin of E v is the cusp labelled by 0 ∈ |Fl |) as227

a proof of the abc conjecture after Mochizuki

the evaluation points of X v labelled by j. Note that the value of Θv in Example 8.8
at a point of Ÿ v lying over an evaluation point labelled by j ∈ |Fl | is in the µ2l -orbit of
{
}
j2
q
v

(√
by calculation Θ̈

j
−qv

)
=

j∈Z such that j≡j in |Fl |2√−2j
j −j /2 √(−1) qv−1 Θ̈( −1)j/2=−j 2 /2
qv

in the notationjof Lemma 7.4 (cf. the formula Θ̈(q Ü ) = (−1) q −1/2 Ü −2 Θ̈(Ü ) in Lemma 7.4). In
particular, the points of X v lying over evaluation points of X v are all defined over
Kv , by the definition of X v → X v (Note that the image of a point in the domain of(covering map,Θ̈)Ÿ,→Ÿ × A1 is rational over Kv , then the point is rational over Kv . cf.
also Assumption (5) of Definition 7.13). We shall refer to the points in X(Kv ) lying
over the evaluation points of X v (labelled by j ∈ |Fl |) as the evaluation points ofX v (labelled by j ∈ |Fl |). We also shall refer to the sections Gv ,→ Πv (= ΠX ) givenv

by the evaluation points (labelled by j ∈ |Fl |) as the evaluation section of Πv  Gv
(labelled by j ∈ |Fl |). Note that, by using Theorem 3.7 (elliptic cuspidalisation) andRemark 6.12.1 (together with Lemma 7.16, Lemma 7.12), we can group-theoreticallyreconstruct the evaluation sections from (an isomorph of) Πv .
Let D> = {D>,w }w∈V be a copy of the tautological D-prime-strip {Dw }w∈V . Write
naturaltemp(Πv )0 −→ B(Kv )0φΘ
v j :=Aut(D>,v ) ◦ (Beval. section−→

labelled by j

B temp (Πv )0 ) ◦ Aut(Dvj )poly: Dvj −→ D>,v .
Note that the homomorphism π1 (Dvj ) → π1 (D>,v ) induced by any constituent of thepoly-morphism φΘ
v j (which is well-defined up to inner automorphisms) is compatible withthe respective outer actions on π1geo (Dvj ) and π1geo (D>,v ) (Here we write π1geo for thegeometric portion of π1 , which can be group-theoretically reconstructed by Lemma 6.2)∼
for some outer isomorphism π1geo (Dvj ) → π1geo (D>,v ) (which is determined up to finite
ambiguity by Remark 6.10.1). We say this fact, in short, as φΘ
v j is compatible with theouter actions on the respective geometric tempered fundamental groups.
Let v ∈ Vgood . WriteφΘv j : Dv j

full poly
∼

→

D>,v

to be the full poly-isomorphism for each j ∈ F>
l ,
poly

Θ
φΘ
j := {φv j }v∈V : Dj −→ D> ,

and
poly

Θ
φΘ
> := {φj }j∈F> : D> −→ D> .
l

228

Go Yamashita

This is called the model base-(or D-)Θ-bridge (Note that this is not a poly-isomorphism).
Note that D> has a natural permutation poly-action by F>l , and that, on the other hand,
the labels ∈ |Fl | (or ∈ LabCusp(D> )) determined by the evaluation sections corresponding to a given j ∈ F>l are fixed by any automorphisms of D> .
Definition 10.18.
Definition 4.6])(D-NF-Bridge, D-Θ-Bridge, and D--Hodge Theatre, [IUTchI,(1) A base-(or D-)NF-bridge is a poly-morphism
poly

† NF
φ>

: † DJ −→ † D} ,where † D} is a category equivalent to the model global object D} , and † DJ is a cupsule of D-prime-strips indexed by a finite set J, such that there exist isomorphisms∼∼† NFD} → † D} , D> → † DJ , conjugation by which sends φNF> 7→ φ> . An isomor(())
poly † }poly ‡ }
∼
‡ NF
‡
†
0
phism of D-NF-bridges † φNF
:
D
−→
D
→
φ
:
D
−→
D
is a
J
J
>
>
†

pair of a capsule-full poly-isomorphism DJ

capsule-full poly∼‡−→

DJ 0 and an Aut († D} )-

poly
∼ ‡

orbit (or, equivalently, an Aut ( D )-orbit) D → D} of isomorphisms, which‡ NF
are compatible with † φNF> , φ> . We define compositions of them in an obvious
manner.
‡}†}(2) A base-(or D-)Θ-bridge is a poly-morphism
† Θ
φ>

poly

: † DJ −→ † D> ,where † D> is a D-prime-strip, and † DJ is a cupsule of D-prime-strips indexed∼∼
by a finite set J, such that there exist isomorphisms D> → † D> , D> → † DJ ,† Θconjugation by which sends φΘisomorphism of D-Θ-bridges> 7→ φ> . An )()(
poly †
poly ‡
∼
† Θ
†
‡ Θ
‡
φ> : DJ −→ D> → φ> : DJ 0 −→ D> is a pair of a capsule-full polycapsule-full poly∼full poly∼‡isomorphism † DJ−→
DJ 0 and the full-poly isomorphism † D> →‡‡ Θ
D> , which are compatible with † φΘ> , φ> . We define compositions of them in an
obvious manner.

(3) A base-(or D-)ΘNF-Hodge theatre (or a D--Hodge theatre) is a collection
(
)
† NF
† Θ
φ> †
D-
†
† } φ> †HT= D ←− DJ −→ D> ,† Θ
where † φNF> is a D-NF-bridge, and φ> is a D-Θ-bridge, such that there exist iso∼∼∼morphisms D} → † D} , D> → † DJ , D> → † D> , conjugation by which sends

a proof of the abc conjecture after Mochizuki

229

† NF
Θ
† Θ
φNF
> 7→ φ> , φ> 7→ φ> . An isomorphism of D--Hodge theatres is a pairof isomorphisms of D-NF-bridges and D-Θ-bridges such that they induce the samebijection between the index sets of the respective capsules of D-prime-strips. We
define compositions of them in an obvious manner.

Proposition 10.19.

(Transport of Label Classes of Cusps via Base-Bridges,
†

φNF
>

†

φΘ
>

[IUTchI, Proposition 4.7]) Let † HT D- = († D} ←− † DJ −→ † D> ) be a D--Hodge
theatre.
bad(1) The structure of D-Θ-bridge † φΘinvolving the evaluation sections> at v ∈ V
determines a bijection∼†
χ : J → F>l .

(2) For j ∈ J, v ∈ Vnon (resp. v ∈ Varc ), we consider the various outer homomor†† }ofphisms π1 († Dvj ) → π1 († D} ) induced by the (v, j)-portion † φNF
v j : Dv j → D† }the D-NF-bridge † φNF> . By considering cuspidal inertia subgroups of π1 ( D ) whose
unique subgroup of index l is contained in the image of this homomorphism (resp.
the closures in π1 († D} ) of the images of cuspidal inertia subgroups of π1 († Dvj )(cf. Definition 10.14 (2) for the group-theoretic reconstruction of cuspidal inertia
subgroups for v ∈ Varc ), these homomorphisms induce a natural isomorphism∼

LabCusp(† D} ) → LabCusp(† Dvj )
∼

†
of F>
l -torsors. These isomorphisms are compatible with the isomorphism LabCusp( Dv j ) →LabCusp(† Dwj ) of F>l -torsors in Proposition 10.15 when we vary v ∈ V. Hence weobtaine a natural isomorphism∼

LabCusp(† D} ) → LabCusp(† Dj )of F>l -torsors.
††
Next, for each j ∈ J, the various v(∈ V)-portions of the j-portion † φΘ
j : Dj → D>of the D-Θ-bridge † φΘ
> determine an isomorphism∼

LabCusp(† Dj ) → LabCusp(† D> )>of F>l -torsors. Therefore, for each j ∈ J, by composing isomorphisms of Fl -torsors† Θ
obtained via † φNF
j , φj , we get an isomorphism
† LC
φj

∼

: LabCusp(† D} ) → LabCusp(† D> )

† LC
of F>
is obtained from † φLC
by the action by † χ(j) ∈ F>1
l -torsors, such that φj
l .

230

Go Yamashita(3) By considering the canonical elements † η v ∈ LabCusp(† Dv ) for v’s, we obtain aunique element[† ] ∈ LabCusp(† D} )∼such that, for each j ∈ J, the natural bijection LabCusp(† D> ) → F>l in Proposi† LC †† LC †††
tion 10.15 sends φj ([ ]) = φ1 ( χ(j) · [ ]) 7→ χ(j). In particular, the element[† ] determines an isomorphism†∼∼

ζ> : LabCusp(† D} ) → J (→ F>
l )

of F>
l -torsors.
Remark 10.19.1. (cf. [IUTchI, Remark 4.5.1]) We consider the group-theoretic
algorithm in Proposition 10.19 (2) for v ∈ V. Here, the morphism π1 († Dvj ) → π1 († D} )is only known up to π1 († D} )-conjugacy, and a cuspidal inertia subgroup labelled by anelement ∈ LabCusp(† D} ) is also well-defined up to π1 († D} )-conjugacy. We have nonatural way to synchronise these indeterminacies. Let J be the unique open subgroup ofindex l of a cuspidal inertia subgroup. A nontrivial fact is that, if we use Theorem 6.11,
then we can factorise J ,→ π1 († D} ) up to π1 († D} )-conjugacy into J ,→ π1 († Dvj ) up toπ1 († Dvj )-conjugacy and π1 († Dvj ) ,→ π1 († D} ) up to π1 († D} )-conjugacy (i.e., factorise
out

out

out

J ,→ π1 († D} ) as J ,→ π1 († Dvj ) ,→ π1 († D} )). This can be regarded as a partial
synchronisation of the indeterminacies.
Proof. The proposition immediately follows from the described algorithms.
The following proposition follows from the definitions:
Proposition 10.20.
(Properties of D-NF-Brideges, D-Θ-Bridges, D--Hodge
theatres, [IUTchI, Proposition 4.8])>‡ NF† NF ‡ NF(1) For D-NF-bridges † φNF> , φ> , the set Isom( φ> , φ> ) is an Fl -torsor.
‡ Θ† NF ‡ NF(2) For D-Θ-bridges † φΘ> , φ> , we have #Isom( φ> , φ> ) = 1.

(3) For D--Hodge theatres † HT D- , ‡ HT D- , we have #Isom(† HT D- , ‡ HT D- ) =1.
† Θ(4) For a D-NF-bridge † φNF> and a D-Θ-bridge φ> , the set{

capsule-full poly-isom.
is an F>l -torsor.

capsule-full poly∼†DJ −→ † DJ 0}

† Θ
by which † φNF> , φ> form a D-  -Hodge theatre

231

a proof of the abc conjecture after Mochizuki>(5) For a D-NF-bridge † φNF> , we have a functorial algorithm to construct, up to Fl indeterminacy, a D--Hodge theatre whose D-NF-bridge is † φNF> .

Definition 10.21.
([IUTchI, Corollary 4.12]) Let † HT D- , ‡ HT D- be D-Hodge theatres. the base-(or D-)ΘNF-link (or D--link)†D

HT D- −→‡HT D-

is the full poly-isomorphism†

D`>full poly
∼
‡

−→

D`>

between the mono-analyticisations of the codomains of the D-Θ-bridges.
Remark 10.21.1.
In D--link, the D` -prime-strips are shared, but not the arithmetically holomorphic structures. We can visualise the “shared” and “non-shared”
relation as follows:†

HT D- − − > † D`> ∼= ‡ D`> > − − ‡ HT D-

We shall refer to this diagram as the étale-picture of D--Hodge theatres. Notethat we have a permutation symmetry in the étale-picture.
We constructed D--Hodge theatres. These are base objects. Now, we begin
constructing the total spaces, i.e., -Hodge theatres, by putting Frobenioids on them.
†

D-

†

}

†

φNF
>

†

†

φΘ
>

We start with the following situation: Let HT= ( D ←− DJ −→ † D> )be a D--Hodge theatre (with respect to the fixed initial Θ-data). Let † HT Θ =({† F v }v∈V , † Fmod ) be a Θ-Hodge theatre, whose associseted D-prime strip is equal to†

D> in the given D--Hodge theatre. We write † F> for the F-prime-strip tautologicallyassociated to (the {† F v }v∈V -portion of) the Θ-Hodge theatre † HT Θ . Note that † D>
can ben identified with the D-prime-strip associated to † F> :
†

†

HT Θ



HT D- / † F>_
/ † D> .

Definition 10.22.
([IUTchI, Example 5.4 (iii), (iv)]) Let † F ~ be a pre-Frobenioid
isomorphic to F ~ († D} ) as in Example 9.5, where † D} is the data in the given D-~
Hodge theatre † HT D- . We write † F } := † F ~ |† D} , and † Fmod:= † F ~ |terminal object in † D~ ,as in Example 9.5.

232

Go Yamashita(1) For δ ∈ LabCusp(† D} ), a δ-valuation ∈ V(† D} ) is a valuation which lies in†the “image” (in the obvious sense) via † φNF> of the unique D-prime-strip Dj of∼
the capsule † DJ such that the bijection LabCusp(† D} ) → LabCusp(† Dj ) induced∼by † φNF
sends δ to the element of LabCusp(† Dj ) → F>jl (cf. Proposition 10.15)>
labelled by 1 ∈ Fl (Note that, if we allow ourselves to use the model object D} ,then a δ-valuation ∈ V(† D} ) is an element, which is sent to an elemento of V±un ⊂∼V(K) under the bijection LabCusp(† D} ) → LabCusp(D} ) induced by a unique∼Aut († D} )-orbit of isomorphisms † D} → D} sending δ 7→ [] ∈ LabCusp(D} )).
(2) For δ ∈ LabCusp(† D} ), by localising at each of the δ-valuations ∈ V(† D} ), from† }e ~× ) in Definition 9.6), we canF (or, from ((† Π~ )rat y † M~ ) = (π1 († D} ) y O
construct an F-prime-strip†

F } |δ

which is well-defined up to isomorphism (Note that the natural projection V±un 
Vmod is not injective, hence it is necessary to think that † F|δ is well-defined only
up to isomorphism, since there is no canonical choice of an element of a fiber of thenatural projection V±un  Vmod ) as follows: For a non-Archimedean δ-valuationv, it is the pv -adic Frobenioid associated to the restrictions to “the open subgroup”
of † Πp0 ∩ π1 († D} ) determined by δ ∈ LabCusp(† D} ) (i.e., corresponding to “X” or
bad
†
“X
−
→”) (cf. Definition 9.6 for Πp0 ). Here, if v lies over an element of Vmod , then we
have to replace the above “open subgroup” by its tempered analogue, which canbe done by reconstructing, from the open subgroup of † Πp0 ∩ π1 († D} ), the semigraph of anabelioids by Remark 6.12.1 (cf. also [SemiAnbd, Theorem 6.6]). For an
Archimedean δ-valuation v, this follows from Proposition 4.8, Lemma 4.9, and the∼isomorphism M~ († D} ) → † M~ in Example 9.5.
(3) For an F-prime-strip ‡ F whose associated D-prime-strip is ‡ D, a poly-morphism
‡

‡

poly

F −→ † F }full poly∼†

is a full poly-isomorphism F −→ F } |δ for some δ ∈ LabCusp(† D} ) (Note thatthe fact that † F } |δ is well-defined only up to isomorphism is harmless here). We
poly

regard such a poly-morphism ‡ F −→ † F } is lying over an induced poly-morphism
poly
poly
‡
D −→ † D} . Note also that such a poly-morphism ‡ F −→ † F } is compatible
with the local and global ∞ κ-coric structures (cf. Definition 9.6) in the following
sense: The restriction of associated Kummer classes determines a collection of polymorphisms of pseudo-monoids{}† ~ rat† ~ poly ‡‡( Π ) y M∞ κ −→ M∞ κv ⊂ M∞ κ×v
v∈V

233

a proof of the abc conjecture after Mochizuki

indexed by V, where the left-hand side († Π~ )rat y † M~
is well-defined up to au∞κ† ~ rattomorphisms induced by the inner automorphisms of ( Π ) , and the right-handside ‡ M∞ κv ⊂ ‡ M∞ κ×v is well-defined up to automorphisms induced by the automorphisms of the F-prime strip ‡ F. For v ∈ Vnon , the above poly-morphism isequivariant with respect to the homomorphisms (‡ Πv )rat → († Π~ )rat (cf. Definipolytion 9.6 (2) for (‡ Πv )rat ) induced by the given poly-morphism ‡ F −→ † F } .
(4) For a capsule E F = {e F} of F-prime-strips, whose associated capsule of D-primestrips is E D, and an F-prime-strip † F whose associated D-prime-strip is † D, a
poly-morphism
E

poly

poly

F −→ † F } (resp. E F −→ † F )
poly

poly

is a collection of poly-morphisms {e F −→ † F } }e∈E (resp. {e F −→ † F}e∈E ). We
poly

poly

consider a poly-morphism E F −→ † F } (resp. E F −→ † F) as lying over the induced
poly

poly

poly-morphism E D −→ † D} (resp. E D −→ † D).
We return to the situation of††HT Θ 

HT

/ † F>_
/ † D> .

D- Definition 10.23.
(Model Θ-Bridge, Model NF-Bridge, Diagonal F-Objects,
Localisation Functors, [IUTchI, Example 5.4 (ii), (v), (i), (vi), Example 5.1 (vii)]) For
j ∈ J, let † Fj = {† Fvj }j∈J be an F-prime-strip whose associated D-prime-strip is equalto † Dj . We also write † FJ := {† Fj }j∈J (i.e., a capsule indexed by j ∈ J).
Let † F ~ be a pre-Frobenioid isomorphic to F ~ († D} ) as in Example 9.5, where† }D is the data in the given D--Hodge theatre † HT D- . We write † F } := † F ~ |† D} ,~and † Fmod:= † F ~ |terminal object in † D~ , as in Example 9.5.
(1) For j ∈ J, we write†polyψjΘ : † Fj −→ † F>

for the poly-morphism (cf. Definition 10.22 (4)) uniquely determined by † φj byRemark 10.10.1. Write
†

poly

Θ
ψ>
:= {† ψjΘ }j∈F> : † FJ −→ † F> .
lΘ
† ΘWe regard † ψ>as lying over † φΘ
> . We shall refer to ψ> as the model Θ-bridge.

234

Go Yamashita

cf. also the following diagram:†††HT Θ &
/ † F>_

 † HT D- / † D> .
8

Fj , † FJ_ΘψjΘ , † ψ>†

Dj , DJ o
†

†

† Θ
φΘ
j , φ>(2) For j ∈ J, we write†

poly

ψjNF : † Fj −→ † F }for the poly-morphism (cf. Definition 10.22 (3)) uniquely determined by † φj byLemma 10.10 (2). Write
†

poly

NF
ψ>
:= {† ψjNF }j∈F> : † FJ −→ † F } .
l

† NF
NF
as lying over † φNF
We regard † ψ>
> . We shall refer to ψ> as the model NFbridge. cf. also the following diagram:
†

†

†

NF
ψjNF , † ψ>&

Fj , † FJ_


Dj , † DJ oF_}/ † D} .
9 † HT D- 
†

†

† NF
φNF
j , φ>(3) Let also † FhJi = {† FvhJi }vhJi ∈VhJi be an F-prime-strip. We write † DhJi for the
associated D-prime-strip to † FhJi . We write Vj := {v j }v∈V . We have a natural∼bijection Vj → V : v j 7→ v. These bijections determine the diagonal subset
VhJi ⊂ VJ :=

∏

Vj ,

j∈J
∼

which admits a natural bijection VhJi → V. Hence we obtain a natural bijection∼
VhJi → Vj for j ∈ J.

235

a proof of the abc conjecture after Mochizuki

We have the full poly-isomorphism†

FhJi

full poly
∼
†

−→

F>

and the “diagonal arrow”†

FhJi −→ † FJ ,full poly∼

which is the collection of the full poly-isomorphisms † FhJi −→ † Fj indexed byj ∈ J. We regard † Fj (resp. † FhJi ) as a copy of † F> “situated on” the constituent
labelled by j ∈ J (resp. “situated in a diagonal fashion on” all the consitutents) of
the capsule † DJ .
We have natural bijections
∼

∼

∼

~
VhJi → Vj → Prime(† Fmod
) → Vmodfor j ∈ J. Write
†

∼

~
~
~
)},
FhJi
:= {† Fmod
, VhJi → Prime(† Fmod

†

∼

~
~
Fj~ := {† Fmod
, Vj → Prime(† Fmod)}~~
for j ∈ J. We regard † Fj~ (resp. † FhJi
) as a copy of † Fmod
“situated on” theconstituent labelled by j ∈ J (resp. “situated in a diagonal fashion on” all the~consitutents) of the capsule † DJ . When we write † FhJi
for the underlying cate† ~† ~gory (i.e., Fmod ) of FhJi by abuse of notation, we have a natural embedding of
categories
∏
† ~
† ~
FhJi ,→ † FJ~ :=
Fj .
j∈J

Note that we do not regard the category † FJ~ as being a (pre-)Frobenioid. We write~~† ~R † ~R
Fj , FhJi for the realifications (Definition 8.4) of † FhJi
, † FhJi
respectively, and∏~R~R
write † FJ := j∈J † Fj .
~Since † Fmod
is defined by the restriction to the terminal object of † D~ , any polypoly

polymorphism † FhJi −→ † F } (resp. † Fj −→ † F } ) (cf. Definition 10.22 (3)) induces,
via restriction (in the obvious sense), the same isomorphism class∼~~(† F } → † F ~ ⊃ )† Fmod−→ † FhJi∼

gl. to loc. †−→~(resp. († F } → † F ~ ⊃ )† Fmod−→ † Fj~

FvhJi

gl. to loc. †

−→

Fvj )

236

Go Yamashita

of restriction functors, for each v hJi ∈ VhJi (resp. v j ∈ Vj ) (Here, for v hJi ∈ Varc
hJi††(resp. v j ∈ Varc
),
we
write
F
(resp.
F)forthecategory
component
of
v hJi
vj
j
the triple, by abuse of notation), i.e., it is independent of the choice (among its†† }F>(resp. † Fj → † F } ). cf. alsol -conjugates) of the poly-morphism FhJi → F
Remark 11.22.1 and Remark 9.6.2 (4) (in the second numeration). We write∼

gl. to loc. †~~(† F } → † F ~ ⊃ )† Fmod−→ † FhJi

−→

∼

FhJi

gl. to loc. †~(resp. († F } → † F ~ ⊃ )† Fmod−→ † Fj~−→

Fj )for the collection of the above isomorphism classes of restriction functors, as v hJi(resp. v j ) ranges over the elements of VhJi (resp. Vj ). By combining j ∈ J, we also
obtain a natural isomorphism classes†

FJ~

gl. to loc. †−→FJof restriction functors. We also obtain their natural realifications
†

~R
FhJi

gl. to loc. † R−→
FhJi ,

Definition 10.24.
tion 5.5])

†

FJ~R

gl. to loc. † R
−→
FJ ,

†

Fj~R

gl. to loc. † R−→
Fj .

(NF-Bridge, Θ-Bridge, -Hodge Theatre, [IUTchI, Defini-(1) an NF-bridge is a collection
(
‡
FJ

‡

NF
ψ>

−→

‡

F

}

99K

‡

F

~

)

as follows:(a) ‡ FJ = {‡ Fj }j∈J is a capsule of F-prime-strip indexed by J. We write ‡ DJ ={‡ Dj }j∈J for the associated capsule of D-prime-strips.
(b) ‡ F } , ‡ F ~ are pre-Frobenioids isomorphic toy ‡ F } , ‡ F ~ in the definition of
the model NF-bridge (Definition 10.23), respectively. We write ‡ D} , ‡ D~ forthe base categories of ‡ F } , ‡ F ~ respectively.
(c) The arrow 99K consists of a morphism ‡ D} → ‡ D~ , which is abstractly equivalent (cf. Section 0.2) to the morphism † D} → † D~ definition of the model∼
NF-bridge (Definition 10.23), and an isomorphism ‡ F } → ‡ F ~ |‡ D} .
NF(d) ‡ ψ>is a poly-morphism which is a unique lift of a poly-morphism ‡ φNF
> :
‡

poly

DJ −→ ‡ D} such that ‡ φNF> forms a D-NF-bridge.

237

a proof of the abc conjecture after Mochizuki‡ NF
Note that we can associate an D-NF-bridge ‡ φNF> to any NF-bridge ψ> . An
isomorphism of NF-bridges

(

1

1

NF
ψ>
1

FJ1 −→ F

}

99K F
1

~

)

∼

(

→

2

2

NF
ψ>
2

FJ2 −→ F

}

99K F
2

~

)

is a triple
1

FJ1

capsule-full poly∼2−→

FJ 2 , F
1

1of a capsule-full poly-isomorphism FJ1
2

}

poly
∼ 2

∼

−→ F } , 1 F ~ −→ 2 F ~capsule-full poly
∼
2

−→

poly
∼

FJ2 (We write DJ1 −→1

DJ2 for the induced poly-isomorphism), a poly-isomorphism F
1

}

poly
∼ 2

−→ F } (We
poly
∼

poly
∼

write 1 D} −→ 2 D} for the induced poly-isomorphism) such that the pair 1 DJ1 −→
1

}

poly
∼ 2

DJ2 and D −→ D} forms a morphism of the associated D-NF-bridges, and∼an isomoprhism 1 F ~ −→ 2 F ~ , such that this triple is compatible (in the obvious
NF 2 NF
sense) with 1 ψ>
, ψ> , and the respective 99K’s. Note that we can associate an
isomorphism of D-NF-bridges to any isomorphism of NF-bridges.
2(2) A Θ-bridge is a collection
(

‡

‡

Θ
ψ>
‡

−→ F> 99K

FJ

‡

)
HT

Θ

as follows:(a) ‡ FJ = {‡ Fj }j∈J is a capsule of F-prime-strips indexed by J We write ‡ DJ ={‡ Dj }j∈J for the associated capsule of D-prime-strips.
(b) ‡ HT Θ is a Θ-Hodge theatre.
(c) ‡ F> is the F-prime-strip tautologically associated to ‡ HT Θ . We use the notation 99K to write this relationship between ‡ F> and ‡ HT Θ . We write ‡ D> forthe D-prime-strip associated to ‡ F> .
polyΘ(d) ‡ ψ>= {‡ ψjΘ }j∈F> is the collection of poly-morphisms ‡ ψjΘ : ‡ Fj −→ ‡ F>l
‡ Θdetermined by a D-Θ-bridge ‡ φΘ
> = { φj }j∈F> by Remark 10.10.1.
l‡ ΘNote that we can associate an D-Θ-bridge ‡ φΘ
> to any Θ-bridge ψ> . An isomorphism of Θ-bridges

(

1

1

Θ
ψ>
1

‡

FJ1 −→ F> 99K HT

)
Θ

∼

→

(

2

2

FJ 2

Θ
ψ>
2

)

−→ F> 99K HT
2

Θ

238

Go Yamashita

is a triple
1

FJ1

capsule-full poly∼2−→

FJ 2 , 1 F>of a capsule-full poly-isomorphism 1 FJ1
poly
} ∼ 2

full poly
∼
2

−→

∼

F> , 1 HT Θ −→ 2 HT Θ

capsule-full poly∼2−→

FJ2 the full poly-isomorphism∼

F −→ F } and an isomoprhism 1 F ~ −→ 2 F ~ of HT -Hodge theatres, such that
Θ 2 Θ
this triple is compatible (in the obvious sense) with 1 ψ>
, ψ> , and the respective
99K’s. Note that we can associate an isomorphism of D-Θ-bridges to any isomorphism of Θ-bridges.
1(3) A ΘNF-Hodge theatre (or -Hodge theatre) is a collection
(
‡ NF
‡ Θ
ψ>
ψ>

‡
HT = ‡ F ~ L99 ‡ F } ←− ‡ FJ −→ ‡ F> 99K
(
where

‡

F

~

‡

L99 F

}

‡

NF
ψ>

)
‡

←− FJ

(
forms an NF-bridge, and

‡

‡

‡

Θ
ψ>

)
HT

Θ

,
)

‡

‡

FJ −→ F> 99K HTΘforms a Θ-bridge, such that the associated D-NF-bridge ‡ φNF> and the associated‡ Θ
D-Θ-bridge φ> form a D--Hodge theatre. An isomorphism of -Hodge theatres is a pair of a morphism of NF-bridge and a morphism of Θ-bridge, which
induce the same bijection between the index sets of the respective capsules of Fprime-strips. We define compositions of them in an obvious manner.
Lemma 10.25.
Corollary 5.6])(Properties of NF-Brideges, Θ-Bridges, -Hodge theatres, [IUTchI,

NF 2 NFΘ 2 Θ(1) For NF-bridges 1 ψ>
, ψ> (resp. Θ-bridges 1 ψ>
, ψ> , resp. -Hodge theatres
1HT , 2 HT ) whose associated D-NF-bridges (resp. D-Θ-bridges, resp. D-D- 2
2 NF
1 Θ 2 Θ1
Hodge theatres) are 1 φNF, HT D- ) respec> , φ> (resp. φ> , φ> , resp. HT
tively, the natural map
2 NF
NF 2 NF
Isom(1 ψ>, ψ> ) → Isom(1 φNF> , φ> )Θ 2 Θ
2 Θ(resp. Isom(1 ψ>, ψ> ) → Isom(1 φΘ
> , φ> ),resp. Isom(1 HT  , 2 HT  ) → Isom(1 HT D- , 2 HT D- ))is bijective.
NFΘ(2) For an NF-bridge ‡ ψ>
and a Θ-bridge ‡ ψ>
, the set{

capsule-full poly-isom.
is an F>l -torsor.

capsule-full poly∼‡
FJ −→ ‡ FJ 0}

NF ‡ Θ
by which ‡ ψ>
, ψ> form a -Hodge theatre

239

a proof of the abc conjecture after Mochizuki

Proof. By using Lemma 10.10 (5), the claim (1) (resp. (2)) follows from Lemma 10.10(1) (resp. (2)).
§ 10.5.

The Additive Symmetry  : Θ±ell -Hodge Theatres and Θell -,Θ± -Bridges.

We begin constructing the additive portion of full Hodge theatres.
Definition 10.26.

([IUTchI, Definition 6.1 (i)]) We shall refer to an element ofFo±as positive (resp. negative) if it is sent to +1 (resp. −1) by the natural surjctionlo±Fl  {±1}.
∼(1) An F±
l -group is a set E with a {±1}-orbit of bijections E → Fl . Hence any±
Fl -group has a natural Fl -module structure.
∼o±±(2) An F±
l -torsor is a set T with an Fl -orbit of bijections T → Fl (Here, Fl 3 (λ, ±1)
is actingg on z ∈ Fl via z 7→ ±z + λ). For an F±l -torsor T , take an bijection∼o±f : T → Fl in the given Fl -orbit, then we obtain a subgroup

Aut+ (T ) (resp. Aut± (T ) )of Aut(Sets) (T ) by transporting the subgroup Fl ∼= {z 7→ z + λ for λ ∈ Fl } ⊂o± ∼Aut(Sets) (Fl ) (resp. Fl= {z 7→ ±z + λ for λ ∈ Fl } ⊂ Aut(Sets) (Fl )) via f .
Note that this subgroup is independent of the choice of f in its Fo±l -orbit. Moreover, any element of Aut+ (T ) is independent of the choice of f in its Fl -orbit,
hence if we consider f up to Fo±l -orbit, then it gives us a {±1}-orbit of bijections∼
Aut+ (T ) → Fl , i.e., Aut+ (T ) has a natural F±l -group structure. We shall refer to±Aut+ (T ) as the Fl -group of positive automorphisms of T . Note that we have
[Aut± (T ); Aut+ (T )] = 2.
The following is an additive counterpart of Definition 10.14([IUTchI, Definition 6.1 (ii), (iii), (vi)]) Let † D = {† Dv }v∈V

Definition 10.27.
be a D-prime-strip.

(1) For v ∈ Vbad (resp. v ∈ Vgood ∩ Vnon ), we can group-theoretically reconstruct
in a functorial manner, from π1 († Dv ), a tempered group (resp. a profinite group)(⊃ π1 († Dv )) corresponding to X v by Lemma 7.12 (resp. by Lemma 7.25). We write†D±v

for its B(−)0 . We have a natural morphism † Dv → † D±v (This corresponds toarc
X → X v )). Similarly, for v ∈ V , we can algorithmically
X v → X v (resp. −
→
v

240

Go Yamashita

reconstruct, in a functorial manner, from † Dv , an Aut-holomorphic orbispace † D±v
corresponding to X v by translating Lemma 7.25 into the theory of Aut-holomorphic
spaces (since −X admits a Kv -core) with a natural morphism † Dv → † D±
v . Write→v†

D± := {† D±v }v∈V .

(2) Recall that we can algorithmically reconstruct the set of conjugacy classes of cuspibad
dal decomposition groups of π1 († Dv ) or π1 († D±, byv ) by Corollary 6.12 for v ∈ V
goodnonCorollary 2.9 for v ∈ V∩ V , and by considering π0 (−) of a cofinal collectionof the complements of compact subsets of the underlying topological space of † Dv
arcor † D±. We say them the set of cusps of † Dv or † D ±v for v ∈ Vv .
For v ∈ V, a ±-label class of cusps of † Dv is the set of cusps of † Dv lying overa single (not necessarily non-zero) cusp of † D±
v . We write
LabCusp± († Dv )for the set of ±-label classes of cusps of † Dv . Note that LabCusp(† Dv ) has a natural
F×l -action. Note also that, for any v ∈ V, we can algorithmically reconstruct a zeroelement† 0
η v ∈ LabCusp± († Dv ),and a canonical element† ±η v ∈ LabCusp± († Dv )which is well-defined up to multiplication by ±1, such that we have † η ±7→ † η vv
under the natural bijection
{
}
∼
± †
† 0
LabCusp ( Dv ) \ { η v } /{±1} → LabCusp(† Dv ).
Hence we have a natural bijection∼

LabCusp± († Dv ) → Fl ,which is well-defined up to multiplication by ±1, and compatible with the bijection∼± †±LabCusp(† Dv ) → F>l in Proposition 10.15, i.e., LabCusp ( Dv ) has a natural Fl group structure. This structure F±l -group gives us a natural surjection
Aut(† Dv )  {±1}by considering the induced automorphism of LabCusp± († Dv ). We write
Aut+ († Dv ) ⊂ Aut(† Dv )

241

a proof of the abc conjecture after Mochizuki

for the kernel of the above surjection, and we shall refer to it as the subgroupof positive automorphisms Write Aut− († Dv ) := Aut(† Dv ) \ Aut+ († Dv ),and weshall refer to it as the set of negative automorphisms. Similarly, for α ∈ {±1}V ,we write
Aut+ († D) ⊂ Aut+ († D) (resp. Autα († D) ⊂ Aut+ († D) )for the subgroup of automorphisms such that any v(∈ V)-component is positive(resp. v(∈ V)-component is positive if α(v) = +1 and negetive if α(v) = −1),and we shall refer to it as the subgroup of positive automorphisms (resp. thesubgroup of α-signed automorphisms).
(3) Let † D}± is a category equivalent to the model global object D}± in Definition 10.3.
Then by Remark 2.9.2, similarly we can define the set of cusps of † D }± and theset of ±-label classes of cusps
LabCusp± († D}± ),which can be identified with the set of cusps of † D}± .
([IUTchI, Definition 6.1 (iv)]) Let † D = {† Dv }v∈V , ‡ D =Definition 10.28.
‡†{ Dv }v∈V be D-prime-strips. For any v ∈ V, a +-full poly-isomorphism Dv‡+-full poly∼−→+-full poly∼‡Dv (resp. † D−→D) is a poly-isomorphism obtained as the Aut+ († Dv )-orbit(resp. Aut+ († D)-orbit) (or equivalently, Aut+ (‡ Dv )-orbit (resp. Aut+ (‡ D)-orbit)) of∼∼
an isomorphism † Dv → ‡ Dv (resp. † D → ‡ D). If † D = ‡ D, then there are precisely two+-full poly∼+-full poly-isomorphisms † Dv−→ † Dv (resp. the set of +-full poly-isomorphisms∼†
Dv → † Dv has a natural bijection with {±1}V ). We shall refer to the +-full polyisomorphism determined by the identity automorphism as positive, and the other one
negative (resp. the +-full poly-isomorphism corresponding to α ∈ {±1}V an α-signed+-full poly-automorphism). A capsule-+-full poly-morphism between capsulesof D-prime-strips†{ Dt }t∈T

capsule-+-full poly∼−→

is a collection of +-full poly-isomorphisms † Dt
ι : T ,→ T 0 .

{‡ Dt0 }t0 ∈T 0+-full poly∼‡−→Dι(t) , relative to some injection

Definition 10.29.
([IUTchI, Definition 6.1 (v)]) As in Definition 10.16, we cangroup-theoretically construct, from the model global object D}± in Definition 10.3, theouter homomorphism(Aut(X K ) ∼=) Aut(D}± ) → GL2 (Fl )/{±1}242

Go Yamashita

determined by EF [l], by considering the Galois action on ∆ab
X ⊗Fl (The first isomorphism
follows from Theorem 3.17). {(Note )}
that the image of the above outer homomorphism∗∗
contains the Borel subgroupof SL2 (Fl )/{±1} since the covering X K  XK
0∗corresponds to the rank one quotient ∆ab
X ⊗ Fl  Q. This rank one quotient determinesa natural surjective homomorphism
Aut(D}± )  F>l ,which can be reconstructed group-theoretically from D}± . We write Aut± (D}± ) ⊂∼Aut(D}± ) → Aut(X K ) for the kernel of the above homomorphism. Note that the sub∼group Aut± (D}± ) ⊂ Aut(D}± ) → Aut(X K ) contains AutK (X K ), and acts transitively
on the cusps of X K . Next, we write Autcusp (D}± ) ⊂ Aut(D}± ) for the subgroup of
automorphisms which fix the cusps of X K (Note that we can group-theoretically reconstruct this subgroup by Remark 2.9.2). Then we obtain natural outer isomorphisms∼∼

AutK (X K ) → Aut± (D}± )/Autcusp (D}± ) → Fo±l ,where the second isomorphism depends on the choice of the cusp  of C K . cf. also thefollowing diagram:
Aut(X K )O?

AutK (X K ) ∼/ Aut(D}± )
O
F>
l

l
(∗ ∗)
0 ~ ⊂ SL2 (Fl )/{±1}?
/ Aut± (D}± )
O
(
Fo±
l

/ / F>

?
Autcusp (D}± ).

1 ~
0 ±

/ / Fo±Gl)∼

If we write Aut+ (D}± ) ⊂ Aut± (D}± ) for the unique subgroup of index 2 containing
Autcusp (D}± ), then the cusp  determines a natural F±l -group structure on the subgroup
Aut+ (D}± )/Autcusp (D}± ) ⊂ Aut± (D}± )/Autcusp (D}± )(corresponding to Gal(X K /XK ) ⊂ AutK (X K )), and a natural F±
l -torsor structure on±}±
LabCusp (D ). Write also
V± := Aut± (D}± ) · V = Autcusp (D}± ) · V ⊂ V(K).
Note also that the subgoup Aut± (D}± ) ⊂ Aut(D}± ) ∼= Aut(X K ) can be identified±
with the subgroup of Aut(X K ) which stabilises V , and also that we can easily showthat V± = V±un (Definition 10.16) (cf. [IUTchI, Remark 6.1.1]).

243

a proof of the abc conjecture after Mochizuki

Remark 10.29.1. Note that Fo±l -symmetry permutes the cusps of X K without±
permuting V (⊂ V(K)), and is of geometric nature, which is suited to construct HodgeArakelov-theoretic evaluation map (Section 11).
>On the other hand, F>
l is a subquotient of Gal(K/F ) and Fl -symmetry permutes±±unvarious F>⊂ VBor (⊂ V(K)), and is of arithmetic nature (cf.
l -translates of V = V[IUTchI, Remark 6.12.6 (i)]), which is suite to the situation where we have to consider
descend from K to Fmod . Such a situation induces global Galois permutations of various
copies of Gv (v ∈ Vnon ) associated to distinct labels ∈ F>l which are only well-defined>
up to conjugacy indeterminacies, hence Fl -symmetry is ill-suited to construct HodgeArakelov-theoretic evaluation map.
Remark 10.29.2.

(cf. [IUTchII, Remark 4.7.6]) One of the important diﬀerences

o±
>
of F>
l -symmetry and Fl -symmetry is that Fl -symmetry does not permute the label 0
with the other labels, on the other hand, Fo±l -symmetry does.
We need to permute the label 0 with the other labels in Fo±l -symmetry to performthe conjugate synchronisation (cf. Corollary 11.16 (1)), which is used to construct “diagonal objects” or “horizontally coric objects” (cf. Corollary 11.16, Corollary 11.17, andCorollary 11.24) or “mono-analytic cores” (In this sense, label 0 is closely related to theunits and additive symmetry. cf. [IUTchII, Remark 4.7.3]),
On the other hand, we need to separate the label 0 from the other labels in F>l symmetry, since the simultaneous excutions of the final multiradial algorithms on objects
in each non-zero labels are compatible with each other by separating from mono-analytic
cores (objects in the label 0), i.e., the algorithm is multiradial (cf. Section 11.1, and§ A.4), and we perform Kummer theory for NF (Corollary 11.23) with F>l -symmetry>(since Fl -symmetry is of arithmetic nature, and suited to the situation involved Galois
group Gal(K/Fmod )) in the NF portion of the final multiradial algorithm. Note alsothat the value group portion of the final multiradial algorithm, which involves thetavalues arising from non-zero labels, need to be separated from 0-labelled objects (i.e.,
mono-analytic cores, or units). In this sense, the non-zero labels are closely related to
the value groups and multiplicative symmetry.
Definition 10.30.
(Model D-Θ± -Bridge, [IUTchI, Example 6.2]) In this definition, we regard Fl as an F±l -group. Let D = {D,v }v∈V , Dt = {Dv t }v∈V be copies of
the tautological D-prime-strip {Dv }v∈V for each t ∈ Fl (Here, we write v t for the pair(t, v)). For each t ∈ Fl , let±φΘvt: Dv t+-full poly∼−→D,v ,

±
φΘ
t

: Dvt

+-full poly
∼

−→

D,v

be the positive +-full poly-isomorphisms respectively, with respect to the identifications

244

Go Yamashita

with the tautological D-prime-strip {Dv }v∈V . Then we put
±

±

poly

Θ
φΘ
± := {φt }t∈Fl : D± := {Dt }t∈Fl −→ D .
±±We shall refer to φΘ± as model base-(or D-)Θ -bridge.
±
We have a natural poly-automorphism −1Fl of order 2 on the triple (D± , D , φΘ
± )as follows: The poly-automorphism −1Fl acts on Fl as multiplication by −1, and induces
poly∼+-full poly∼

the poly-morphisms Dt −→ D−t (t ∈ Fl ) and D
−→ D determined by the +full poly-automorphism whose sign at every v ∈ V is negative, with respect to theidentifications with the tautological D-prime-strip {Dv }v∈V . This −1Fl is compatible±Vwith φΘ± in the obvious sense. Similarly, each α ∈ {±1} determines a natural poly±±
automorphism αΘ of order 1 or 2 as follows: The poly-automorphism αΘ acts on Fl
as the identity and the α-signed +-full poly-automorphism on Dt (t ∈ Fl ) and D . This±±αΘ is compatible with φΘ± in the obvious sense.
Definition 10.31.
(Model D-Θell -Bridge, [IUTchI, Example 6.3]) In this definition, we regard Fl as an F±l -torsor. Let Dt = {Dv t }v∈V be a copy of the tautologicalD-prime-strip {Dv }v∈V for each t ∈ Fl , and write D± := {Dt }t∈Fl as in Definition 10.30.
Let D}± be the model global object in Definition 10.3. In the following, fix an isomor∼
phism LabCusp± (D}± ) → Fl of F±l -torsor (cf. Definition 10.29). This identification∼}±induces an isomorphism Aut± (D /Autcusp (D}± ) → Fo±of groups For v ∈ Vbadl(resp. v ∈ Vgood ∩ Vnon , resp. v ∈ Varc ), we write}±φΘ•,v : Dv −→ Dellfor the natural morphism correpsonding to X v → X v → X K (resp. →X → Xv → XK,
−
v

∼

}±

resp. a tautological morphism Dv = −X → Xv → X(D→v, v) (cf. also Definition 10.11(1), (2)).
Write
poly

}±
}±
) ◦ φΘ
φΘ,•,v ◦ Aut+ (Dv 0 ) : Dv 0 −→ Dv 0 := Autcusp (D
ell

ell

and
φΘ
0
Since φΘ
0

ell

ell

poly

:= {φvΘ0 }v∈V : D0 −→ D}± .
ellis stable under the action of Autcusp (D}± ), we obtain a poly-morphism
ell

φΘ
t

:= (action of t) ◦ φΘ
0

ell

poly

: Dt −→ D}± ,∼by post-composing a lift of t ∈ Fl ∼= Aut+ (D}± )/Autcusp (D}± ) (⊂ Fo±= Aut± (D}± )l/Autcusp (D}± )) to Aut+ (D}± ). Hence we obtain a poly-morphism
ell

φΘ
±

poly

:= {φΘ
}t∈Fl : D± −→ D}±
t
ell

245

a proof of the abc conjecture after Mochizuki

from a capsule of D-prime-strip to the global object D}± (cf. Definition 10.11 (3)). Thisis called the model base-(or D-)Θell -bridge.
Note that each γ ∈ Fo±
gives us a natural poly-automorphism γ± of D± as follows:l
The automorphism γ± acts on Fl via the usual action of Fo±
on Fl , and induces thel+-full poly∼+-full poly-isomorphism Dt−→ Dγ(t) whose sign at every v ∈ V is equal to thesign of γ. In this way, we obtain a natural poly-action of Fo±
on D± . On the other
l
o±
}±
}± ∼
hand, the isomorphism Aut± (D )/Autcusp (D ) → Fl determines a natural polyell
action of Fo±on D}± . Note that φΘis equivariant with respect to these natural
±
l
o±
}±
poly-actions of Fl on D± and D . Hence we obtain a natural poly-action of Fo±lell
on (D± , D}± , φΘ).
±
Definition 10.32.
Definition 6.4])(D-Θ± -Bridge, D-Θell -Bridge, D--Hodge Theatre, [IUTchI,(1) A base-(or D-)Θ± -bridge is a poly-morphism
† Θ±
φ±

poly

: † DT −→ † D ,where † D is a D-prime-strip, and † DT is a cupsule of D-prime-strips indexed∼ †∼ †by an F±l -group T , such that there exist isomorphisms D → D , D± → DT ,∼whose induced morphism Fl → T on the index sets is an isomorphism of F±
l Θ±
† Θ±
groups, and conjugation by which sends φ± 7→ φ± . An isomorphism of D))((±
poly ‡
poly †
∼
‡
†
‡ Θ±
0
:
D
−→
D
is a pair of a
:
D
−→
D
→
φ
Θ± -bridges † φΘ
T

T

±
±
capsule-+full poly∼∼
capsule-+-full poly-isomorphism † DT −→ ‡ DT 0 whose induced morphism T → T 0
on the index sets is an isomorphism of F±l -groups, and a +-full-poly isomorphism+-full poly
±
∼
‡ Θ±
†
‡
D
→
D , which are compatible with † φΘ± , φ± . We define compositionsof them in an obvious manner.
(2) A base-(or D-)Θell -bridge is a poly-morphism† Θellφ±poly: † DT −→ † D}± ,where † D}± is a category equivalent to the model global object D}± , and † DT
is a cupsule of D-prime-strips indexed by an F±l -torsor T , such that there exist∼ †∼}± ∼ † }±isomorphisms D→ D , D± → DT , whose induced morphism Fl → T on
the index sets is an isomorphism of F±l -torsors, and (conjugation by which sends
)
poly

∼

ell
† Θ
φΘ
7→ † φΘ
φ± : † DT −→ † D}± →±± . An isomorphism of D-Θ -bridges
capsule-+-full poly
(
)
poly
∼
‡ Θellφ± : ‡ DT 0 −→ ‡ D}± is a pair of a capsule-+-full poly-isomorphism † DT −→ ‡ DT 0
ell

ell

ell

246

Go Yamashita∼

whose induced morphism T → T 0 on the index sets is an isomorphism of F±l † }±‡ }±torsors, and an Autcusp ( D )-orbit (or, equivalently, an Autcusp ( D )-orbit)
†

}±

poly
∼ ‡

‡ Θ
D
→ D}± of isomorphisms, which are compatible with † φΘ± , φ± . We
define compositions of them in an obvious manner.
ell

ell

(3) A base-(or D-)Θ±ell -Hodge theatre (or a D--Hodge theatre) is a collection
)
(
ell
±
†

HT D- =

†

†

†

φΘ
±

φΘ
±

D ←− † DT −→ † D}±

,

±

† Θ
±
where T is an F±
is a D-Θell -bridge, and † φΘ
± is a D-Θ -bridge, suchl -group, φ±∼∼∼that there exist isomorphisms D}± → † D}± , D± → † DT , D → † D , conjugation
ell
ell
Θ±
† Θ±
by which sends φΘ
7→ † φΘ±± , φ± 7→ φ± . An isomorphism of D--Hodge
theatres is a pair of isomorphisms of D-Θell -bridges and D-Θ± -bridges such thatthey induce the same poly-isomorphism of the respective capsules of D-prime-strips.
We define compositions of them in an obvious manner.
ell

The following proposition is an additive analogue of Proposition 10.33, and follows
by the same manner as Proposition 10.33:(Transport of ±-Label Classes of Cusps via Base-Bridges,

Proposition 10.33.

†[IUTchI, Proposition 6.5]) Let HTtheatre.
(1) The D-Θell -bridge † φΘ±ell

† Θell
ζvt

D-

†

†

±

φΘ
±

†

†

ell

φΘ
±

= ( D ←− DT −→ † D}± ) be a D--Hodge

induces an isomorphism∼: LabCusp± († Dvt ) → LabCusp± († D}± )of F±l -torsors of ±-label classes of cusps for each v ∈ V, t ∈ T . Moreover, thecomposite∼† ΘellΘell −1† Θellξvt ,wt := († ζw)◦(
ζvt ) : LabCusp± († Dvt ) → LabCusp± († Dwt )t† Θis an isomorphism of F±l -groups for w ∈ V. By these identifications ξv t ,wt of± †F±l -groups LabCusp ( Dv t ) when we vary v ∈ V, we can write
ellLabCusp± († Dt )for them, and we can write the above isomorphism as an isomorphism† Θellζtof F±
l -torsors.

∼: LabCusp± († Dt ) → LabCusp± († D}± )247

a proof of the abc conjecture after Mochizuki±(2) The D-Θ± -bridge † φΘ
± induces an isomorphism
† Θ±
ζv t

∼

: LabCusp± († Dvt ) → LabCusp± († D,v )of F±
l -groups of ±-label classes of cusps for each v ∈ V, t ∈ T . Moreover, thecomposites† Θ±Θ±† Θell† Θ± −1ξ,v,w := († ζw
)
◦
ξ
◦
(
ζv 0 )
v
,w
0
0
0

∼

: LabCusp± († D,v ) → LabCusp± († D,w ),∼† Θ±
Θ± −1 † Θ±† Θ±ξ,vt ,wt := († ζw
)
◦
ξ
◦
(
ζvt ) : LabCusp± († Dvt ) → LabCusp± († Dwt ),v,wt(Here we write 0 for the zero element of the F±
l -group T ) are isomorphisms of
ell
±
† Θ±
Fl -groups for w ∈ V, and we also have ξvt ,wt = † ξvΘt ,wt . By these identifications± †† Θ±
ξ,v,w of F±l -groups LabCusp ( D,v ) when we vary v ∈ V, we can write

LabCusp± († D )±for them, and the various † ζvΘt ’s, and † ζvΘt ’s determine a single (well-defined)
isomorphism∼† Θellζt: LabCusp± († Dt ) → LabCusp± († D )
ell

of F±
l -groups.
(3) We have a natural isomorphism†∼

ζ± : LabCusp± († D}± ) → T

† Θ
of F±
(0) ∈ LabCusp± († D}± ),l -torsors, by considering the inverse of the map T 3 t 7→ ζt± †where we write 0 for the zero element of the F±
l -group LabCusp ( Dt ). Moreover,the composite
ell

±

∼

±

(† ζ0Θ )−1 ◦ († ζtΘ ) ◦ († ζtΘ )−1 ◦ († ζ0Θ ) : LabCusp± († D0 ) → LabCusp± († D0 )
ell

ell

is equal to the action of († ζ0Θ )−1 ((† ζ± )−1 (t)).
ell(4) For α ∈ Aut± († D}± )/Autcusp († D}± ), if we replece † φΘby α ◦ † φΘ±
± , then the† Θell† Θellresulting “ ζt ” is related to the original ζtby post-composing with the imageof α via the natural bijection
ell

ell

∼
Aut± († D}± )/Autcusp († D}± ) → Aut± (LabCusp± († D}± ))(∼
= Fo±
l )

(cf. also Definition 10.29).
The following is an additive analogue of Proposition 10.20, and it follows from thedefinitions:

248

Go Yamashita

Proposition 10.34.
(Properties of D-Θ± -Brideges, D-Θell -Bridges, D--Hodge
theatres, [IUTchI, Proposition 6.6])
±

±

±

±

‡ Θ
† Θ
‡ Θ
V
(1) For D-Θ± -bridges † φΘ± , φ± , the set Isom( φ± , φ± ) is a {±1}×{±1} -torsor,
where the first factor {±1} (resp. the second factor {±1}V ) corresponds to the poly±
automorphism −1Fl (resp. αΘ ) in Definition 10.30.
o±‡ Θ† NF ‡ NF(2) For D-Θell -bridges † φΘ± , φ± , the set Isom( φ> , φ> ) is an Fl -torsor, and‡ NF ∼
we have a natural isomorphism Isom(† φNF(T, T 0 ) of Fo±> , φ> ) = IsomF±
l l -torsors
torsors.
ellell(3) For D--Hodge theatres † HT D- , ‡ HT D- , the set Isom(† HT D- , ‡ HT D- ) isan {±1}-torsor, and we have a natural isomorphism Isom(† HT D- , ‡ HT D- ) ∼=0
IsomF± -groups (T, T ) of {±1}-torsors.
l

±

{

ell
† Θ
(4) For a D-Θ± -bridge † φΘ
± and a D-Θ -bridge φ± , the set

capsule-+-full poly-isom.

ellcapsule-+-full poly∼†DT −→ † DT 0±

† Θ
by which † φΘ± , φ±

ell}
form a D- -Hodge theatre

is an Fo±× {±1}V -torsor, where the first factor Fo±(resp. the subgroup {±1} ×llo±V{±1} ) corresponds to the Fl in (2) (resp. to the {±1}×{±1}V in (1)). Moreover,the first factor can be regarded as corresponding to the structure group of the Fo±
l 0
torsor IsomF± -torsors (T, T ).
lo±(5) For a D-Θell -bridge † φΘ
± , we have a functorial algorithm to construct, up to Fl ell
indeterminacy, a D--Hodge theatre whose D-Θell -bridge is † φΘ
± .
ellDefinition 10.35.
([IUTchI, Corollary 6.10]) Let † HT D- , ‡ HT D- be D-Hodge theatres. the base-(or D-)Θ±ell -link (or D--link)†D

HT D- −→‡HT D-

is the full poly-isomorphism†

D`>full poly
∼
‡

−→

D`>

between the mono-analyticisations of the D-prime-strips constructed in Lemma 10.38
in the next subsection.
Remark 10.35.1. In D--link, the D` -prime-strips are shared, but not the arithmetically holomorphic structures. We can visualise the “shared” and “non-shared”
relation as follows:†` ∼‡ `HT D- − − > † D>
= D> > − − ‡ HT D-249

a proof of the abc conjecture after Mochizuki

We shall refer to this diagram as the étale-picture of D--Hodge theatres. Notethat we have a permutation symmetry in the étale-picture.
(Θ±-Bridge, Θell -Bridge, -Hodge Theatre, [IUTchI, Deifi-

Definition 10.36.
nition 6.11])(1) A Θ± -bridge is a poly-morphism
†

±

poly

Θ
ψ±
: † FT −→ ‡ F ,where † F is an F-prime-strip, and † FT is a cupsule of F-prime-strips indexed by anpoly±† Θ± †
F±
φ± : DT −→ † D .
l -group T , which lifts (cf. Lemma(10.10 (2)) a D-Θ -bridge
)
(
)
poly
poly
∼
Θ±
Θ±
An isomorphism of Θ± -bridges † ψ±
: † FT −→ † F → ‡ ψ±
: ‡ FT 0 −→ ‡ F
poly
∼ ‡

†

†

poly
∼ ‡

is a pair of poly-isomorphisms FT −→ FT 0 and F −→ F , which lifts a mor±‡ Θ±phism between the associated D-Θ± -bridges † φΘ± , φ± . We define compositionsof them in an obvious manner.
(2) A Θell -bridge
†

Θ
ψ±

ell

poly

: † FT −→ † D}± ,where † D}± is a category equivalent to the model global object D}± in Definition 10.3, and † FT is a capsule of F-prime-strips indexed by an F±l -torsor T , is apoly

D-Θell -bridge † φΘ±

: † DT −→ † D}± , where † DT is the associated capsule of D)
(
poly
∼
Θell
: † FT −→ † D}± →prime-strips to † FT . An isomorphism of Θell -bridges † ψ±
poly
poly
(
)
poly ‡ }±∼ ‡‡ Θell‡†† }± ∼
00
ψ± : FT −→ D
is a pair of poly-isomorphisms FT −→ FT and D
−→
ell

‡

D}± , which determines a morphism between the associated D-Θell -bridges † φΘ± ,‡ Θellφ± . We define compositions of them in an obvious manner.
ell(3) A Θ±ell -Hodge theatre (or a -Hodge theatre) is a collection
(
)
±
ell
†

±

HT



=

†

†

Θ
ψ±

†

†

Θ
ψ±

F ←− FT −→ † D}±,ΘΘ
is a Θell -bridge, such that the associated
where † ψ±
is a Θ± -bridge, and † ψ±±ell
† ΘellD-Θ± -bridge † φΘ
form a D--Hodge the± and the associated D-Θ -bridge φ±atre. An isomorphism of -Hodge theatres is a pair of a morphism of Θ± bridge and a morphism of Θell -bridge, which induce the same bijection between therespective capsules of F-prime-strips. We define compositions of them in an obvious
manner.
ell

250

Go Yamashita

The following lemma follows from the definitions:Lemma 10.37.
(Properties of Θ± -Brideges, Θell -Bridges, -Hodge theatres,[IUTchI, Corollary 6.12])
±

±

Θ
Θ
Θ
Θ
(1) For Θ± -bridges 1 ψ±
, 2 ψ±(resp. Θell -bridges 1 ψ±
, 2 ψ±, resp. -Hodge the 21±
atres HT , HT ) whose associated D-Θ -bridges (resp. D-Θell -bridges, resp.
±
D- 2
2 Θ±
1 Θell 2 Θell
1
D--Hodge theatres) are 1 φΘ, HT D- )± , φ± (resp. φ± , φ± , resp. HT
respectively, the natural map
ell

±

±

±

ell

ell

ell

ell

±

Θ
Θ
2 Θ
Isom(1 ψ±, 2 ψ±) → Isom(1 φΘ
± , φ± )
ell

Θ
Θ
2 Θ
(resp. Isom(1 ψ±, 2 ψ±) → Isom(1 φΘ
± , φ± ),resp. Isom(1 HT  , 2 HT  ) → Isom(1 HT D- , 2 HT D- ))is bijective.
Θ(2) For a Θ± -bridge ‡ ψ±{
capsule-+-full poly-isom.

±Θand a Θell -bridge ‡ ψ±
, the setell

capsule-+-full poly∼‡
FT −→ ‡ FT 0

}
±

Θ
Θ
by which ‡ ψ±, ‡ ψ±

ell

form a -Hodge theatre

is an Fo±×{±1}V -torsor. Moreover, the first factor can be regarded as correspondl0
ing to the structure group of the Fo±l -torsor IsomF± -torsors (T, T ).
l

§ 10.6.

Θ±ell NF-Hodge Theatres — An Arithmetic Analogue of theUpper Half Plane.

In this subsection, we combine the multiplicative portion of Hodge theatre and the
additive portion of Hodge theature to obtain full Hodge theatre.
Lemma 10.38.

(From (D-)Θ± -Bridge To (D-)Θ-Bridge, [IUTchI, Definition 6.4
±

poly

†
†
(i), Proposition 6.7, Definition 6.11 (i), Remark 6.12 (i)]) Let † φΘ
± : DT −→ D (resp.
†

±

poly

Θ
ψ±
: † FT −→ † F ) be a D-Θ± -bridge (resp. Θ± -bridge). We write†

D|T | (resp.

†

F|T | )

for the l± -capsule (cf. Section 0.2 for l± ) of D-prime-strips (resp. F-prime-strips)
obtained from l-capsule † DT (resp. † FT ) of D-prime-strips (resp. F-prime-strips) byforming the quotient |T | of the index set T by {±1}, and identifying the components ofthe cupsule † DT (resp. † FT ) in the same fibers of T  |T | via the components of the
±
† Θ±
† Θ±
† Θ±
poly-morphism † φΘ
± = { φt }t∈T (resp. ψ± = { ψt }t∈T ) (Hence each component

251

a proof of the abc conjecture after Mochizukiof † D|T | (resp.
write

†

F|T | ) is only well-defined up to a positive automorphism). We also†

DT > (resp.

†

FT > )

for the l> -capsule determined by the subset T > := |T | \ {0} of non-zero elements of |T |.
±±We identify † D0 (resp. † F0 ) with † D (resp. † F ) via † φΘ
(resp. † ψ0Θ ), and0
we write † D> (resp. † F> ) for the resulting D-prime-strip (resp. F-prime-strip) (i.e.,>= {0, }). For v ∈ Vgood , we replace the +-full poly-morphism at v-component of† Θ±
Θ±φ± (resp. † ψ±
) by the full poly-morphism. For v ∈ Vbad , we replace the +-full±† Θ±poly-morphism at v-component of † φΘ
± (resp. ψ± ) by the poly-morphism determinedby (group-theoretically reconstructed) evaluation section as in Definition 10.17 (resp. by
the poly-morphism lying over (cf. Definition 10.23 (1), (2), and Remark 10.10.1) the
poly-morphism determined by (group-theoretically reconstructed) evaluation section as
in Definition 10.17). Then we algorithmically obtain a D-Θ-bridge (resp. a potion of
Θ-bridge)
poly
† Θ †
φ> : DT > −→ † D>

(resp.

†

poly

Θ †
ψ>
: FT > −→ † F> )in a functorial manner. cf. also the following:†

D0 , † D
†
Dt , † D−t (t 6= 0)†
DT |T \{0}

7→
7→
7→

†

†
D> ,
F0 , † F
†
D|t| , † Ft , † F−t (t 6= 0)††
DT > ,
FT |T \{0}

7→
7→
7→

†

F> ,
F|t|
†
FT > ,†

where we write |t| for the image of t ∈ T under the surjection T  |T |.
Definition 10.39.

poly([IUTchI, Remark 6.12.2]) Let † FT −→ † F be a Θ± -bridge,
polywhose associated D-Θ± -bridge is † DT −→ † D . Then we have a group-theoreticallypolyfunctorial algorithm for constructing a D-Θ-bridge † DT > −→ † D> from the D-Θ± poly
poly
bridge † DT −→ † D by Lemma 10.38. Suppose that this D-Θ-bridge † DT > −→ † D>
polyarises as the D-Θ-bridge associated to a Θ-bridge ‡ FJ −→ ‡ D> 99K ‡ HT Θ , where
J = T >:†

poly

FT −→ † F_†DT −→ D poly †

poly

‡

poly

FJ −→ ‡ D> 99K ‡ HT Θ_/ † DT > −→ D> .
poly †

poly

Then the poly-morphism ‡ FJ −→ ‡ F> lying over † DT > −→ † D> is completely determined (cf. Definition 10.23 (1), (2), and Remark 10.10.1). Hence we can regard
poly

this portion ‡ FJ −→ ‡ F> of the Θ-bridge as having been constructed via the functorial algorithm of Lemma 10.38. Moreover, by Lemma 10.25 (1), the isomorphisms

252

Go Yamashita

between Θ-bridges have a natural bijection with the the isomorphisms between thepoly“‡ FJ −→ ‡ F> ”-portion of Θ-bridges.
poly
In this situation, we say that the Θ-bridge ‡ FJ −→ ‡ D> 99K ‡ HT Θ (resp. D-Θpoly

poly

bridge † DT > −→ † D> ) is glued to the Θ± -bridge † FT −→ † F (resp. D-Θ± -bridge†poly

DT −→ † D ) via the functorial algorithm in Lemma 10.38. Note that, by Proposition 10.20 (2) and Lemma 10.25 (1), the gluing isomorphism is unique.
Definition 10.40.
nition 6.13])(D-  -Hodge Theatre, -Hodge Theatre, [IUTchI, Defi-(1) A base-(or D-)Θ±ell NF-Hodge theatre † HT D- is a tripe of a D--Hodge
theatre † HT D- , a D--Hodge theatre † HT D- , and the (necessarily unique) gluing
isomorphism between † HT D- and † HT D- . We define an isomorphism of D- 
-Hodge theatres in an obvious manner.
(2) A Θ±ell NF-Hodge theatre † HT  is a tripe of a -Hodge theatre † HT  , a
-Hodge theatre † HT  , and the (necessarily unique) gluing isomorphism between†
HT  and † HT  . We define an isomorphism of -Hodge theatres in an
obvious manner.
§ 11.

Hodge-Arakelov-theoretic Evaluation Maps.
§ 11.1.

Radial Environments.

In inter-universal Teichmüller theory, not only the existence of functorial grouptheoretic algorithms, but also the contents of algorithms are important. In this subsection, we introduce important notions of coricity, uniradiality, and multiradiality for thecontents of algorithms.
Definition 11.1.

(Radial Environment, [IUTchII, Example 1.7, Example 1.9])(1) A radial environment is a triple (R, C, Φ), where R, C are groupoids (i.e., categories in which all morphisms are isomorphisms) such that all objects are isomorphic, and Φ : R → C is an essentially surjective functor (In fact, in our mind, we
expect that R and C are collections of certain “type of mathematical data” (i.e.,
species), and Φ is “algorithmically defined” functor (i.e., mutations). In this survey, we avoid the rigorous formulation of the language of species and mutations (cf.
[IUTchIV, §3]), and we just assume that R, C to be as above, and Φ to be a functor.
cf. also Remark 3.4.4 (2)). We shall refer to C as a coric category an object of Ca coric data, R a radial category an object of R a radial data, and Φ a radialalgorithm.

a proof of the abc conjecture after Mochizuki

253(2) We shall refer to Φ as multiradial, if Φ is full. We shall refer to Φ as uniradial,
if Φ is not full. We shall refer to (R, C, Φ) as multiradial environment (resp.
uniradial environment), if Φ is multiradial (resp. uniradial).
Note that, if Φ is uniradial, then an isomoprhism in C does not come from an
isomorphism in R, which means that an object of R loses a portion of rigidityby Φ, i.e., might be subject to an additional indeterminacy (From another pointof view, the liftability of isomorphism, i.e., multiradiality, makes possible doing akind of parallel transport from another radial data via the associated coric data. cf.
[IUTchII, Remark 1.7.1]).
(3) Let (R, C, Φ) be a radial environment. Let † R be another groupoid in which all
objects are isomorphic, † Φ : † R → C an essentially surjective functor, and ΨR :R → † R a functor. We shall refer to ΨR as multiradially defined or multiradial(resp. uniradially defined or uniradial) if Φ is multiradial (resp. uniradial) andif the diagram
ΨR †
/ R
R
}
}
}}
Φ
}
†
 ~}}} ΦCis 1-commutative. We shall refer to ΨR as corically defined (or coric), if ΨR has
a factorisation ΞR ◦ Φ, where ΞR : C → † R is a functor, and if the above diagram
is 1-commutative.
(4) Let (R, C, Φ) be a radial environment. Let E be another groupoid in which all
objects are isomorphic, and Ξ : R → E a functor. We write
Graph(Ξ)
for the category whose objects are pairs (R, Ξ(R)) for R ∈ Ob(R), and whosemorphisms are the pairs of morphisms (f : R → R0 , Ξ(f ) : Ξ(R) → Ξ(R0 )). We
shall refer to Graph(Ξ) as the graph of Ξ. We have a commutative diagram/ Graph(Ξ)
vv
vv
Φ
v
v
vvΦ
 v
z v Graph(Ξ)
C,
R

ΨΞ

of natural functors, where ΨΞ : R 7→ (R, Ξ(R)) and ΦGraph(Ξ) : (R, Ξ(R)) 7→ Φ(R).
Remark 11.1.1.
([IUTchII, Example 1.7 (iii)]) A crucial fact on the consequenceof the multiradiality is the following: For a radial environment (R, C, Φ), we write R×C R

254

Go Yamashita

for the category whose objects are triple (R1 , R2 , α), where R1 , R2 ∈ Ob(R), and α is an∼
isomorphism Φ(R1 ) → Φ(R2 ), and whose morphisms are morphisms of triples definedin an obvious manner. Then the switching functor
R ×C R → R ×C R : (R1 , R2 , α) 7→ (R2 , R1 , α−1 )preserves the isomorphism class of objects of R ×C R, if Φ is multiradial, since any∼
object (R1 , R2 , α) in R ×C R is isomorphic to the object (R1 , R1 , id : Φ(R1 ) → Φ(R1 )).
This means that, if the radial algorithm is multiradial, then we can switch two radial
data up to isomorphism.
Ultimately, in the final multiradial algorithm, we can “switch”, up to isomorphism, the theta values (more precisely, Θ-pilot object, up to mild indeterminacies)2“{‡ q j }1≤j≤l> ” on the right-hand side of (the final update of) Θ-link to the theta valvues (more precisely, Θ-pilot object, up to mild indeterminacies) “{† q j }1≤j≤l> ” on the2v

left-hand side of (the final update of) Θ-link, which is isomorphic to ‡ q (more precisely,v

q-pilot object, up to mild indeterminacies) by using the Θ-link compatibility of the final
multiradial algorithm (Theorem 13.12 (3)):{‡ q j }N1≤j≤l>
2v! {† q j }N1≤j≤l>!!

2v

Then we cannot distinguish {‡ q j }1≤j≤l> from ‡ q
2

v

v

∼
=

‡ N

q

v

up to mild indeterminacies (i.e.,(Indet ↑), (Indet →), and (Indet xy)), which gives us a upper bound of height function(cf. also Appendix A).
Example 11.2.
(1) A classical example is holomorphic structures on R2 :†_CforgetR2 oforget ‡ C,

where R is the category of 1-dimensional C-vector spaces and isomorphisms of Cvector spaces, C is the category of 2-dimensional R-vector spaces and isomorphismsof R-vector spaces, and Φ sends 1-dimensional C-vector spaces to the underlying
R-vector spaces. Then the radial environment (R, C, Φ) is uniradial. Note thatthe underlying R2 is shared (i.e., coric), and that we cannot see one holomorphic
structure † C from another holomorphic structure ‡ C.

255

a proof of the abc conjecture after Mochizuki

Next, we replace R by the category of 1-dimensional C-vector spaces † C equipped∼with the GL2 (R)-orbit of an isomorphism † C → R2 (for a fixed R2 ). Then the
resulting radial environment (R, C, Φ) is tautologically multiradial:∼(† C → R2 x GL2 (R))_
forgetR2 o∼ (‡ C →R2 x GL (R)).
2forgetNote that the underlying R2 is shared (i.e., coric), and that we can describe thediﬀerence between one holomorphic structure † C and another holomorphic structure‡
C in terms of the underlying analytic structure R2 .
(2) An arithmetic analogue of the above example is as follows: As already explainedin Section 3.5, the absolute Galois group Gk of an MLF k has an automorphism
which does not come from any automorphism of fields (at least in the case where
the residue characteristic is 6= 2), and one “dimension” is rigid, and the other“dimension” is not rigid, hence we consider Gk as a mono-analytic structure. On
the other hand, from the arithmetic fundamental group ΠX of hyperbolic orbicurve
X of strictly Belyi type over k, we can reconstruct the field k (Theorem 3.17),
hence we consider ΠX as an arithmetically holomorphic structure, and the quotient(ΠX )Gk (group-theoretically reconstructable by Corollary 2.4) as the underlying
mono-analytic structure. For a fixed hyperbolic orbicurve X of strictly Belyi typeover an MLF k, let R be the category of topological groups isomorphic to ΠX
and isomorphisms of topological groups, and C the category of topological groups
isomorphic to Gk and isomorphisms of topological groups, and Φ be the functor
which sends Π to the group-theoretically reconstructed quotien (Π )G. Then theradial environment (R, C, Φ) is uniradial:
†

_Π

†

∃



∃

G∼
= Gk ∼= ‡G o ‡ Π.

Next, we replace R by the category of topological groups isomorphic to ΠX equipped∼with the full-poly isomorphism G → Gk , where (Π )G is the group-theoretic reconstructed quotient. Then the resulting radial environment (R, C, Φ) is tautologi-

256

Go Yamashita

cally multiradial:††( Π G

full poly
∼

_

→

Gk )


†

full polyG∼=full poly

Gk

∼
=

‡

 (‡ Π  † G

Gofull poly
∼

→

Gk ).

cf. also the following table (cf. [Pano, Fig. 2.2, Fig. 2.3]):

coric

underlying analytic str.

R2

G

uniradial

holomorphic str.

CΠmultiradial∼

C → R x GL2 (R )holomorphic str. described2

2

full poly∼Π/∆ −→ Gin terms of underlying coric str.

In the final multiradial algorithm (Theorem 13.12), which admits mild indeterminacies, we describe the arithmetically holomorphic structure on one side of (the final
update of) Θ-link from the one on the other side, in terms of shared mono-analytic
structure.
Definition 11.3.
([IUTchII, Definition 1.1, Proposition 1.5 (i), (ii)]) Let MΘ∗ =ΘΘ(· · · ← MM ← MM 0 ← · · · ), be a projective system of mono-theta environments de). For each N , by Coroltermined by X v (v ∈ Vbad ), where MΘ
, DMΘ
, sΘ
M = (ΠMΘ
MΘ
M
M
M
lary 7.22 (3) and Lemma 7.12, we can functorially group-theoretically reconstruct, fromMΘN , a commutative diagram

Πtemp
MΘ
;
ON
v
v
v
v
vv
v
v
?
,v 

/ ∆tempµN (MΘ
)N
MΘ
N

Gv (MΘ
)8
OO N fNf NkVkNVVVVVV8pp
NNN VVVVV
ppp
VVVV
NNN
p
p
VVVV
p
N
p
VVVV
N
p
p 





temp
temp
temp
/Π
/Π
/ Πtemp (MΘ )
//Π
(MΘ
(MΘ
(MΘ
N)
N)
N)
X
C
Y
X
O N
O
O
O
?
/ / ∆temp (MΘ )  YN?
/ ∆temp (MΘ )  XN?
/ ∆temp (MΘ )  XN?
/ ∆temp (MΘ )
C

N

257

a proof of the abc conjecture after Mochizukiof topological groups, which is an isomorph of

Πtemp
[µN ]
Y
;w O
ww
ww
w
ww
?
- w w
temp/∆Y [µN ]µN

Gv iSiw; ; OO cGc GSGSSSSSw
GG SSSS
w
GG
ww
SSSS
w
GG
w
SSSS
w w
S





/ / Πtemp/ Πtemp/ Πtemp/ Πtemp
Y
X
CO
X
O
O
O
?
/ / ∆temp  ?
/ ∆temp  

Y

X

?
/ ∆temp  X?
/ ∆temp .
C

For each N , by Theorem 7.23 (1), we can also functorially group-theoretically reconstruct an isomorph (l∆Θ )(MΘN ) of the internal cyclotome and the cyclotomic rigidity
isomorphism∼Θ(l∆Θ )(MΘN ) ⊗ (Z/N Z) → µN (MN ).
The transition morphisms of the resulting projective system {· · · ← Πtemp
(MΘ
M) ←
X
Πtemp
(MΘ
M 0 ) ← · · · } are all isomorphism. We identify these topological groups via
Xthese transition morphisms, and we write Πtemp(MΘ∗ ) for the resulting topological
X
temp
temp
temp
temp
group. Similarly, we define Gv (MΘ
(MΘ
(MΘ
(MΘ
(MΘ
∗ ), ΠY∗ ), ΠX∗ ), ΠC∗ ), ∆Y
∗ ),
temp
temp
temp
temp
Θ
Θ
∆temp
(MΘ
(MΘ
(MΘ
(MΘ
(MΘ
∗ ), ∆X∗ ), ∆C∗ ), (l∆Θ )(M∗ ) from Gv (M∗ ), ΠYN ), ∆Y
N ),
X
Θ
Θ
Θ
∆temp
(MΘ
b (M∗ ) := limN µN (MN ), then we obN ), (l∆Θ )(MN ) respectively. We write µZ
X←−tain a cyclotomic rigidity isomorphism∼Θ(l∆Θ )(MΘb (M∗ ).
∗ ) → µZ

Proposition 11.4.
(Multiradial Mono-theta Cyclotomic Rigidity, [IUTchII, Corollary 1.10]) Let Πv be the tempered fundamental group of the local model objects X v forv ∈ Vbad in Definition 10.2 (1), and (Πv )Gv the quotient group-theoretically reconstructed by Lemma 6.2.
(1) Let C ` be the category whose objects are
G y O×µ (G),
where G is a topological group isomorphic to Gv , O×µ (G) is the group-theoreticallyreconstructed monoid by Proposition 5.2 (Step 1) and Definition 8.5 (1), and whose∼morphisms (G y O×µ (G)) → (G0 y O×µ (G0 )) are pairs of the isomorphism∼
G → G0 of topological groups, and an Isomet(G)-multiple of the functorially group∼
theoretically reconstructed isomorphism O×µ (G) → O×µ (G0 ) from the isomorphism∼
G → G0 .

258(

Go Yamashita(2) Let RΘ be the category whose objects are triples
Π y µZb (MΘ∗ (Π)) ⊗ Q/Z ,

GyO

×µ

(G) ,

poly
×µ
αµ,×µ : (Π y µZb (MΘ(G))|Π
∗ (Π)) ⊗ Q/Z) −→ (G y O

where Π is a topological group isomorphic to Πv , the topological group (Π )G is
the quotient group-theoretically reconstructed by Lemma 6.2, we write (−)|Π for the
restriction via Π  G, we write µZb (MΘ∗ (Π)) for the external cyclotome (cf. justafter Theorem 7.23) of the projective system of mono-theta environment MΘ
∗ (Π)
group-theoretically reconstructed from Π by Corollary 7.22 (2) (Note that such a
projective system is uniquely determined, up to isomorphism, by the discrete rigidity(Theorem 7.23 (2))), and αµ,×µ is the composite××µµZb (MΘ(Π)∗ (Π)) ⊗ Q/Z ,→ O (Π)  O

poly∼→ O×µ (G)of ind-topological modules equipped with topological group actions, where the firstarrow is given by the composite of the tautological Kummer map for MΘ
∗ (Π) andthe inverse of the isomorphism induced by the cyclotomic rigidity isomorphism of
mono-theta environment (cf. the diagrams in Proposition 11.7 (1), (4)), the second
arrow is the natural surjection and the last arrow is the poly-isomorphism induced
full poly∼by the full poly-isomorphism Π/∆ → G (Note that the composite of the above
diagram is equal to 0), and whose morphisms are pairs (fΠ , fG ) of the isomorphism∼
00ΘfΠ : (Π y µZb (MΘ
b (M∗ (Π ))⊗Q/Z) of ind-topological mod∗ (Π))⊗Q/Z) → (Π y µZ∼
ules equipped with topological group actions induced by an isomorphism Π → Π0 of
topological groups with an Isomet(G)-multiple of the functorially group-theoretically∼Θ0reconstructed isomorphism µZb (MΘ
b (M∗ (Π )) ⊗ Q/Z, and the iso∗ (Π)) ⊗ Q/Z → µZ∼
morphism fG : (G y O×µ (G)) → (G0 y O×µ (G0 )) of ind-topological modules∼
equipped with topological group actions induced by an isomorphism G → G0 of topological groups with an Isomet(G)-multiple of the functorially group-theoretically re∼constructed isomorphism O×µ (G) → O×µ (G0 ) (Note that these isomorphisms are0
automatically compatible αµ,×µ and αµ,×µ
in an obvious sense).
(3) Let ΦΘ : RΘ → C ` be the essentially surjective functor, which sends (Π y µZb (MΘ∗ (Π))⊗×µ×µQ/Z, G y O (G), αµ,×µ ) to G y O (G), and (fΠ , fG ) to fG .
(4) Let E Θ be the category whose objects are the cyclotomic rigidity isomorphismsof mono-theta environments∼(l∆Θ )(Π) → µZb (MΘ∗ (Π))
reconstructed group-theoretically by Theorem 7.23 (1), where Π is a topologicalgroup isomorphic to Πv , the cyclomotmes (l∆Θ )(Π) and µZb (MΘ∗ (Π)) are the internal and external cyclotomes respectively group-theoretically reconstructed from Π

)
,

259

a proof of the abc conjecture after Mochizuki∼

by Corollary 7.22 (1), and whose morphisms are pair of isomorphisms (l∆Θ )(Π) →∼Θ0(l∆Θ )(Π0 ) and µZb (MΘ
b (M∗ (Π )) which are induced functorially group∗ (Π)) → µZ∼
theoretically reconstructed from an isomorphism of topological groups Π → Π0 .
(5) Let ΞΘ : RΘ → E Θ be the functor, which sends (Π y µZb (MΘ∗ (Π)) ⊗ Q/Z, G y×µO (G), αµ,×µ ) to the cyclotomic rigidity isomorphisms of mono-theta environ∼ments (l∆Θ )(Π) → µZb (MΘ∗ (Π)) reconstructed group-theoretically by Theorem 7.23(1), and (fΠ , fG ) to the isomorphism functorially group-theoretically reconstructed∼from Π → Π0 .
Then the radial environment (RΘ , C ` , ΦΘ ) is multiradial, and ΨΞΘ is multiradially defined, where ΨΞΘ the naturally defined functorΨΞΘ/ Graph(ΞΘ )
ss
ss
s
ΦΘ
s
 yssssΦGraph(ΞΘ )C`RΘ

by the construction of the graph of ΞΘ .
Proof. By noting that the composition in the definition of αµ,×µ is 0, and thatfull poly∼

we are considering the full poly-isomorphism Π/∆ −→ G, not the tautological single∼
isomorphism Π/∆ → G, the proposition immediately from the definitions.
Remark 11.4.1.
†

Let see the diagram†
Π y µZb (MΘ∗ ( Π)) ⊗ Q/Z_(† G y O×µ († G)) ∼= (‡ G y O×µ (‡ G)) o ‡ Π y µ (MΘ (‡ Π)) ⊗ Q/Z,b
Z∗

by dividing into two portions:††Π/† ∆ −→ G

0

_Π

†

full poly
∼


Go

‡

‡

 ‡ Π.

full poly∼Π/ ∆ −→ G

O

µ


×µ o

‡
0

µ,

On the left-hand side, by “loosening” (cf. taking GL2 (R)-obit in Exapmle 11.2) the∼∼natural single isomorphisms † Π/† ∆ → G, ‡ Π/‡ ∆ → G by the full poly-isomorphisms(This means that the rigidification on the underlying mono-analytic structure G by

260

Go Yamashita

the arithmetically holomorphic structure Π is resolved), we make the topological group
portion of the functor Φ full (i.e., multiradial).
On the right-hand side, the fact that the map µ → O×µ is equal to zero makes theind-topological module portion of the functor Φ full (i.e., multiradial). This means thatit makes possible to “simultaneously perform” the algorithm of the cyclotomic rigidity
isomorphism of mono-theta environment without making harmfull eﬀects on other radial
data, since the algorithm of the cyclotomic rigidity of mono-theta environment uses
only µ-portion (unlike the one via LCFT uses the value group portion as well), andthe µ-portion is separated from the relation with the coric data, by the fact that thehomomorphism µ → O×µ is zero.
For the cyclotomic rigidity via LCFT, a similarly defined radial environment isuniradial, since the cyclotomic rigidity via LCFT uses the value group portion as well,and the value group portion is not separated from the coric data, and makes harmfull
eﬀects on other radial data. Even in this case, we replace O (−) by O× (−), and web × -indeterminacy on the cyclotomic rigidity, then it is tautologically multiradial
admit Z
as seen in the following proposition:
Proposition 11.5.
(Multiradial LCFT Cyclotomic Rigidity with Indeterminacies, [IUTchII, Corollary 1.11]) Let Πv be the tempered fundamental group of the local
model objects X v for v ∈ Vbad in Definition 10.2 (1), and (Πv )Gv the quotientgroup-theoretically reconstructed by Lemma 6.2.
(1) Let C ` be the same category as in Proposition 11.4.
(2) Let RLCFT be the category whose objects are triples()
Π y O (Π) , G y Ogcp (G) , α,×µ , ,where Π is a topological group isomorphic to Πv , the topological group (Π )Gis the quotient group-theoretically reconstructed by Lemma 6.2, O (Π) is the indtopological monoid determined by the ind-topological field group-theoretically reconstructed from Π by Corollary 3.19 and αµ,×µ is the following diagram:b× -orbit
Z



g
c
p

(Π y O (Π)) ,→ (Π y O (Π))

poly
∼

→

(G y Ogcp (G))|Π ←- (G y O× (G))|Π  (G y O×µ (G))|Πof ind-topological monoids equipped with topological group actions determined by thefull poly∼b × -orbit of the poly-morphism determined by the full poly-morphism Π/∆ −→Z
G, where ∆ := ker(Π  G) and the natural homomorphisms, where Ogcp (Π) :=limJ⊂Π : open (O (Π)gp )J (resp. Ogcp (G) := limJ⊂G : open (O (G)gp )J ), and whose
−→
−→

261

a proof of the abc conjecture after Mochizuki∼

morphisms are pairs (fΠ , fG ) of the isomorphism fΠ : (Π y O (Π)) → (Π0 yO (Π0 )) of ind-topological monoids equipped with topological group actions induced∼
by an isomorphism Π → Π0 of topological groups with an Isomet(G)-multiple of∼the functorially group-theoretically reconstructed isomorphism O (Π) → O (Π0 ),∼and the isomorphism fG : (G y Ogcp (G)) → (G0 y Ogcp (G0 )) of ind-topological∼groups equipped with topological group actions induced by an isomorphism G → G0 of
topological groups with an Isomet(G)-multiple of the functorially group-theoretically∼reconstructed isomorphism Ogcp (G) → Ogcp (G0 ) (Note that these isomorphisms are0
automatically compatible α,×µ and α,×µin an obvious sense).
(3) Let ΦLCFT : RLCFT → C ` be the essentially surjective functor, which sends (Π yO (Π), G y Ogcp (G), α,×µ ) to G y O×µ (G), and (fΠ , fG ) to the functorially∼group-theoretically reconstructed isomorphism (G y O×µ (G)) → (G0 y O×µ (G0 )).
b × -orbit (= the full(4) Let E LCFT be the category whose objects are the pairs of the Z
poly-isomorphism, cf. Remark 3.19.2 in the case of O× )poly∼

µZb (G) → µZb (O× (G))
of cyclotomic rigidity isomorphisms via LCFT reconstructed group-theoreticallyby Remark 3.19.2 (for M = O× (G)), and the Aut(G)-orbit (which comes fromfull poly∼

the full poly-isomorphism Π/∆ −→ G)
poly∼µZb (G) → (l∆Θ )(Π)
of the isomorphism obtained as the composite of the cyclotomic rigidity isomor∼
phism via positive rational structure and LCFT µZb (G) → µZb (Π) group-theoretically∼reconstructed by Remark 6.12.2 and the cyclotomic rigidity isomorphism µZb (Π) →(l∆Θ )(Π) group-theoretically reconstructed by Remark 9.4.1, where Π is a topological group isomorphic to Πv , the topological group (Π )G is the quotient grouptheoretically reconstructed by Lemma 6.2, and (l∆Θ )(Π) is the internal cyclotome
group-theoretically reconstructed from Π by Corollary 7.22 (1), and whose mor∼∼phisms are triple of isomorphisms µZb (G) → µZb (G0 ), µZb (O× (G)) → µZb (O× (G0 ))∼and (l∆Θ )(Π) → (l∆Θ )(Π0 ) which are induced functorially group-theoretically re∼constructed from an isomorphism of topological groups Π → Π0 .
(5) Let ΞLCFT : RLCFT → E LCFT be the functor, which sends (Π y O (Π), G y
Ogcp (G), α,×µ ) to the pair of group-theoretically reconstructed isomorphisms, and∼(fΠ , fG ) to the isomorphism functorially group-theoretically reconstructed from Π →
Π0 .

262

Go Yamashita

Then the radial environment (RLCFT , C ` , ΦLCFT ) is multiradial, and ΨΞLCFT is multiradially defined, where ΨΞLCFT the naturally defined functor
ΨΞLCFT/ Graph(ΞLCFT )
ooo
o
LCFT
o
Φ
o
oooΦGraph(ΞLCFT )
o
 o
o
w
C`
RLCFT

by the construction of the graph of ΞLCFT .
Definition 11.6.
([IUTchII, Remark 1.4.1 (ii)]) Recall that we have hyperbolic
orbicurves X v  X v  C v for v ∈ Vbad , and a rational pointµ− ∈ X v (Kv )Z(i.e., “−1” in Grigm /qX v . cf. Definition 10.17). The unique automorphism ιX of X vof order 2 lying over ιX (cf. Section 7.3 and Section 7.5) corresponds to the unique∆temp-outer automorphism of Πtemp
over Gv of order 2. We also write ιX for the
X
X
v

v

latter automorphism by abuse of notation. We also have tempered coverings Ÿ v 
Y v  X v . Note that we can group-theoretically reconstruct Πtemp, Πtempfrom ΠX byYŸ
Corollary 7.22 (1) and the description of Ÿ

v

v

v

 Y . We write Πtemp(Π), Πtemp(Π) for the
Y
Ÿ
v
v

reconstructed ones from a topological group Π isomorphic to ΠẌ , respectively. Sincev

Kv contains µ4l , there exist rational points(µ− )Ÿ ∈ Ÿ v (Kv ), (µ− )X ∈ X v (Kv ),such that (µ− )Ÿ 7→ (µ− )X → µ− . Note that ιX fixes the Gal(X v /X v )-orbit of (µ− )X ,since ιX fixes µ− , hence ιX fixes (µ− )X , since Aut(X v ) ∼= µl × {±1} by Remark 7.12.1(Here, ιX corresponds to the second factor of µl × {±1} since l 6= 2). Then it follows
that there exists an automorphismιŸof Ÿ of order 2 lifting ιX , which is uniquely determined up to lZ-conjugacy and composition with an element ∈ Gal(Ÿ /Y ) ∼= µ2 , by the condition that it fixes the Gal(Ÿ /Y )v

v

v

v

orbit of some element (“(µ− )Ÿ ” by abuse of nonation) of the Gal(Ÿ v /X v )(∼= lZ × µ2 )orbit of (µ− )Ÿ . We also write ιŸ for the corresponding ∆Ÿtemp -outer automorphism ofv

Πtemp
by abuse of notation. We shall refer to ιŸ as an inversion automorphism asŸvwell. We write ιŸ for the automorphism of Ÿv induced by ιŸ .

263

a proof of the abc conjecture after Mochizuki

We writeDµ− ⊂ Πtemp
Ÿv

for the decomposition group of (µ− )Ÿ , which is well-defined up to ∆temp-conjugacy.
Ÿv

Hence Dµ− is determined by ιŸ up to ∆temp
Yv -conjugacy. We shall refer to the pairs(

ιŸ ∈ Aut(Ÿ v ) , (µ− )Ÿ

)

)
(
temp
temp
, or
ιŸ ∈ Aut(ΠŸ )/Inn(∆Ÿ ) , Dµ−vv

as a pointed inversion automorphism. Recall that an étale theta function of standard type is defined by the condition on the restriction to Dµ− is in µ2l (Definition 7.7
and Definition 7.14).
Proposition 11.7.
(Multiradial Constant Multiple Rigidity, [IUTchII, CorolΘ`Θlary 1.12]) Let (R , C , Φ ) be the multiradial environment defined in Proposition 11.4.
(1) There is a functorial group-theoretic algorithm to reconstruct, from a topologicalgroup Π isomorphic to Πtemp(v ∈ Vbad ), the following commutative diagram:Xv

O× (Π) ∪ O× (Π) · ∞ θ(Π) / ∞ H 1 (Πtemp (Π), (l∆Θ )(Π))
Ÿ

∼
=



×
Θ
Θ
(Π))
∪
O
(M
O× (MΘ
∗ (Π)) · ∞ θ env (M∗ (Π))∗∼= Cycl. Rig. Mono-Th. in Prop.11.4

/ ∞ H 1 (Πtemp (MΘ(Π)),µb (MΘ
∗ (Π))),∗
Z

Ÿ

where we put, for a topological group Π isomorphic to Πtemp(resp. for a projecXv

temp
tive system MΘ(Π) (resp.
∗ of mono-theta environments determined by X v ), ΠŸ
temp
Πtemp
(MΘ
reconstructed from Πtemp(Π) by Defini∗ )) to be the isomorph of ΠŸŸŸtion 11.6 (resp. from Πtemp(MΘ∗ ) by Definition 11.3 and the descrption of Ÿ  Y ),
Ÿ
and
∞H

∞H

1

1

(Π), (l∆Θ )(Π)) :=(ΠtempŸΘ(Πtemp(MΘb (M∗ )) :=∗ ), µZŸlim−→

H 1 (Πtemp(Π) ×Π J, (l∆Θ )(Π)),
Ÿ

lim
−→

Θ
(MΘ
H 1 (Πtempb (M∗ )),∗ ) ×Π J, µZ
ŸJ⊂Π : open, of fin. index

J⊂Π : open, of fin. indexand∞ θ(Π) (⊂ ∞ H

1

temp
1
Θ
(Πtemp(Π), (l∆Θ )(Π)) (resp. ∞ θenv (MΘ(MΘ
b (M∗ )) )∗ ) (⊂ ∞ H (ΠŸ∗ ), µZŸ264

Go Yamashita

for the subset of elements for which some positive integer multiple (if we consider
multiplicatively, some positive integer power) is, up to torsion, equal to an elementof the subset
temp1Θθ(Π) (⊂ H 1 (Πtemp(Π), (l∆Θ )(Π)) (resp. θenv (MΘ(MΘ
b (M∗ )) )∗ ) (⊂ H (ΠŸ∗ ), µZŸ

of the µl -orbit of the reciprocal of lZ × µ2 -orbit η̈ Θ,lZ×µ2 of an l-th root of theétale theta function of standard type in Section 7.3 (resp. corresponding to theµl -orbit of the reciprocal of (lZ × µ2 )-orbit η̈ Θ,lZ×µ2 of an l-th root of the étaletheta function of standard type in Section 7.3, via the cyclotomic rigidity isomor∼Θphism (l∆Θ )(MΘb (M∗ ) group-theoretically reconstructed by Theorem 7.23
∗ ) → µZ(1), where we write (l∆Θ )(MΘ∗ ) for the internal cyclotome of the projective systemΘM∗ of mono-theta environments group-theoretically reconstructd by Theorem 7.23(1)) (Note that these can functorially group-theoretically reconstructed by the constant multiple rigidity (Proposition 11.7)), and we define
O× (MΘ
∗ (Π))
to be the submodule such that the left vertical arrow is an isomorphism. We also
put
O× ∞ θ(Π) := O× (Π) · ∞ θ(Π),×ΘΘ
O× ∞ θenv (MΘ∗ (Π)) := O (M∗ (Π)) · ∞ θ env (M∗ (Π)).

(2) There is a functorial group-theoretic algorithmΠ 7→ {(ι, D)}(Π),
which construct, from a topological group Π isomorphic to Πtemp, a collectionXvof pairs (ι, D), where ι is a ∆temp(Π)(:= Πtemp(Π) ∩ ∆)-outer automorphism of
Ÿ
Ÿ
Πtemp
(Π), and D ⊂ Πtemp(Π) is a ∆temp(Π)-conjgacy class of closed subgroupsŸŸŸ
corresponding to the pointed inversion automorphisms in Definition 11.6. We shallrefer to each (ι, D) as a pointed inversion automorphism as well. For a pointed
inversion automorphism (ι, D), and a subset S of an abelian group A, if ι acts on
Im(S → A/Ators ), then we write S ι := {s ∈ S | ι(s mod Ators ) = s mod Ators }.
(3) Let (ι, D) be a pointed inversion automorphism reconstructed in (1). Then the restriction to the subgroup D ⊂ Πtemp(Π) gives us the following commutative diagram:Ÿ{O× ∞ θ(Π)}ι
ι
{O× ∞ θenv (MΘ∗ (Π))}/ O× (Π)()⊂ ∞ H 1 (Π, (l∆Θ )(Π))∼= Cycl. Rig. Mono-Th. in Prop.11.4

/ O× (MΘ∗ (Π))()⊂ ∞ H 1 (Π, µZb (MΘ
(Π))
,
∗

265

a proof of the abc conjecture after Mochizuki

where we put∞H1(Π, (l∆Θ )(Π)) :=lim−→

H 1 (J, (l∆Θ )(Π)),

lim−→

H 1 (J, µZb (MΘ∗ (Π))).

J⊂Π : open, of fin. index∞H1(Π, µZb (MΘ∗ (Π))) :=J⊂Π : open, of fin. index

Note that the inverse image of the torsion elements via the upper (resp. lower) horiι
zontal arrow in the above commutative diagram is equal to ∞ θ(Π)ι (resp. ∞ θenv (MΘ∗ (Π)) ).
In particular, we obtain a functorial algorithm of constructing splittingsΘιµΘ
O×µ (Π) × {∞ θ(Π)ι /Oµ (Π)}, O×µ (MΘ∗ (Π)) × {∞ θ env (M∗ (Π)) /O (M∗ (Π))}

of {O× ∞ θ(Π)}ι /Oµ (Π)ιµΘ(resp. {O× ∞ θenv (MΘ∗ (Π))} /O (M∗ (Π)) ).

×µ(4) For an object (Π y µZb (MΘ(G), αµ,×µ ) of the radial category∗ (Π)) ⊗ Q/Z, G y O
ΘR , we assign• the projective system MΘ
∗ (Π) of mono-theta environments,• the subsets O× (Π) ∪ O× ∞ θ(Π) (⊂ ∞ H 1 (Πtemp(Π), (l∆Θ )(Π))), and
Ÿ
temp
×
Θ
1
Θ
O× (MΘ(MΘb (M∗ (Π)))) in (1),∗ (Π))∪O ∞ θ env (M∗ (Π)) (⊂ ∞ H (ΠŸ∗ (Π)), µZ• the splittings O×µ (Π) × {∞ θ(Π)ι /Oµ (Π)}, andιµΘΘO×µ (MΘ∗ (Π)) × {∞ θ env (M∗ (Π)) /O (M∗ (Π))} in (3), and• the diagram
∼
∼
µ
Θ
µ
×
×µ
µZb (MΘ(Π)∗ (Π))⊗Q/Z → O (M∗ (Π)) → O (Π) ,→ O (Π)  O

poly∼→ O×µ (G),

where the first arrow is induced by the tautological Kummer map for MΘ
∗ (Π),
the second arrow is induced by the vertical arrow in (1), the third and thefourth arrow are the natural injection and surjection respectively (Note that thecomposite is equal to 0), and the last arrow is the poly-isomorphism inducedby the full poly-isomorphism Π/∆

full poly∼→G.

Then this assignment determines a functor Ξenv : RΘ → E env , and the naturalfunctor ΨΞenv : RΘ → Graph(Ξenv ) is multiradially defined.
Proof. Proposition immediately follows from the described algorithms.
Remark 11.7.1.
tions:†∞ θ( Π)cf. also the following étale-pictures of the étale theta func-− − > G y O×µ (G) x Isomet(G) > − − ∞ θ(‡ Π)

266

Go YamashitaΘ †∞ θ env (M∗ ( Π))‡− − > G y O×µ (G) x Isomet(G) > − − ∞ θenv (MΘ∗ ( Π))

Note that the object in the center is a mono-analytic object, and the objects in the left
and in the right are holomorphic objects, and that we have a permutation symmetry
in the étale-picture, by the multiradiality of the algorithm in Proposition 11.7 (cf. also
Remark 11.1.1).
Remark 11.7.2.

([IUTchII, Proposition 2.2 (ii)]) The subsetθι (Π) ⊂ θ(Π) (resp. ∞ θι (Π) ⊂ ∞ θ(Π) )determines a specific µ2l (O(Π))-orbit (resp. Oµ (Π)-orbit) within the unique (lZ × µ2l )orbit (resp. each (lZ × µ)-orbit) in the set θ(Π) (resp. ∞ θ(Π)).
§ 11.2.

Hodge-Arakelov-theoretic Evaluation and Gaussian Monoids at
Bad Places.

In this subsection, we perform Hodge-Arakelov-theoretic evaluation, and construct
Gaussian monoids for v ∈ Vbad (Note that the case for v ∈ Vbad plays a central role).
Recall that Corollary 7.22 (2) reconstructs a mono-theta environment from a topological group (“Π 7→ M”) and Theorem 8.14 reconstructs a mono-theta environment
from a tempered Frobenioid (“F 7→ M”). First, we transport theta classes θ and thetheta evaluations from a group-theoretic situation to a mono-theta-theoretic situationvia (“Π 7→ M”) and the cyclotomic rigidity for mono-theta environments, then, via(“F 7→ M”), a Frobenioid-theoretic situation can access to the theta evaluation (cf. also
[IUTchII, Fig. 3.1]):
F
/Mo
Π
θ, eval /θ

F-Theoretic Theta Monoidsenv, evalenv ,Kummer /M-Theoretic Theta Monoids
Galois Evaluation

F-Theoretic Gaussian Monoids oM-Theoretic Gaussian Monoids.

(Kummer)−1 , or forget

Note also that, from the view point of the scheme-theoretic Hodge-Arakelov theory
and p-adic Hodge theory (cf. Appendix A), the evaluation maps correspond, in somesense, to the comparison map, which sends Galois representations to filtered ϕ-modules
in the p-adic Hodge theory.
Definition 11.8.

([IUTchII, Remark 2.1.1, Proposition 2.2, Definition 2.3])

267

a proof of the abc conjecture after Mochizuki(1) For a hyperbolic orbicurve (−)v over Kv , we write Γ(−) for the dual graph of thespecial fiber of a stable model. Note that each of maps
ΓŸ

/ ΓY

ΓX


ΓŸ


/ ΓY ,


ΓX

induces a bijection on vertices, since the covering X v  X v is totally ramified at
the cusps. We writeΓIX ⊂ ΓXfor the unique connected subgraph of ΓX , which is a tree and is stabilised by ιX(cf. Section 7.3, Section 7.5, and Definition 11.6), and contains all vertices of ΓX .
We writeΓ•X ⊂ ΓI
X
for the unique connected subgraph of ΓX , which is stabilised by ιX and contains precisely one vertex and no edges. Hence if we write labels on ΓX by {−l> , . . . , −1, 0, 1, . . . , l> },where 0 is fixed by ιX , then ΓI
X is obtained by removing, from ΓX , the edge connecting the vertices labelled by ±l> , and Γ•X consists only the vertex labelled by
0. From Γ•X ⊂ ΓI
X (⊂ ΓX ), by taking suitable connected components of inverse
images, we obtain finite connected subgraphsΓ•X ⊂ ΓIX ⊂ ΓX ,Γ•Ÿ ⊂ ΓI⊂ ΓŸ ,ŸΓ•Ÿ ⊂ ΓI⊂ ΓŸ ,Ÿ

which are stabilised by respective inversion automorphisms ιX , ιŸ , ιŸ (cf. Section 7.3, Section 7.5, and Definition 11.6). Note that each ΓI(−) maps isomorphicallyI
to ΓX .
(2) WriteΠv• := Πtemp
X , Γ•v

X⊂ ΠvI := Πtemp
X , ΓIv

X⊂ Πv (= Πtemp)
Xv

for Σ := {l} in the notation of Corollary 6.9 (i.e., H = ΓIX ), Note that we haveΠvI ⊂ Πtemp∩ Πv = Πtemp. Note also that ΠvI is well-defined up to Πv -conjugacy,Yv
Yv

and after fixing ΠvI , the subgroup Πv• ⊂ ΠvI is well-defined up to ΠvI -conjugacy.
Moreover, note that we may assume that Πv• , ΠvI and ιŸ have been chosen so thatsome representative of ιŸ stabilises Πv• and ΠvI . Finally, note also that, from Πv ,we can functorially group-theoretically reconstruct the data (Πv• ⊂ ΠvI ⊂ Πv , ιŸ )up to Πv -conjugacy, by Remark 6.12.1.

268

Go Yamashita(3) We write
temp
temp
temp
temp
±
∆v := ∆temp, ∆±, ∆cor, Πcorv := ∆Xv := ∆Cv , Πv := ΠXv := ΠCv
Xv

v

v(Note also that we can group-theoretically reconstruct these groups from Πv byd for the profinite completion in thisLemma 7.12). We also use the notation (−)
subsection. We also putΠ±(Πv• ) ⊂ Π±
(ΠvI ) ⊂ Π±vI := NΠ±v.
v• := NΠ±v
v
Note that we have
∼
∼
∼
∼
±
±
±
∼
Π±
v• /Πv• → ΠvI /ΠvI → Πv /Πv → ∆v /∆v → Gal(X v /X v ) = Z/lZ,

and
±
Π±
v• ∩ Πv = Πv• , ΠvI ∩ Πv = ΠvI ,since Πv• and ΠvI are normally terminal in Πv , by Corollary 6.9 (6).
bb±(4) A ±-label class of cusps of Πv (resp. of Π±v , resp. of Πv , resp. of Πv )bb±is the set of Πv -conjugacy (resp. Π±v -conjugacy, resp. Πv -conjugacy, resp. Πv b
conjugacy) classes of cuspidal inertia subgroups of Πv (resp. of Π±v , resp. of Πv ,
±
±
b±
b±
resp. of Πv ) whose commensurators in Πv (resp. in Πv , resp. in Πv , resp. inb ± ) determine a single Π± -conjugacy (resp. Π± -conjugacy, resp. Π
b ± -conjugacy,
Π
v
v
v
v
±
±
b±
b±
resp. Π-conjugacy)
classofsubgroups
inΠ(resp.
in
Π
,
resp.
in Π
v
v
v
v , resp.
±b v ). (Note that this is group-theoretic condition. Note also that such a set of
in Π
b
b±
Πv -conjugacy (resp. Π±v -conjugacy, resp. Πv -conjugacy, resp. Πv -conjugacy) classis of cardinality 1, since the covering X v  X v is totally ramified at cusps (or thecovering X v  X v is trivial).) We write± b± b±
LabCusp± (Πv ) (resp. LabCusp± (Π±v ), resp. LabCusp (Πv ), resp. LabCusp (Πv ) )bfor the set of ±-label classes of cusps of Πv (resp. of Π±v , resp. of Πv , resp. of±±b v ). Note that LabCusp (Πv ) can be naturally identified with LabCusp± († Dv ) inΠ
Definition 10.27 (2) for † Dv := Btemp (Πv )0 , and admits a group-theoretically reconstructable natural action of F×l , a group-theoretically reconstructable zero element±† 0
η v ∈ LabCusp (Πv ) = LabCusp± († Dv ), and a group-theoretically reconstructable±-canonical element † η ±∈ LabCusp± (Πv ) = LabCusp± († Dv ) well defined up tov
multiplication by ±1.

(5) An element t ∈ LabCusp± (Πv ) determines a unique vertex of ΓI
X (cf. Corollary 6.9I(4)). We write Γ•t
X ⊂ ΓX for the connected subgraph with no edges whose uniquea proof of the abc conjecture after Mochizuki

269

vertex is the vertex determined by t. Then by a functorial group-theoretic algorithm,
Γ•t
X gives us a decomposition groupΠv•t ⊂ ΠvI ⊂ Πv
well-defined up to ΠvI -conjugacy. We also writeΠ±(Πv•t ).
v•t := NΠ±v∼(Note that we have a natural isomorphism Π±v•t /Πv•t → Gal(X v /X v ) by Corollary 6.9 (6)).
± b±×(6) The images in LabCusp± (Π±v ) (resp. LabCusp (Πv )) of the Fl -action, the zeroelement † η 0v , and ±-canonical element † η ±of LabCusp± (Πv ) in the above (4), viav±b±
the natural outer injection Πv ,→ Π±v (resp. Πv ,→ Πv ), determine a natural Fl ± b±
torsor structure (cf. Definition 10.26 (2)) on LabCusp± (Π±v ) (resp. LabCusp (Πv )).
±±b cor b ±b±Moreover, the natural action of Πcorv /Πv (resp. Πv /Πv ) on Πv (resp. Πv )preserves this F±
l -torosr structure, thus, determines a natural outer isomorphism± ∼ o±
corb ± ∼ o±b cor
Πv /Πv = Fl (resp. Πv /Πv = Fl ).
b v ) is not normal in Πcorb cor
(resp. Π
Here, note that, even though Πv (resp. Πvv ), thebcuspidal inertia subgroups of Πv (resp. Πv ) are permuted by the conjugate action
±
b cor
b±
(resp. Πof Πcorv ), since, for a cuspidal inertia subgroup I in Πv (resp. Πv ), wev
b v = I l ) (Here, we write multiplicatively in the notation
have I ∩ Πv = I l (resp. I ∩ Π
cor
b±
b cor
I l ), and Π±(resp. Πv (resp. Πv ) is normal in Πvv ) ([IUTchII, Remark 2.3.1]).
Lemma 11.9.

([IUTchII, Corollary 2.4]) Let t ∈ LabCusp± (Πv ). Write

temp
±
±
∆v•t := ∆v ∩ Πv•t , ∆±, ∆v¨•t := ∆v ∩ Πv¨•t ,•t := Πv•t ∩ ΠŸ
v•t := ∆v ∩ Πv•t , Πv¨
v

temp
±
±
∆vI := ∆v ∩ ΠvI , ∆±, ∆vÏ := ∆v ∩ ΠvÏ .
vI := ∆v ∩ ΠvI , Πv Ï := ΠvI ∩ ΠŸ
v

Note that we have[Πv•t : Πv¨•t ] = [ΠvI : ΠvÏ ] = [∆v•t : ∆v¨•t ] = [∆vI : ∆vÏ ] = 2,
±
±
±
[Π±
v•t : Πv•t ] = [ΠvI : ΠvI ] = [∆v•t : ∆v•t ] = [∆vI : ∆vI ] = l.

(1) Let It ⊂ Πv be a cuspidal inertia subgroup which belongs to the ±-label class t suchγb±that It ⊂ ∆v•t (resp. It ⊂ ∆vI ). For γ ∈ ∆v , we write (−) for the conjugationb±γ(−)γ −1 by γ. Then for γ 0 ∈ ∆v , the following are equivalent:±0(a) γ 0 ∈ ∆±v•t (resp. γ ∈ ∆vI ),270

Go Yamashita
00(b) Itγγ ⊂ Πγv•t (resp. Itγγ ⊂ ΠγvI ),
0

0

γ
γ
(c) Itγγ ⊂ (Π±(resp. Itγγ ⊂ (Π±vI ) ).
v•t )b±(2) In the situation of (1), write δ := γγ 0 ∈ ∆v , then any inclusion00Itδ = Itγγ ⊂ Πγv•t = Πδv•t (resp. Itδ = Itγγ ⊂ ΠγvI = ΠδvI )as in (1) completely determines the following data:(a) a decomposition group Dtδ := NΠδv (Itδ ) ⊂ Πδv¨•t (resp. Dtδ := NΠδv (Itδ ) ⊂ ΠδvÏ ),δ(b) a decomposition group Dµδ − ⊂ ΠδvÏ , well-defined up to (Π±vI ) -conjugacy (or,δequivalently (∆±vI ) -conjugacy), corresponding to the torsion point µ− in Definition 11.6.
δδ(c) a decomposition group Dt,µ⊂ Πδv¨•t (resp. Dt,µ⊂ ΠδvÏ ), well-defined up to
−
−
± δ
± δ
δ
(Π±
v•t ) -conjugacy (resp. (ΠvI ) -conjugacy) (or equivalently, (∆v•t ) -conjugacy
δ(resp. (∆±vI ) -conjugacy)), that is, the image of an evaluation section corresponding to µ− -translate of the cusp which gives rise to Itδ .

Moreover, the construction of the above data is compatible with conjugation by arb±
bitrary δ ∈ ∆v as well as with the natural inclusion Πv•t ⊂ ΠvI , as we vary the
non-resp’d case and resp’d case.
(3) (Flo± -symmetry) The construction of the data (2a), (2c) is compatible with conjugab cor b ± ∼ b cor b ± ∼ o±b cortion by arbitrary δ ∈ Πv , hence we have a ∆v /∆v → Πv /Πv → Fl -symmetryon the construction.
Proof. We show (1). The implications (a) ⇒ (b) ⇒ (c) are immediately follow
from the definitions. We show the implication (c) ⇒ (a). We may assume γ = 1 without
0
γ0
±
±
±
loss of generality. Then the condition Itγ ⊂ Π±v•t ⊂ Πv (resp. It ⊂ ΠvI ⊂ Πv )implies γ 0 ∈ ∆±v by Theorem 6.11 (“profinite conjugates vs. tempered conjugates”). By
0
d
b±
b±
Corollary 6.9 (4), we obtain γ 0 ∈ ∆v•t (resp. γ ∈ ∆vI ), where we write (−) for theb±closure in ∆v (which is equal to the profinite completion, by Corollary 6.9 (2)). Then
±
±
±
0
±
b±
b±
we obtain γ 0 ∈ ∆v•t ∩ ∆v = ∆v•t (resp. γ ∈ ∆vI ∩ ∆v = ∆vI ) by Corollary 6.9 (3).
(2) follows from Theorem 3.7 (elliptic cuspidalisation) and Remark 6.12.1 (together
with Lemma 7.16, Lemma 7.12) (cf. also Definition 10.17). (3) follows immediately fromthe described algorithms.
We write(l∆Θ )(ΠvÏ )271

a proof of the abc conjecture after Mochizuki

for the subquotient of ΠvÏ determined by the subquotient (l∆Θ )(Πv ) of Πv (Note that∼the inclusion ΠvÏ ,→ Πv induces an isomorphism (l∆Θ )(ΠvÏ ) → (l∆Θ )(Πv )). We writeΠv  Gv (Πv ), ΠvÏ  Gv (ΠvÏ )for the quotients determined by the natural surjection Πv  Gv (Note that we canfunctorially group-theoretically reconstruct these quotients by Lemma 6.2 and Definition 11.8 (2)).
Proposition 11.10.
Corollary 2.6])(Π-theoretic Theta Evaluation, [IUTchII, Corollary 2.5,

0(1) Let Itδ = Itγγ ⊂ ΠδvÏ ⊂ ΠγvI = ΠδvI be as in Lemma 11.9 (2). Then the restrictionof the ιγ -invariant sets θι (Πγv ), ∞ θι (Πγv ) of Remark 11.7.2 to the subgroup ΠγvÏ ⊂Πtemp(Πv )(⊂ Πv ) gives us µ2l -, µ-orbits of elementsŸθι (ΠγvÏ ) ⊂ ∞ θι (ΠγvÏ ) ⊂ ∞ H 1 (ΠγvÏ , (l∆Θ )(ΠγvÏ )) :=

lim
−→

b Π
b v : openJ⊂γb
H 1 (ΠγvÏ ×Π
b v J, (l∆Θ )(Πv Ï )).

δ
The further restriction of the decomposition groups Dt,µin Lemma 11.9 (2) gives−
us µ2l -, µ-orbits of elementsθt (ΠγvÏ ) ⊂ ∞ θt (ΠγvÏ ) ⊂ ∞ H 1 (Gv (ΠγvÏ ), (l∆Θ )(ΠγvÏ )) :=for each t ∈ LabCusp±(Πγv )

conj. by γ
∼

−→

lim
−→
γ

JG ⊂Gv (ΠvÏ ) : open

H 1 (JG , (l∆Θ )(ΠγvÏ )),LabCusp± (Πv ). Since the sets θt (ΠγvÏ ),γt∞ θ (Πv Ï ) depend only on the label |t| ∈ |Fl |, we write

θ|t| (ΠγvÏ ) := θt (ΠγvÏ ),

∞θ

|t|

(ΠγvÏ ) := ∞ θt (ΠγvÏ ).

γb±(2) If we start with an arbitrary ∆v -conjugate Πv Ï of Πv Ï , and we consider the resultδb±ing µ2l -, µ-orbits θ|t| (Πγ ), ∞ θ|t| (Πγ ) arising from an arbitrary ∆v -conjugate It
v Ï

v Ï

conj. by γ∼of It contained in ΠγvÏ , as t runs over LabCusp± (Πγv ) −→ LabCusp± (Πv ), thenwe obtain a group-theoretic algorithm to construct the collections of µ2l -, µ-orbits
{
}
{
}
γ
|t|
θ|t| (ΠγvÏ )
(Π
,
θ
)
,
∞
v Ï
|t|∈|Fl ||t|∈|Fl |which is functorial with respect to the isomorphisms of topological groups Πv , andγ1b±compatible with the independent conjugacy actions of ∆=b±v on the sets {It }γ1 ∈Π
v
γ1
γ2
γ2
{It }γ1 ∈∆and {ΠvÏ }γ2 ∈Π= {ΠvÏ }γ2 ∈∆
b±
b±
b±
v
v
v

272

Go Yamashita(3) The γ-conjugate of the quotient ΠvÏ  Gv (ΠvÏ ) determines subsets(∞ H 1 (Gv (ΠγvÏ ), (l∆Θ )(ΠγvÏ )) ⊃) O× (ΠγvÏ ) ⊂ ∞ H 1 (ΠγvÏ , (l∆Θ )(ΠγvÏ )),O× θι (ΠγvÏ ) := O× (ΠγvÏ )θι (ΠγvÏ ) ⊂ O× ∞ θι (ΠγvÏ ) := O× (ΠγvÏ )∞ θι (ΠγvÏ ) ⊂ ∞ H 1 (ΠγvÏ , (l∆Θ )(ΠγvÏ )),which are compatible with O× (−), O× ∞ θι (−) in Proposition 11.7, respectively, relative to the first restriction operation in (1). We put
O×µ (ΠγvÏ ) := O× (ΠγvÏ )/Oµ (ΠγvÏ ).

(4) In the situation of (1), we take t to be the zero element. Then the set θt (ΠγvÏ ) (resp.
γγtµ∞ θ (Πv Ï )) is equal to µ2l (resp. µ). In particular, by taking quotietn by O (Πv Ï ),δ
the restriction to the decomposition group Dt,µ(where t is the zero element) gives−

us splittings
O×µ (ΠγvÏ ) × {∞ θι (ΠγvÏ )/Oµ (ΠγvÏ )}of O× ∞ θι (ΠγvÏ )/Oµ (ΠγvÏ ), which are compatible with the splittings of Proposition 11.7
(3), relative to the first restriction operation in (1):label 00x
/ O×µ (Πγ )v Ï/ O× ∞ θι (Πγ )/Oµ (Πγ )
v Ï

v Ï

/ ∞ θι (Πγ )/Oµ (Πγ )
v Ï

v Ï

/ 0.

Remark 11.10.1. (principle of Galois evaluation, [IUTchII, Remark 1.12.4]) Let
us consider some “mysterious evaluation algorithm” which constructs theta values froman abstract theta function, in general. It is natural to require that this algorithm
is compatible with taking Kummer classes of the “abstract theta function” and the“theta values”, and that this algorithm extend to coverings on both input and output
data. Then by the natural requirement of functoriality with respect to the Galois
groups on either side, we can conclude that the “mysterious evaluation algorithm” infact arises from a section G → ΠŸ (Π) of the natural surjection ΠŸ (Π)  G, as in
Proposition 11.10. We shall refer to this as the principle of Galois evaluation.
Moreover, from the point of view of Section Conjecture, we expect that this sections
arise from geometric points (as in Proposition 11.10).
Remark 11.10.2. ([IUTchII, Remark 2.6.1, Remark 2.6.2]) It is important that
we perform the evaluation algorithm in Proposition 11.10 (1) by using single base point,i.e., connected subgraph ΓI
X ⊂ ΓX , and that the theta values
θ|t| (ΠvÏ γ) ⊂ H 1 (Gv (ΠvÏ γ), (l∆Θ )(ΠvÏ γ))

a proof of the abc conjecture after Mochizuki

273

live in the cohomology of single Galois group Gv (ΠvÏ γ) with single cyclotome (l∆Θ )(ΠγvÏ )coeﬃcient for various |t| ∈ |Fl |, since we want to consider the collection of the thetavalues for |t| ∈ |Fl |, not as separated objects, but as “connected single object”, by synchronising indeterminacies via Fo±l -symmetry, when we construct Gaussian monoids
via Kummer theory (cf. Corollary 11.17).
Remark 11.10.3.

([IUTchII, Remark 2.5.2]) WriteΠ}± := ΠX K , ∆}± := ∆X K .

b±Recall that, using the global data ∆}± (∼=∆v ), we write ±-labels on local objects in
a consistent manner (Proposition 10.33), where the labels are defined in the form ofb±conjugacy classes of It . Note that ∆}± (∼= ∆v ) is a kind of “ambient container” of±b v -conjugates of both It and ∆vÏ . On the other hand, when we want to vary v,∆
the topological group ΠvÏ is purely local (unlike the label t, or conjugacy classes ofb±It ), and cannot be globalised, hence we have the independence of the ∆}± (∼= ∆v )conjugacy indeterminacies which act on the conjugates of It and ∆vÏ . Moreover, sinceb cor
b cor b ± ∼ o± deos not have a splitting, the ∆b cor
the natural surjection ∆v  ∆v /∆v = Flv -outercor b ± ∼ o±}± ∼ b ±baction of ∆v /∆v = Fl in Lemma 11.9 (3) induces independent ∆= ∆v -conjugacy
indeterminacies on the subgroups It for distinct t.
Remark 11.10.4. ([IUTchII, Remark 2.6.3]) We explain the choice of ΓI⊂ ΓŸ .
Ÿ
0
Let Γ ⊂ ΓŸ be a finite subgraph. Then
(1) For the purpose of getting single base point as explained in Remark 11.10.2, thesubgraph Γ0 should be connected.
(2) For the purpose of getting the crucial splitting in Proposition 11.10 (4), the subgraphΓ0 should contain the vertex of label 0.
(3) For the purpose of making the final height inequality sharpest (cf. the calculationsin the proof of Lemma 1.10), we want to maximise the value
{ }
1 ∑
2
0
min
,
j∈Γ , j≡j in |Fl | j
#Γ0
>
j∈Fl

where we identified ΓŸ with Z. Then we obtain #Γ0 ≥ l> , since the above function
is non-decreasing when #Γ0 grows, and constant for #Γ0 ≥ l> .
(4) For the purpose of globalising the monoids determined by theta values, via global
realified Frobenioids (cf. Section 11.4), such a manner that the product formula
should be satisfied, the set {j ∈ Γ0 , j ≡ j in |Fl |} should consist of only one element

274

Go Yamashita

for each j ∈ F>l , because the independent conjugacy indeterminacies explained in
Remark 11.10.3 are incompatible with the product formula, if the set has more than
two elements.
Then the only subgraph satisfying (1), (2), (3), (4) is ΓI.
ŸΘΘFor a projective system MΘ
∗ = (· · · ← MM ← MM 0 ← · · · ) of mono-theta environtemp
Θ
∼
, DMΘ
, sΘ
), putments such that ΠX (MΘ∗ ) = Πv , where MM = (ΠMΘ
MΘ
M
M
M

:= lim ΠMΘ
.
ΠMΘ
∗
M
←−
M

Note that we have a natural homomorphism ΠMΘ→ Πtemp(MΘ∗ ) of topological groups
X∗
whose kernel is equal to the external cyclotome µZb (MΘ∗ ), and whose image correpsonds
temp
to ΠY . We write
v

ΠMΘ

∗Ï

⊂ ΠMΘ
∗I

⊂ ΠMΘ
∗

temp
respectively, andfor the inverse image of ΠvÏ ⊂ ΠvI ⊂ Πv ∼= ΠX (MΘ∗ ) in ΠMΘ∗ΘΘΘµZb (MΘ∗Ï ), (l∆Θ )(M∗Ï ), Πv Ï (M∗Ï ), Gv (M∗Ï )and thefor the subquotients of ΠMΘdetermined by the subquotient µZb (MΘ∗ ) of ΠMΘ
∗
∗
temp
temp
temp
Θ
Θ
Θ
∼
subquotients (l∆Θ )(ΠX (M∗ )), ΠvÏ , and Gv (ΠX (M∗ )) of Πv = ΠX (M∗ ). Notethat we obtain a cyclotomic rigidity isomorphism of mono-theta environment∼Θ(l∆Θ )(MΘb (M∗Ï )∗Ï ) → µZ∼

by restricting the cyclotomic rigidity isomorphism of mono-theta environment (l∆Θ )(MΘ∗)→Θ
µZb (M∗ ) in Proposition 11.4 to ΠMΘ (Definition [IUTchII, Definition 2.7]).
∗Ï

Corollary 11.11.
(M-theoretic Theta Evaluation, [IUTchII, Corollary 2.8]) Let
be a projective system of mono-theta environments with Πtemp(MΘ∗ ) = Πv . We
X
write
MΘ
∗

γ
(MΘ
∗)

for the projective system of mono-theta environments obtained via transport of structure∼from the isomorphism Πv → Πγv given by the conjugation by γ.
0(1) Let Itδ = Itγγ ⊂ ΠδvÏ ⊂ ΠγvI = ΠδvI be as in Lemma 11.9 (2). Then by using thecyclomotic rigidity isomorphisms of mono-theta environment
∼

∼

γ
Θ γ
Θ γ
Θ γ
(l∆Θ )((MΘb ((M∗Ï ) ), (l∆Θ )((M∗ ) ) → µZb ((M∗ ) )∗Ï ) ) → µZ

a proof of the abc conjecture after Mochizuki

275(cf. just before Corollary 11.11), we replace H 1 (−, (l∆Θ )(−)) by H 1 (−, µZb (−)) in
Proposition 11.10. Then the ιγ -invariant subsets θι (Πγv ) ⊂ θ(Πγv ), ∞ θι (Πγv ) ⊂γγ∞ θ(Πv ) determines ι -invariant subsets
ι
γ
Θ γ
Θ γ
Θ γ
θιenv ((MΘ∗ ) ) ⊂ θ env ((M∗ ) ), ∞ θ env ((M∗ ) ) ⊂ ∞ θ env ((M∗ ) ).

The restriction of these subsets to ΠvÏ ((MΘ)γ ) gives us µ2l -, µ-orbits of elements
∗Ï
ι
γ
Θ γ
1
Θ γ
Θ γ
θιenv ((MΘb ((M∗Ï ) )),∗Ï ) ) ⊂ ∞ θ env ((M∗Ï ) ) ⊂ ∞ H (Πv Ï ((M∗Ï ) ), µZ1Θ γb)γ ), −) := limJ⊂
where ∞ H 1 (ΠvÏ ((MΘb v J, −). The
b v ; open H (Πv Ï ((M∗Ï ) ) ×Π∗Ï−→ b Π
δ
further restriction to the decomposition groups Dt,µin Lemma 11.9 (2) gives us−µ2l -, µ-orbits of elements
t
γ
Θ γ
1
Θ γ
Θ γ
θtenv ((MΘb ((M∗Ï ) )),∗Ï ) ) ⊂ ∞ θ env ((M∗Ï ) ) ⊂ ∞ H (Gv ((M∗Ï ) ), µZ)γ ), −) := limJ ⊂G ((MΘ )γ ) : open H 1 (JG , −), for eachwhere we write ∞ H 1 (Gv ((MΘ∗Ï−→ G v ∗Ï
±conj. by γ∼t ∈ LabCusp−→ LabCusp± (Πv ). Since the sets θtenv ((MΘ)γ ), ∞ θtenv ((MΘ
)γ )
∗Ï
∗Ï
depend only on the label |t| ∈ |Fl |, we write(Πγv )

t
γ
Θ γ
θ|t|
((MΘ
∗Ï ) ) := θ env ((M∗Ï ) ),
env

|t|
t
Θ γ
Θ γ
∞ θ env ((M∗Ï ) ) := ∞ θ env ((M∗Ï ) ).

Θ γ
Θ
b±
(2) If we start with an arbitrary ∆v -conjugate Πv Ï ((M∗Ï ) ) of Πv Ï (M∗Ï ), and we|t||t|
consider the resulting µ2l -, µ-orbits θenv ((MΘ)γ ), ∞ θenv ((MΘ)γ ) arising from
∗Ï
∗Ï
δ
Θ γ
b±
an arbitrary ∆) ), as t runs overv -conjugate It of It contained in Πv Ï ((M∗Ï

conj. by γ∼

LabCusp± (Πγv ) −→ LabCusp± (Πv ), then we obtain a group-theoretic algorithm
to construct the collections of µ2l -, µ-orbits
{
{
}
}
|t|
|t|
Θ γ
Θ γ
θenv ((M∗Ï ) ),,∞ θ env ((M∗Ï ) )|t|∈|Fl ||t|∈|Fl |which is functorial with respect to the projective system MΘ∗ of mono-theta environb±
ments, and compatible with the independent conjugacy actions of ∆v on the sets
γ1
γ1
Θ γ2
Θ γ2
= {It }γ1 ∈∆and {ΠvÏ ((M∗Ï ) )}γ2 ∈Π= {ΠvÏ ((M∗Ï ) )}γ2 ∈∆{It }γ1 ∈Π
b±
b±
b±
b±
v
v
v
v
(3) In the situation of (1), we take t to be the zero element. By using the cyclomotic
rigidity isomorphisms in (1) we replace (l∆Θ )(−) by µZb (−) in Proposition 11.10,
then we obtain splittings
ι
γ
Θ γ
µ
Θ γ
O×µ ((MΘ∗Ï ) ) × {∞ θ env ((M∗Ï ) )/O ((M∗Ï ) )}

276

Go Yamashita

of O× ∞ θιenv ((MΘ)γ )/Oµ ((MΘ)γ ), which are compatible with the splittings of Propo∗Ï∗Ï∼Θ
sition 11.7 (3) (with respect to any isomorphism MΘ
∗ → M∗ (Πv )), relative to thefirst restriction operation in (1):label 00v
/ O×µ ((MΘ )γ )∗Ï/ O× ∞ θι ((MΘ )γ )/Oµ ((MΘ )γ )
∗Ï
∗Ï
env

/ ∞ θι ((MΘ )γ )/Oµ ((MΘ )γ )
∗Ï
∗Ï
env

Remark 11.11.1.
(Theta Evaluation via Base-field-theoretic Cyclotomes, [IUTchII,
Corollary 2.9, Remark2.9.1]) If we use the cyclotomic rigidity isomorphisms∼∼µZb (Gv (Πv )) → (l∆Θ )(Πv ), µZb (Gv (ΠγvÏ )) → (l∆Θ )(ΠγvÏ )determined by the composites of the cyclotomic rigidity isomorphism via positive∼
rational structure and LCFT “µZb (G) → µZb (Π)” group-theoretically reconstructed∼by Remark 6.12.2 and the cyclotomic rigidity isomorphism “µZb (Π) → (l∆Θ )(Π)” grouptheoretically reconstructed by Remark 9.4.1 and its restriction to ΠγvÏ (like Proposition 11.5; however, we allow indeterminacies in Proposition 11.5), instead of using∼
the cyclomotic rigidity isomorphisms of mono-theta environment (l∆Θ )((MΘ)γ ) →∗Ï∼γ
Θ γµZb ((MΘ)γ ), (l∆Θ )((MΘb ((M∗ ) ), then we functorially group-theoretically∗ ) ) → µZ
∗Ïobtain the following similar objects with similar compatibility as in Corollary 11.11:
ιγ -invariant subsets
θιbs (Πγv ) ⊂ θbs (Πγv ), ∞ θιbs (Πγv ) ⊂ ∞ θbs (Πγv ).
The restriction of these subsets to ΠγvÏ gives us µ2l -, µ-orbits of elementsθιbs (Πγ∗Ï ) ⊂ ∞ θιbs (Πγ∗Ï ) ⊂ ∞ H 1 (ΠγvÏ , µZb (Gv (Πγ∗Ï ))),γ
1bwhere ∞ H 1 (ΠγvÏ , −) := limJ⊂b v J, −). The further restriction to theb v : open H (Πv Ï ×Π−→ b Π
δ
decomposition groups Dt,µin Lemma 11.9 (2) gives us µ2l -, µ-orbits of elements−

θtbs (ΠγvÏ ) ⊂ ∞ θtbs (ΠγvÏ ) ⊂ ∞ H 1 (Gv (ΠγvÏ ), µZb (Gv (ΠγvÏ ))),
where ∞ H 1 (Gv (ΠγvÏ ), −) := limJ ⊂G (Πγ ) : open H 1 (JG , −), for each t ∈ LabCusp± (Πγv )−→ G v vÏ
conj. by γ∼−→ LabCusp± (Πv ). Since the sets θtbs (ΠγvÏ ), ∞ θtbs (ΠγvÏ ) depend only on the label|t| ∈ |Fl |, we write
θ|t|(ΠγvÏ ) := θtbs (ΠγvÏ ),bs

Hence the collections of µ2l -, µ-orbits
{
}
γ
|t|
θbs (ΠvÏ )|t|∈|Fl |

γ
γ
|t|
t
∞ θ bs (Πv Ï ) := ∞ θ bs (Πv Ï ).

{
,

}

γ
|t|
,
∞ θ bs (Πv Ï )|t|∈|Fl |

/ 0.

277

a proof of the abc conjecture after Mochizuki

and splittings
O×µ (ΠγvÏ )bs × {∞ θιbs (ΠγvÏ )/Oµ (ΠγvÏ )bs }of O× ∞ θιbs (ΠγvÏ )/Oµ (ΠγvÏ )bs (Here, we write O×µ (−)bs , O× (−)bs , Oµ (−)bs for the objects corresponding to O×µ (−), O× (−), Oµ (−), respectively, via the cyclotomic rigidity
isomorphism):label 00x
/ O×µ (Πγ )bs/ O× ∞ θι (Πγ )/Oµ (Πγ )bs

v Ï

bs

v Ï

v Ï

/ ∞ θι (Πγ )/Oµ (Πγ )bs
bs

v Ï

v Ï

/ 0.

Note that we use the value group portion in the construction of the cyclotomic rigidity
isomorphism via positive rational structure and LCFT (cf. the final remark in Remark 6.12.2). Therefore, the algorithm in this remark (unlike Corollary 11.11) is only
uniradially defined (cf. Proposition 11.5 and Remark 11.4.1).
On the other hand, the cyclotomic rigidity isomorphism via positive rational structure and LCFT has an advantage of having the natural surjectionb
H 1 (Gv (−), µZb (Gv (−)))  Z
in (the proof of) Corollary 3.19 (cf. Remark 6.12.2), and we use this surjection to
construct some constant monoids (cf. Definition 11.12 (2)).
Definition 11.12.

(M-theoretic Theta Monoids, [IUTchII, Proposition 3.1]) Let

temp
∼
MΘ
(MΘ
∗ be a projective system of mono-theta environments with ΠX∗ ) = Πv .

(1) (Split Theta Monoids) We put{
Ψenv (MΘ∗ ) :=

}

temp
ι
×
Θ
Θ N
1
Θ
Ψιenv (MΘ(MΘb (M∗ )))∗ ) := O (M∗ ) · θ env (M∗ ) (⊂ ∞ H (ΠŸ∗ ), µZ{Θ∞ Ψenv (M∗ ) :=

,
ι

}

temp
ι
ι
Θ
×
Θ
Θ N
1
Θ
(MΘ
∞ Ψenv (M∗ ) := O (M∗ ) · ∞ θ env (M∗ ) (⊂ ∞ H (ΠŸb (M∗ )))∗ ), µZ

These are functorially group-theoretically reconstructed collections of submonoids of
temp
temp
1
Θ
(MΘ
(MΘ
∞ H (ΠŸb (M∗ )) equipped with natrural conjugation actions of ΠX∗ ), µZ∗ ),together with the splittings up to torsion determined by Corollary 11.11 (3). We
ιΘ
shall refer to each of Ψιenv (MΘ∗ ), ∞ Ψenv (M∗ ) as a mono-theta-theoretic thetamonoid.
(2) (Constant Monoids) By using the cyclotomic rigidity isomorphism via positive rab via the surjection
tional structure and LCFT, and taking the inverse image of Z ⊂ Z

.
ι

278

Go Yamashitab (cf. Remark 11.11.1) for Gv (MΘ ) := Gv (Πtemp (MΘ )),H 1 (Gv (−), µZb (Gv (−)))  Z∗∗X
we obtain a functorial group-theoretic reconstructiontemp
1
ΘΨcns (MΘ(MΘb (M∗ ))∗ ) ⊂ ∞ H (ΠŸ∗ ), µZof an isomorph of OF , equipped with a natural conjugate action by Πtemp
(MΘ
∗ ).
X
v

We shall refer to Ψcns (MΘ∗ ) as a mono-theta-theoretic constant monoid.
Definition 11.13.

([IUTchII, Example 3.2])(1) (Split Theta Monoids) Recall that, for the tempered Frobenioid F v (cf. Example 8.8), the choice of a Frobenioid-theoretic theta function Θv ∈ O× (OŸbirat ) (cf.
v

Example 8.8) among the µ2l (OŸbirat )-multiples of the AutDv (Ÿ v )-conjugates of Θvv

determines a monoid OCΘ (−) on DvΘ (cf. Definition 10.5 (1)) Suppose, for simplicv
ity, the topological group Πv arises from a universal covering pro-object A∞ ofΘ
Dv . Then for AΘ∞ := A∞ × Ÿ v ∈ pro-Ob(Dv ) (cf. Definition 10.5 (1)), we obtain
submonoids
N
Q≥0
×
Θ
birat
ΨFvΘ ,id := OCΘ (AΘ⊂ ∞ ΨFvΘ ,id := OC×Θ (AΘ|AΘ
⊂ O× (OAΘ ).
∞ ) = OC Θ (A∞ )·Θv |AΘ∞ )·Θv
∞
∞
∞
v

v

v

For the various conjugates Θαof Θv for α ∈ AutDv (Ÿ v ), we also similarly obtainv
submonoids
biratΨFvΘ ,α ⊂ ∞ ΨFvΘ ,α ⊂ O× (OA
Θ ).
∞
Write

{
}
ΨFvΘ := ΨFvΘ ,α

{
,

α∈Πv

∞ ΨFvΘ ,α :=∞ ΨFvΘ ,α

}
,
α∈Πv

where we use the same notation α, by abuse of notation, for the image of α via thesurjection Πv  AutDv (Ÿ v ). Note that we have a natural conjugation action of Πv≥0on the above collections of submonoids. Note also that ΘQ|AΘ
gives us splittings∞v

up to torsion of the monoids ΨFvΘ ,α , ∞ ΨFvΘ ,α (cf. splΘv in Definition 10.5 (1)), which
are compatible with the Πv -action. Note that, from F v , we can reconstruct thesecollections of submonoids with Πv -actions together with the splittings up to torsion
up to an indeterminacy arising from the inner automorphismsof Πv (cf. Section 8.3.
cf. also the remark given just before Theorem 8.14). We shall refer to each of ΨFvΘ ,α ,∞ ΨFvΘ ,α as a Frobenioid-theoretic theta monoid.
(2) (Constant Monoids) Similarly, the pre-Frobenioid structure on Cv = (F v )base-field ⊂F v gives us a monoid OCv (−) on Dv . We put

ΨCv := OCv (AΘ∞ ),a proof of the abc conjecture after Mochizuki

279

which is equipped with a natural Πv -action. Note that, from F v , we can reconstructΠv y ΨCv , up to an indeterminacy arising from the inner automorphisms of Πv .
We shall refer to ΨCv as a Frobenioid-theoretic constant monoid.
Proposition 11.14.
(F-theoretic Theta Monoids, [IUTchII, Proposition 3.3])
Θ∼
Let M∗ be a projective system of mono-theta environments with Πtemp(MΘ∗ ) = Πv .
X
Θ
†
†
=
Suppose that MΘ∗ arises from a tempered Frobenioid F v in a Θ-Hodge theatre HT({† F w }w∈V , † Fmod ) by Theorem 8.14 (“F 7→ M”):

Θ †MΘ∗ = M∗ ( F v ).

(1) (Split Theta Monoids) Note that, for an object S of F v such that µlN (S) ∼=∼Z/lN Z, and (l∆Θ )S ⊗ Z/N Z = Z/N Z as abstract groups, the exterior cyclotome†µZb (MΘb (S) = limN µN (S), where µN (S) ⊂∗ ( F v )) corresponds to the cyclotome µZ←−
×
O (S) ⊂ Aut† F (S) (cf. [IUTchII, Proposition 1.3 (i)]). Then by the Kummervmaps, we obtain collections of Kummer isomorphisms
Kum
∼

Kum
∼

Ψ† FvΘ ,α → Ψιenv (MΘ
∗ ),

ι
Θ
∞ Ψ† FvΘ ,α → ∞ Ψenv (M∗ ),of monoids, which is well-defined up to an inner automorphism and compatible(MΘwith both the respective conjugation action of Πtemp∗ ), and the splittings up
X
to torsion on the monoids, under a suitable bijection of lZ-torsors between “ι” inDefinition 11.8, and the images of “α” via the natural surjection Πv  lZ:
“ ι”s

∼

←→ “ Im(α)”s.

(2) (Constant Monoids) Similary, using the correspondence between the exterior cy†
clotome µZb (MΘ
b (S) = limN µN (S), we obtain Kum∗ ( F v )) and the cyclotome µZ←−
mer isomorphisms
Kum∼Ψ† Cv → Ψcns (MΘ∗)
for constant monoids, where † Cv := († F v )base-field , which is well-defined up to
an inner automorphism, and compatible with the respective conjugation actions of
(MΘ
Πtemp
∗ ).
X
Proof. Proposition follows from the definitions.
In the following, we often use the abbreviation (∞) (−) for a description like both(−) and ∞ (−).

280

Go Yamashita

Proposition 11.15.
(Π-theoretic Theta Monoids, [IUTchII, Proposition 3.4])
Θ∼
Let M∗ be a projective system of mono-theta environments with Πtemp(MΘ∗ ) = Πv .
X
Θ††Suppose that MΘ=∗ arises from a tempered Frobenioid F v in a Θ-Hodge theatre HT({† F w }w∈V , † Fmod ) by Theorem 8.14 (“F 7→ M”):

Θ †MΘ∗ = M∗ ( F v ).

We consider the full poly-isomorphismMΘ∗ (Πv )

full poly
∼

→

†
MΘ
∗ ( F v)of projective systems of mono-theta environments.
∼(1) (Multiradiality of Split Theta Monoids) Each isomorphism β : MΘ∗ (Πv ) →Θ †M∗ ( F v ) of projective system of mono-theta environmens induces compatible collections of isomorphisms
β
∼
temp
Θ
Πv → ΠX (M∗ (Πv )) →
∼

†
Πtemp
(MΘ
∗ ( F v ))
X

y

†
Πtemp
(MΘ
∗ ( F v ))
X

=

y

Θ
(∞) Ψenv (M∗ (Πv ))

β
∼

y

†
→ (∞) Ψenv (MΘ∗ ( F v ))

Kum−1
∼

→

(∞) Ψ† FvΘ ,which are compatible with the respective splittings up to torsion, andβ∼ΘGv → Gv (M∗ (Πv )) →∼y†
Gv (MΘ
∗ ( F v ))=†Gv (MΘ
∗ ( F v ))

y
β
∼

†
×
Ψenv (MΘ→ Ψenv (MΘ∗ (Πv ))
∗ ( F v ))

y
Kum−1
∼
×

→

Ψ×
†F Θ .
v

Moreover, the functorial algorithm
Πv 7→ (Πv y (∞) Ψenv (MΘ∗ (Πv )) with splittings up to torsion),
which is compatible with arbitrary automorphisms of the pair†×µ
Gv (MΘ
:= (Ψ† FvΘ )× /torsions∗ ( F v )) y (Ψ† FvΘ )arisen as Isomet-multiples of automorphisms induced by automorphisms of the pair†×
Gv (MΘ
∗ ( F v )) y (Ψ† FvΘ ) , relative to the above displayed diagrams, is multiradially defined in the sense of the natural functor “ΨGraph(Ξ) ” of Proposition 11.7.

281

a proof of the abc conjecture after Mochizuki∼Θ †(2) (Uniradiality of Constant Monoids) Each isomorphism β : MΘ∗ (Πv ) → M∗ ( F v )of projective system of mono-theta environmens induces compatible collections of
isomorphisms
β
∼

∼

temp
†
Πv → Πtemp(MΘ(MΘ∗ (Πv )) → ΠX
∗ ( F v ))XyΨcns (MΘ∗ (Πv ))

†
Πtemp
(MΘ
∗ ( F v ))
X

=

y

y
−1

β
∼

†
→ Ψcns (MΘ∗ ( F v ))

Kum
∼

→

Ψ† Cv ,

and
β
∼

∼

Θ †
Gv → Gv (MΘ∗ (Πv )) → Gv (M∗ ( F v ))y=†
Gv (MΘ
∗ ( F v ))

y

y
−1

β
∼

×
†
×
Ψcns (MΘ→ Ψcns (MΘ∗ (Πv ))
∗ ( F v ))

Kum
∼

Ψ×
†C .
v

→

Moreover, the functorial algorithm
Πv 7→ (Πv y Ψcns (MΘ∗ (Πv ))),
which fails to be compatible (Note that we use the cyltomic rigidity isomorphism via
rational positive structure and LCFT and the surjection H 1 (Gv (−), µZb (Gv (−))) b to construct the constant monoid, which use the value group portion as well) withZ
automorphisms of the pair†×µ
Gv (MΘ
:= (Ψ† Cv )× /torsions∗ ( F v )) y (Ψ† Cv )†×induced by automorphisms of the pair Gv (MΘ∗ ( F v )) y (Ψ† Cv ) , relative to theabove displayed diagrams, is uniradially defined.

Proof. Proposition follows from the definitions.
Corollary 11.16.
(M-theoretic Gaussian Monoids, [IUTchII, Corollary 3.5])
Θ∼
Let M∗ be a projective system of mono-theta environments with Πtemp(MΘ∗ ) = Πv .
X
For t ∈ LabCusp± (Πtemp(MΘ∗ )), we write (−)t for copies labelled by t of various objects
X
functorially constructed from MΘ
∗ (We use this convention after this corollary as well).
(1) (Conjugate Synchronisation) If we regard the cuspidal inertia subgroups ⊂
temp
Πtemp
(MΘ
(MΘ
∗ ) corresponding to t as subgroups of cuspidal inertia subgroups of ΠX
∗ ),
X
temp
then the ∆temp (MΘ )-outer action of Fo± ∼(MΘ )/∆temp (MΘ ) on Πtemp (MΘ )
=∆
X

∗

l

C

∗

induces isomorphisms between the pairsΘ
Gv (MΘ∗ )t y Ψcns (M∗ )t

X

∗

X

∗

282

Go Yamashita

of a labelled ind-topological monoid equipped with the action of a labelled topologicalgroup for distinct t ∈ LabCusp± (Πtemp(MΘ∗ )). We shall refer to these isomorphisms
X-symmetrising isomorphisms.When we identify these objects labelled by
as Fo±
lt and −t via a suitable Fo±l -symmetrising isomorphism, we write (−)|t| for the
resulting object labelled by |t| ∈ |Fl |. We write(−)h|Fl |i∏for the object determined by the diagonal embedding in |t|∈Fl (−)|t| via suitableo±Fo±l -symmetrising isomorphisms (Note that, thanks to the Fl -symmetrising isomorphisms, we can construct the diagonal objects). Then by Corollary 11.11, we
obtain a collection of compatible morphismsΘΘ(Πtemp(MΘ∗ ) ←-) Πv Ï (M∗Ï )  Gv (M∗Ï )h|Fl |i
X

y

y
diag
∼

Θ
Ψcns (MΘ∗ ) → Ψcns (M∗ )h|Fl |i ,which are compatible with Fo±l -symmetrising isomorphisms and well-defined up to(MΘ
an inner automorphism of Πtemp∗ ) (i.e., this inner automorphism indetermiXnacy, which a priori depends on |t| ∈ |Fl |, is independent of |t| ∈ |Fl |).
(2) (Gaussian Monoids) We shall refer to an element of the set
>

l
θFenv
:=

∏
|t|∈F>
l

⊂
θ|t|
env

∏

Ψcns (MΘ∗ )|t||t|∈F>
l>

as a value-profile (Note that this set has of cardinality (2l)l ). Then by using Fo±
l symmetrising isomorphisms and Corollary 11.11, we obtain a functorial algorithm
to construct, from MΘ
∗ , two collections of submonoids
Ψgau (MΘ∗ ) :=Θ∞ Ψgau (M∗ ) :=NΘ ×
Ψξ (MΘ∗ ) := Ψcns (M∗ )hF> i · ξ ⊂



∞

l

∏
|t|∈F>lΘ ×Q≥0Ψξ (MΘ⊂∗ ) := Ψcns (M∗ )hF> i · ξlΨcns (MΘ∗ )|t|,
ξ : value profile∏|t|∈F>lΨcns (MΘ∗ )|t|,
ξ : value profileΘ
where each Πξ (MΘ-action. We shall refer∗ ) is equipped with a natural Gv (M∗Ï )hF>l i
ΘΘto each of Ψξ (M∗ ), ∞ Ψξ (M∗ ) as a mono-theta-theoretic Gaussian monoid.

283

a proof of the abc conjecture after Mochizuki

The restriction operations in Corollary 11.11 give us a collection of compatible evaluation isomorphisms(MΘ(Πtemp∗ ) ←-)
X

ΠvÏ (MΘ
)
∗Ï

δ
Dt,µ
’s
−

>
) }
L99 {Gv (MΘ∗Ï |t| |t|∈F
l

y

y

ι
Θ
(∞) Ψenv (M∗ )

eval
∼

→

Θ
(∞) Ψξ (M∗ ),which is well-defined up to an inner automorphism of Πtemp(MΘ∗ ) (Note that up
X
to single inner automorphism by Fo±l -symmetrising isomorphisms), where we write) on the factor labelled by |t| of
L99 for the compatibility of the action of Gv (MΘ∗Ï |t|Θ
the ∞ Ψξ (M∗ ). We write
eval
∼

Θ
(∞) Ψenv (M∗ ) →Θ(∞) Ψgau (M∗ )for these collections of compatible evaluation morphisms induced by restriction.
(3) (Constant Monoids and Splittings) The diagonal-in-|Fl | submonoid Ψcns (MΘ∗ )h|Fl |ican be seen as a grpah between the constant monoid Ψcns (MΘ∗ )0 labelled by the zero>Θelement 0 ∈ |Fl | and the diagonal-in-Fl submonoid Ψcns (M∗ )hF> i , hence determineslan isomorphism
diag
∼

Θ
Ψcns (MΘ∗ )0 → Ψcns (M∗ )hF> ilof monoids, which is compatible with respective labelled Gv (MΘ)-actions. More∗Ïover, the restriction operations to zero-labelled evaluation points (cf. Corollary 11.11)
give us a splitting up to torsion×ΘNΨξ (MΘ∗ ) = Ψcns (M∗ )hF> i · ξ ,
l

Θ
×
Θ
Q≥0
∞ Ψξ (M∗ ) = Ψcns (M∗ )hF> i · ξlof each of the Gaussian monoids, which is compatible with the splitting up to torsionof Definition 11.12 (1), with respect to the restriction isomorphisms in the third
display of (2).
Proof. Corollary follows from the definitions.
Corollary 11.17.

(F-theoretic Gaussian Monoids, [IUTchII, Corollary 3.6]) Let

temp
∼
MΘ
(MΘ
∗ be a projective system of mono-theta environments with ΠX∗ ) = Πv . SupΘ††pose that MΘ=∗ arises from a tempered Frobenioid F v in a Θ-Hodge theatre HT({† F w }w∈V , † Fmod ) by Theorem 8.14 (“F 7→ M”):
Θ †MΘ∗ = M∗ ( F v ).

284

Go Yamashita(1) (Conjugate Synchronisation) For each t ∈ LabCusp± (Πtemp(MΘ∗ )) the KumX
mer isomorphism in Proposition 11.14 (2) determines a collection of compatible
morphisms
ΘΘ(MΘ(Πtemp∗ ) ) Gv (M∗ )t  Gv (M∗Ï )tXy(Ψ† Cv )tyKum∼→ Ψcns (MΘ∗ )t ,

which are well-defined up to an inner automorphism (which is independent of t ∈tempo±LabCusp± (Πtemp(MΘ(MΘ∗ ))) of ΠX∗ ), and Fl -symmetrising isomorphisms beXΘtween distinct t ∈ LabCusp± (Πtemp(MΘ∗ )) induced by the ∆X (M∗ )-outer action of
X
Fo± ∼= ∆C (MΘ )/∆X (MΘ ) on ΠX (MΘ ).
∗

l

∗

∗

(2) (Gaussian Monoids) For each value-profile ξ, we write∏
ΨFξ († F v ) ⊂ ∞ ΨFξ († F v ) ⊂(Ψ† Cv )|t||t|∈F>lΘ
for the submonoid determined by the monoids Ψξ (MΘ∗ ), ∞ Ψξ (M∗ ) in Corollary 11.16
Kum∼(2), respectively, via the Kummer isomorphism (Ψ† Cv )|t| → Ψcns (MΘ∗ )|t| in (1).
Write
{
}
{
}
†
†
†
†
ΨFgau ( F v ) := ΨFξ ( F v ), ∞ ΨFgau ( F v ) := ∞ ΨFξ ( F v )ξ : value profileξ : value profile

where each ΠFξ († F v ) is equipped with a natural Gv (MΘ∗ )hF> i -action. We shall referlto each of ΠFξ († F v ), ∞ ΠFξ († F v ) as a Frobenioid-theoretic Gaussian monoid.
Then by composing the Kummer isomorphism in (1) and Proposition 11.14 (1), (2)with the restriction isomorphism of Corollary 11.16 (2), we obtain a diagram ofcompatible evaluation ismorphisms
ΠvÏ (MΘ)∗Ï

ΠvÏ (MΘ
)
∗Ï

=

y

Θ
(∞) Ψ† Fv,α

y

Kum
∼

→ (∞) Ψιenv (MΘ
∗)

δ
Dt,µ
’s

∼

−

>
) }
L99 {Gv (MΘ∗Ï |t| |t|∈F

y

eval
∼

→

→

l

{Gv (MΘ∗ )|t| }|t|∈F>
y

Kum−1
∼

→

Θ
(∞) Ψξ (M∗ )l†(∞) ΨFξ ( F v ),which is well-defined up to an inner automorphism of Πtemp(MΘ∗ ) (Note that up toXsingle inner automorphism by Fo±l -symmetrising isomorphisms), where L99 is thesame meaning as in Corollary 11.16 (2). We write(∞) Ψ† FvΘ

Kum
∼

→

Θ
(∞) Ψenv (M∗ )

eval
∼

→

Θ
(∞) Ψgau (M∗ )

Kum−1
∼

→

for these collections of compatible evaluation morphisms.

†(∞) ΨFgau ( F v )

,

285

a proof of the abc conjecture after Mochizuki(3) (Constant Monoids and Splittings) By the same manner as in Corollary 11.16(3), the diagonal submonoid (Ψ† Cv )h|Fl |i determines an isomorphismdiag∼(Ψ† Cv )0 → (Ψ† Cv )hF> ilof monoids, which is compatible with respective labelled Gv (MΘ∗ )-actions. Moreover,the splittings in Corollary 11.16 (3) give us splittings up to torsion
NΨFξ († F v ) = (Ψ×† C )hF> i · Im(ξ) ,
v
l

×
†
Q≥0
∞ ΨFξ ( F v ) = (Ψ† Cv )hF> i · Im(ξ)
l(Here we write Im(−) for the image of Kum−1 ◦ eval ◦ Kum in (2)) of each of theGaussian monoids, which is compatible with the splitting up to torsion of Definition 11.12 (1), with respect to the restriction isomorphisms in the third display of(2).
Proof. Corollary follows from the definitions.
Remark 11.17.1. ([IUTchIII, Remark 2.3.3 (iv)]) It seems interesting to note thatthe cyclotomic rigidity of mono-theta environments admits Fo±-symmetry, contrarylj2
to the fact that the theta functions, or the theta values q v ’s do not admit Fo±l symmetry. This is because the construction of the cyclotomic rigidity of mono-theta
environments only uses the commutator structure [ , ] (in other words, “curvature”) of
the theta group (i.e., Heisenberg group), not the theta function itself.
Remark 11.17.2. (Π-theoretic Gaussian Monoids, [IUTchII, Corollary 3.7, Remark 3.7.1]) If we formulate a “Gaussian analogue” of Proposition 11.15, then the resulting algorithm is only uniradially defined, since we use the cyclotomic rigidity isomorphism via rational positive structure and LCFT (cf. Remark 11.11.1 Proposition 11.15(2)) to construct constant monoids. In the theta functions level (i.e., “env”-labelledobjects), it admits multiradially defined algorithms; however, in the theta values level(i.e., “gau”-labelled objects), it only admits uniradially defined algorithms, since weneed constant monoids as containers of theta values (Note also that this container isholomorphic container, since we need the holomorphic structures for the labels and Fo±l synchronising isomorphisms). Later, by using the theory of log-shells, we will modify
such a “Gaussian analogue” algorithm (cf. below) of Proposition 11.15 into a multiradially defined algorithm after admitting mild indeterminacies (i.e., (Indet ↑), (Indet →),and (Indet xy)) (cf. Theorem 13.12 (1), (2)).
A precise formulation of a “Gaussian analogue” of Proposition 11.15 is as follows:
temp
∼
(MΘ
Let MΘ∗ ) = Πv .
∗ be a projective system of mono-theta environments with ΠXΘ††Suppose that MΘ=∗ arises from a tempered Frobenioid F v in a Θ-Hodge theatre HT286

Go Yamashita({† F w }w∈V , † Fmod ) by Theorem 8.14 (“F 7→ M”):
Θ †MΘ∗ = M∗ ( F v ).

We consider the full poly-isomorphismMΘ∗ (Πv )

full poly
∼

→

†
MΘ
∗ ( F v)(† F v ) for MΘ
forof projective systems of mono-theta environments. We write MΘ
∗Ï
∗Ï
Θ †
Θ
Θ
Θ
Θ
MΘ
∗ = M∗ ( F v ). For M∗ = M∗ (Πv ), we identify Πv Ï (M∗Ï ) and Gv (M∗Ï ) with∼

ΠvÏ and Gv (ΠvÏ ) respectively, via the tautological isomorphisms ΠvÏ (MΘ) → ΠvÏ ,∗Ï∼ΘGv (M∗Ï ) → Gv (ΠvÏ ).
∼Θ †(1) Each isomorphism β : MΘ∗ (Πv ) → M∗ ( F v ) of projective system of mono-theta
environments induces compatible collections of evaluation isomorphisms

Π

δ
Dt,µ
’s
−
L99
{Gv (Πv ÏyιΘ(∞) Ψenv (M∗ (Πv ))

eval
∼
→

β
∼
∼
†
→ {Gv (MΘ († F ))|t| })}→{Gv (MΘ>∗ ( F v ))|t| }|t|∈F>v Ï |t| |t|∈F>
v
∗Ï
|t|∈F
l
l
l
y
y
y
−1
β
Kum
∼
∼
Θ †
Θ
†
→
→
(∞) Ψξ (M∗ ( F v ))(∞) Ψξ (M∗ (Πv ))(∞) ΨFξ ( F v ),andGv (ΠvÏ )
y

β
∼

diag
∼

(† F v ))hF> i→ Gv (ΠvÏ )hF> i → Gv (MΘ
∗Ï
l

l

y

β
eval
ι
Θ
× ∼
Θ
× ∼
Ψenv (M∗ (Πv )) → Ψξ (M∗ (Πv )) →∼→†Gv (MΘ
∗ ( F v ))hF> ily†×Ψξ (MΘ
∗ ( F v ))

y
Kum−1
∼

→

ΨFξ († F v )× ,where L99 is the same meaning as in Corollary 11.16 (2).
(2) (Uniradiality of Gaussian Monoids) The functorial algorithms
Πv 7→ (Gv (ΠvÏ ) y Ψgau (MΘ∗ (Πv )) with splittings up to torsion),Πv 7→ (∞ Ψgau (MΘ∗ (Πv )) with splittings up to torsion),
which fails to be compatible (Note that we use the cyltomic rigidity isomorphism via
rational positive structure and LCFT and the surjection H 1 (Gv (−), µZb (Gv (−))) b to construct the constant monoid, which use the value group portion as well) withZ
automorphisms of the pair††×µ
Gv (MΘ
:= ΨFξ († F v )× /torsions∗ ( F v ))hF> i y ΨFξ ( F v )l††×induced by automorphisms of the pair Gv (MΘ∗ ( F v )) y ΨFξ ( F v ) , relative tothe above displayed diagrams in (1), is uniradially defined.

287

a proof of the abc conjecture after Mochizuki§ 11.3.

Hodge-Arakelov-theoretic Evaluation and Gaussian Monoids at
Good Places.

In this subsection, we perform analogues of Hodge-Arakelov-theoretic evaluation,and construction of Gaussian monoids for v ∈ Vgood .
Let v ∈ Vgood . For v ∈ Vgood ∩ Vnon (resp. v ∈ Varc ), putcor⊂ Π±v := ΠX v ⊂ Πv := ΠCv

Πv := Π X→−v

(resp. Uv := −
X
→

v

cor
⊂ U±
v := Xv ⊂ Uv := Cv ),where −
X , Xv , and Cv are Aut-holomorphic orbispaces (cf. Section 4) associated to −
X ,
→
→
v
v
o±
cor
± ∼
±
cor ∼
X , and Cv , respectively. Note that we have Π /Π = F(resp. Gal(U /U ) =vvo±Fl ). We also write

v

v

l

v

±
±
cor
cor
cor
∆v ⊂ Πv  Gv (Πv ), ∆±v ⊂ Πv  Gv (Πv ), ∆v ⊂ Πv  Gv (Πv )(resp.

Dv` (Uv ))the natural quotients and their kernels (resp. the split monod), which can be grouptheoretically reconstructed by Corollary 2.4 (resp. which can be algorithmically re∼constructed by Proposition 4.5). Note that we have natural isomorphisms Gv (Πv ) →∼cor ∼Gv (Π±v ) → Gv (Πv ) → Gv .
Proposition 11.18.
(Π-theoretic (resp. Aut-hol.-theoretic) Gaussian Monoids
goodnon
at v ∈ V∩V(resp. at v ∈ Varc ), [IUTchII, Proposition 4.1, Proposition 4.3])(1) (Constant Monoids) By Corollary 3.19 (resp. by definitions), we have a functorial group-theoretic algorithm to construct, from the topological group Gv (resp.
from the split monoid Dv` ), the ind-topological submonoid equipped with Gv -action(resp. the topological monoid)
Gv y Ψcns (Gv ) ⊂ ∞ H 1 (Gv , µZb (Gv )) :=(resp.

lim−→

J⊂Gv : open

Ψcns (Dv` ) := O (Cv` )H 1 (J, µZb (Gv ))),which is an isomorph of (Gv y OF ), (resp. an isomorph of OF ). Thus, we obtainvv
a functroial group-theoretic algorithm to construct, from the topological group Πv(resp. from the Aut-holomorphic space Uv ), the ind-topological submonoid equippedwith Gv (Πv )-action (resp. the topological monoid)
Gv (Πv ) y Ψcns (Πv ) := Ψcns (Gv (Πv )) ⊂ ∞ H 1 (Gv (Πv ), µZb (Gv (Πv )))1⊂ ∞ H 1 (Π±b (Gv (Πv ))) ⊂ ∞ H (Πv , µZb (Gv (Πv )))v , µZ

288

Go Yamashita(resp.

Ψcns (Uv ) := Ψcns (Dv` (Uv ))),where ∞ H 1 (Gv (Πv ), −) := limJ⊂G (Π ) : open H 1 (J, −), ∞ H 1 (Π±v , −) := lim−→−→J⊂Gv (Πv ) : open
v
v
1
1
±
1
H (Πv ×Gv (Πv ) J, −), and ∞ H (Πv , −) := limJ⊂G (Π ) : open H (Πv ×Gv (Πv ) J, −).
−→vv(2) (Mono-analytic Semi-simplifications) By Definition 10.6, we have the functorial algorithm to construct, from the topological group Gv (resp. from the split
monoid Dv` ), the topological monoid equipped with the distinguished element`

logGv (pv ) ∈ R≥0 (Gv ) := (R`≥0 )v , (resp. logD (pv ) ∈ R≥0 (Dv` ) := (R`≥0 )v , )(cf. “logDΦ (pv )” in Definition 10.6) and a natural isomorphism
∼

× R
`
ΨR
cns (Gv ) := (Ψcns (Gv )/Ψcns (Gv ) ) → (R≥0 )v(resp.

∼

`
`
` × R
`
ΨR
cns (Dv ) := (Ψcns (Dv )/Ψcns (Dv ) ) → (R≥0 )v)of the monoids (cf. Proposition 5.2 (resp. Proposition 5.4)). Write×`Ψsscns (Gv ) := Ψcns (Gv ) × (R≥0 )v(resp.

`` ×`Ψsscns (Dv ) := Ψcns (Dv ) × (R≥0 )v),

which we consider as semisimplified version of Ψcns (Gv ) (resp. Ψcns (Dv` )). We also
write
ss
×
×
Ψss
cns (Πv ) := Ψcns (Gv (Πv )), Ψcns (Πv ) := Ψcns (Gv (Πv )) , R≥0 (Πv ) := R≥0 (Gv (Πv ))(resp.

ss
`
×
`
×
`
Ψss
cns (Uv ) := Ψcns (Dv (Uv )), Ψcns (Uv ) := Ψcns (Dv (Uv )) , R≥0 (Uv ) := R≥0 (Dv (Uv ))just as in (1).
(3) (Conjugate Synchronisation) If we regard the cuspidal inertia subgroups ⊂ Πv±
corresponding to t as subgroups of cuspidal inertia subgroups of Π±v , then the ∆v o± ∼
±
±
cor
∼
outer action of Fo±= ∆cor= Gal(U±v /∆v on Πv (resp. the action of Flv /Uv )
lon the various Gal(Uv /U±v )-orbits of cusps of Uv ) induces isomorphisms between
the pairs (resp. between the labelled topological monoids)
Gv (Πv )t y Ψcns (Πv )t(resp.

Ψcns (Uv )t)of the labelled ind-topological monoid equipped with the action of the labelled topological group for distinct t ∈ LabCusp± (Πv ) := LabCusp± (B(Πv )0 ) (resp. t ∈

),

289

a proof of the abc conjecture after Mochizuki

LabCusp± (Uv )) (cf. Definition 10.27 (1) (resp. Definition 10.27 (2)) for the definition of LabCusp± (−)). We shall refer to these isomorphisms as Fo±-symmetrisinglisomorphisms. These symmetrising isomorphisms determine diagonal submonoids∏∏
Ψcns (Πv )h|Fl |i ⊂
Ψcns (Πv )|t| , Ψcns (Πv )hF> i ⊂Ψcns (Πv )|t| ,l|t|∈|Fl ||t|∈F>lwhich are compatible with the respective labelled Gv (Πv )-actions∏∏(resp. Ψcns (Uv )h|Fl |i ⊂
Ψcns (Uv )|t| , Ψcns (Uv )hF> i ⊂Ψcns (Uv )|t|l|t|∈|Fl |),|t|∈F>
land an isomorphism
diag
∼

diag
∼

Ψcns (Πv )0 → Ψcns (Πv )hF> i (resp.

Ψcns (Uv )0 → Ψcns (Uv )hF> i

l

)

l

of ind-topological monoids, which is compatible with the respective labelled Gv (Πv )actions (resp. of topological monoids).
(4) (Theta and Gaussian Monoids) Write
{
}
Πv
Πv
×
Ψenv (Πv ) := Ψcns (Πv ) × R≥0 · log (pv ) · log (Θ)(resp.

{}
Ψenv (Uv ) := Ψcns (Uv )× × R≥0 · logUv (pv ) · logUv (Θ)),

where logΠv (pv ) · logΠv (Θ) (resp. logUv (pv ) · logUv (Θ)) is just a formal symbol, and
{
(
) }
Πv
×
2
Ψgau (Πv ) := Ψcns (Πv )hF> i × R≥0 · j · log (pv )
j
l
∏
∏
⊂
Ψss
Ψcns (Πv )×
cns (Πv )j =j × R≥0 (Πv )jj∈F>l(resp.

j∈F>
l

{
(
) }
Uv
2
R≥0 · j · log (pv )
j
∏
∏
×
ss
⊂
Ψcns (Uv )j =
Ψcns (Uv )j × R≥0 (Uv )jΨgau (Uv ) := Ψcns (Uv )×
×
hF>
l i
j∈F>
l

)

j∈F>
l

where logΠv (pv ) (resp. logUv (pv )) is just a formal symbol, and R≥0 · (−) is defined by the R≥0 -module structures of R≥0 (Πv )j ’s (resp. R≥0 (Uv )j ’s). Note that
we need the holomorphic structures for the labels and Fo±l -synchronising isomorphisms. In particular, we obtain a functorial group-theoretically algorithm to construct, from the topological group Πv (from the Aut-holomorphic space Uv ), the290

Go Yamashita

theta monoid Ψenv (Πv ) (resp. Ψenv (Uv )), the Gaussian monoid Ψgau (Πv ) (resp.
Ψgau (Uv )) equipped with natural Gv (Πv )-actions and splittings (resp. equipped withnatural splittings), and the formal evaluation isomorphism
Ψenv (Πv )(resp.

eval∼→Ψenv (Uv )Ψgau (Πv ) : logΠv (pv ) · logΠv (Θ) 7→ (j 2 · logΠv (pv ))j

eval
∼

→

Ψgau (Uv ) : logUv (pv ) · logUv (Θ) 7→ (j 2 · logUv (pv ))j),which restricts to the identity on the respective copies of Ψcns (Πv )× (resp. Ψcns (Uv )× ),and is compatible with the respective Gv (Πv )-actions and the natural splittings (resp.
compatible with the natural splittings).
Remark 11.18.1.
([IUTchII, Remark 4.1.1 (iii)]) Similarly as in Proposition 11.15
and Remark 11.17.2, the construction of the monoids Ψcns (Πv ) (resp. Ψcns (Uv )) is
uniradial, and the constructions of the monoids Ψsscns (Πv ), Ψenv (Πv ), and Ψgau (Πv )ss(resp. Ψcns (Uv ), Ψenv (Uv ), and Ψgau (Uv )), and the formal evaluation isomorphism
eval
∼

eval
∼

Ψenv (Πv ) → Ψgau (Πv ) (resp. Ψenv (Uv ) → Ψgau (Uv )) are multiradial. Note that,the latter ones are constructed by using holomorphic structures; however, these can be
described via the underlying mono-analytic structures (cf. also the table after Example 11.2).
Proof. Proposition follows from the definitions and described algorithms.
Proposition 11.19.
(F-theoretic Gaussian Monoids at v ∈ Vgood ∩ Vnon (resp.
at v ∈ Varc ), [IUTchII, Proposition 4.2, Proposition 4.4]) For v ∈ Vgood ∩ Vnon (resp.
v ∈ Varc ), let † F v = † Cv (resp. † F v = († Cv , † Dv = † Uv , † κv )) be a pv -adic Frobenioid(resp. a triple) in a Θ-Hodge theatre † HT Θ = ({† F w }w∈V , † Fmod ). We assume (forsimplicity) that the base category of † F v is equal to Btemp († Πv )0 ). We write
Gv († Πv ) y Ψ† F

(resp.
v

Ψ† F := O († Cv ))v

for the ind-topological monoid equipped with Gv († Πv )-action (resp. the topological monoid)
determined, up to inner automorphism arising from an element of † Πv by † F v , and†

Gv y Ψ† Fv` (resp.

Ψ† Fv` := O († Cv` ))for the ind-topological monoid equipped with † Gv -action (resp. the topological monoid)
determined, up to inner automorphism arising from an element of † Gv by the v-component† `Fv of F ` -prime-strip {† Fw` }w∈V determined by the Θ-Hodge theatre † HT Θ .

291

a proof of the abc conjecture after Mochizuki(1) (Constant Monoids) By Remark 3.19.2 (resp. by the Kummer structure † κv ),we have a unique Kummer isomorphism
Ψ† F

Kum
∼

→ Ψcns († Πv ) (resp.

v

Ψ† F

Kum
∼

→ Ψcns († Uv )v)

of ind-topological monoids with Gv († Πv )-action (resp. of topological monoids).
b × -orbit (resp. a(2) (Mono-analytic Semi-simplifications) We have a unique Z
unique {±1}-orbit)

Ψ×
†F `

“Kum”
b× -orbit, polyZ∼Ψcns († Gv )× (resp.

→

v

Ψ×
†F `

“Kum”
{±1}-orbit, poly∼→v

Ψcns († Dv` )×)of isomorphisms of ind-topological groups with † Gv -action (resp. of topologicalgroups), and a unique isomorphism×RΨR† F ` := (Ψ† F ` /Ψ† ` )
F
v
v

“Kum”
∼

→

v

×
†
R
R
ΨR
cns ( Gv ) (resp. Ψ† Fv` := (Ψ† Fv` /Ψ† F ` )
v

“Kum”
∼

→

† `
ΨR
cns ( Dv ) )of monoids, which sends the distinguished element of ΨR† F ` determined by the uniquev

generator (resp. by pv = e = 2.71828 · · · , i.e., the element of the complex Archimedeanfield which gives rise to Ψ† F whose natural logarithm is equal to 1) of Ψ† Fv` /Ψ×† F ` to
v

v

Gv
†
R † `
the distinguished element of ΨR(pv ) ∈
cns ( Gv ) (resp. Ψcns ( Dv )) determined by log`
DR≥0 († Gv ) (resp. log v (pv ) ∈ R≥0 († Dv` )). In particular, we have a natural poly-isomorphism×RΨss† F ` := Ψ† ` ×Ψ† F `
F
v
v
v

“Kum”
poly
∼

→

×
†
ss
R
Ψss
cns ( Gv ) (resp. Ψ† Fv` := Ψ† F ` ×Ψ† Fv`
v

“Kum”
poly
∼

→

† `
Ψss
cns ( Dv ) )of ind-topological monoids (resp. topological monoids) which is compatible with thenatural splittings (We can regard these poly-isomorphisms as analogues of Kummerssisomorphism). We write Ψss:= Ψss:= Ψss† F ` (resp. Ψ† F
† F ` ), hence we have a
†F
v

v

v

v

tautological isomorphism
Ψss
†F

tauto
∼
v

→

Ψss
†F `
v

(resp.

Ψss
†F

tauto
∼

→

v

Ψss
).
†F `
v

(3) (Conjugate Synchronisation) The Kummer isomorphism in (1) determines acollection of compatible Kummer isomorphisms
Kum∼(Ψ† F )t → Ψcns († Πv )tv(resp.

Kum∼(Ψ† F )t → Ψcns († Uv )t
v

),

292

Go Yamashita

which are well-defined up to an inner automorphism of † Πv (which is independentof t ∈ LabCusp± († Πv )) for t ∈ LabCusp± († Πv ) (resp. t ∈ LabCusp± († Uv )), and± †Fo±l -symmetrising isomorphisms between distinct t ∈ LabCusp ( Πv ) (resp. t ∈o± ∼ † cor † ±LabCusp± († Uv )) induced by the † ∆±= ∆v / ∆v on † Π±v -outer action of Flv
o± ∼† ± † cor†
† ±(resp. the action of Fl = Gal( Uv / Uv ) on the various Gal( Uv / Uv )-orbits ofcusps of † Uv ). These symmetrising isomorphisms determine an isomorphism(Ψ† F )0
v

diag
∼

→

(Ψ† F )hF> i (resp.
lv(Ψ† F )0
v

diag
∼

→

(Ψ† F )hF> i

)

l

v

of ind-topological monoids (resp. topological monoids), which are compatible withthe respective labelled Gv († Πv )-actions.
(4) (Theta and Gaussian Monoids) We write
ΨFgau († F v )Ψ† FvΘ ,(resp.

Ψ† FvΘ ,ΨFgau († F v ))for the monoids with Gv († Πv )-actions and natural splittings, determined by Ψenv († Πv ),Ψgau († Πv ) in Proposition 11.18 (4) respectviely, via the isomorphisms in (1), (2),and (3). Then the formal evaluation isomorphism of Proposition 11.18 (4) gives us
a collection of evaluation isomorphismsΨ† FvΘ
(resp. Ψ† FvΘ

Kum
∼

→

Kum
∼

→

†

Ψenv ( Πv )†

Ψenv ( Πv )

eval
∼

→

eval
∼

→

†

Ψgau ( Πv )†

Ψgau ( Πv )

Kum−1
∼

→

Kum−1
∼

→

ΨFgau († F v )ΨFgau († F v ) ),which restrict to the identity or the isomorphism of (1) or the inverse of the iso†×
morphism of (1) on the various copies of Ψ×† F , Ψcns ( Πv ) , and are compatiblevwith the various natural actions of Gv († Πv ) and natural splittings.
§ 11.4.

Hodge-Arakelov-theoretic Evaluation and Gaussian Monoids in
the Global Case.

In this subsection, we globalise the constructions in Section 11.2 (v ∈ Vbad ) andin Section 11.3 (v ∈ Vgood ) via global realified Frobenioids (cf. also Remark 10.9.1).
o±
We can globalise the local Fo±
l -symmetries to a global Fl -symmetry, thanks to theglobal {±1}-synchronisation in Proposition 10.33 (cf. also Proposition 10.34 (3)).
This is a -portion of constructions in -Hodge theatres. In the final multiradialalgorithm, we use this -portion to construct Θ-pilot object (cf. Proposition 13.7 andDefinition 13.9 (1)), which gives us a -line bundle (cf. Definition 9.7) (of negative large
degree) through an action on mono-analytic log-shells (cf. Corollary 13.13).

a proof of the abc conjecture after Mochizuki

293

Next, we also perform NF-counterpart (cf. Section 9) of Hodge-Arakelov-theoreticevaluation. This is a -portion of constructions in -Hodge theatres. In the final
×
multiradial algorithm, we use this -portion to construct actions of copies of “Fmod”on mono-analytic log-shells (cf. Proposition 13.11 (2)), through which we convert -line
bundles into -line bundles (cf. the category equivalence (Convert) just after Definition 9.7) and vice versa (cf. Corollary 13.13).
(Π-theoretic Monoids associated to D--Hodge Theatres, [IUTchII,

Corollary 11.20.
Corollary 4.5]) Let
†

HT

D-

†

†

±

φΘ
±

†

†

ell

φΘ
±

= ( D ←− DT −→ † D}± )be a D--Hodge theatre, and‡D = {‡ Dv }v∈Va D-prime-strip. We assume, for simplicity, that ‡ Dv = B temp (‡ Πv )0 for v ∈ Vnon .
We write ‡ D` = {‡ Dv` }v∈V for the associated D` -prime-strip to ‡ D, and ssume that‡ `Dv = B temp (‡ Gv )0 for v ∈ Vnon .
(1) (Constant Monoids) By Definition 11.12 (2) for v ∈ Vbad and Proposition 11.18(1) for v ∈ Vgood , we obtain a functorial algorithm, with respect to the D-prime-strip‡
D, to construct the assignment
{
}
Θ ‡
Θ ‡

(M
(
Π
))
y
Ψ
(M
(
Π
))
v ∈ Vbad ,
G

v
cns
v
v
∗
∗
{
}
Ψcns (‡ D) : V 3 v 7→ Ψcns (‡ D)v :=Gv (‡ Πv ) y Ψcns (‡ Πv )v ∈ Vgood ∩ Vnon ,Ψcns (‡ Dv )v ∈ Varc ,where Ψcns (‡ D)v is well-defined only up to a ‡ Πv -conjugacy indeterminacy for v ∈Vnon .
(2) (Mono-analytic Semi-simplifications) By Proposition 11.18 (2) for v ∈ Vgood
and the same group-theoretic algorithm for v ∈ Vbad (Here, we write Ψcns (Πv ) :=`
Ψcns (MΘ∗ (Πv ))), we obtain a functorial algorithm, with respect to the D -prime-strip‡ `D , to construct the assignment{ ‡ G y Ψss (‡ G )} v ∈ Vnon ,vv
cnsss ‡ `
ss ‡ `
Ψcns ( D ) : V 3 v 7→ Ψcns ( D )v :=Ψss (‡ D` )v ∈ Varc ,
cns

v

‡ `
‡
where Ψsscns ( D )v is well-defined only up to a Gv -conjugacy indeterminacy for‡ `v ∈ Vnon . Each Ψss
cns ( D )v is equipped with a splitting‡ `ss ‡ ` ×‡ `Ψsscns ( D )v = Ψcns ( D )v × R≥0 ( D )v

294

Go Yamashita

and each R≥0 (‡ D` )v is equipped with a distinguished element‡`

log D (pv ) ∈ R≥0 (‡ D` )v .
If we regard ‡ D` as constructed from ‡ D, then we have a functorial algorithm, withrespect to the D-prime-strip ‡ D, to construct isomorphisms∼

ss ‡ ` ×Ψcns (‡ D)×
v → Ψcns ( D )v∼

for each v ∈ V, which are compatible with Gv (‡ Πv ) → ‡ Gv -actions for v ∈ Vnon .
By Definition 10.6 (“D-version”), we also obtain a functorial algorithm, with respect
to D` -prime-strip ‡ D` , to construct a (pre-)Frobenioid
D (‡ D` )isomorphism to the model object Cmod in Definition 10.4, equipped with a bijection∼

Prime(D (‡ D` )) → V,and localisation isomorphisms‡

ρD ,v : ΦD (‡ D` ),v

gl. to loc.
∼−→

R≥0 (‡ D` )vof topological monoids.
(3) (Conjugate Synchronisation) We put†ζ := † ζ± ◦ † ζ0Θ

ell

∼

±

◦ (ζ0Θ )−1 : LabCusp± († D ) → T(cf. Proposition 10.33). The various local Fo±
l -actions in Corollary 11.16 (1) and
Proposition 11.18 (3) induce isomorphisms between the labelled dataΨcns († D )tfor distinct t ∈ LabCusp± († D ). We shall refer to these isomorphisms as Fo±lsymmetrising isomorphisms (Note that the global {±1}-synchronisation established by Proposition 10.33 is crucial here). These Fo±l -symmetrising isomoro±
phisms are compatible with the (doubly transitive) Fl -action on the index set T ofellwith respect to † ζ, hence determine diagonal submonoids
the D-Θell -bridge † φΘ±∏∏Ψcns († D )h|Fl |i ⊂
Ψcns († D )|t| , Ψcns († D )hF> i ⊂Ψcns († D )|t| ,l|t|∈|Fl ||t|∈F>
land an isomorphism
†

diag
∼

Ψcns ( D )0 → Ψcns († D )hF> ilconsisting of the local isomorphisms in Corollary 11.16 (3) and Proposition 11.18
(3).

295

a proof of the abc conjecture after Mochizuki(4) (Local Theta and Gaussian Monoids) By Corollary 11.16 (2), (3) and Proposition 11.18 (4), we obtain a functorial algorithm, with respect to the D-prime-strip†
D , to construct the assignments†(∞) Ψenv ( D ) : V 3 v 7→


Θ †
Θ †

>

{Gv (M∗ ( Πv ))}j∈Fl y (∞) Ψenv (M∗ ( Πv ))†{Gv († Πv )}j∈F> y (∞) Ψenv († Πv )(∞) Ψenv ( D )v :=
l



‡
(∞) Ψenv ( Uv )v ∈ Vbad ∩ Vnon ,v ∈ Vgood ∩ Vnon ,v ∈ Varc ,and†(∞) Ψgau ( D ) : V 3 v 7→


Θ †
Θ †

>

{Gv (M∗ ( Πv ))}j∈Fl y (∞) Ψgau (M∗ ( Πv ))†{Gv († Πv )}j∈F> y (∞) Ψgau († Πv )(∞) Ψgau ( D )v :=
l



‡
Ψ
(
U
)
v
(∞) gauv ∈ Vbad ∩ Vnon ,v ∈ Vgood ∩ Vnon ,v ∈ Varc ,where we write ∞ Ψenv († Πv ) := Ψenv († Πv ) (resp. ∞ Ψenv († Uv ) := Ψenv († Uv )) and
good
†
†
†
†
∩ Vnon∞ Ψgau ( Πv ) := Ψgau ( Πv ) (resp. ∞ Ψgau ( Uv ) := Ψgau ( Uv )) for v ∈ V(resp. v ∈ Varc ) and (∞) Ψenv († D )v ’s, (∞) Ψgau († D )v ’s are equipped with natural
splittings, and compatible evaluation isomorphisms
eval
∼

†
(∞) Ψenv ( D ) →†(∞) Ψgau ( D )constructed by Corollary 11.16 (2) and Proposition 11.18 (4).
(5) (Global Realified Theta and Gaussian Monoids) We have a functorial algorithm, with respect to the D` -prime-strip † D` , to construct a (pre-)Frobenioid
Denv († D` )as a coply of the Frobenioid D († D` ) of (2) above, multiplied a formal symbol† `log D (Θ), equipped with a bijection∼

Prime(Denv († D` )) → V,and localisation isomorphisms
ΦDenv († D` ),v

gl. to loc.
∼−→

Ψenv († D` )Rv

of topological monoids. We have a functorial algorithm, with respect to the D` prime-strip † D` to construct a (pre-)Frobenioid∏
Dgau († D` ) ⊂D († D` )j
j∈F>
l

296

Go Yamashita

whose divisor and rational function monoids are determined by the weighted diagonal( j 2 )j∈F> , equipped with a bijectionl∼Prime(Dgau († D` )) → V,and localisation isomorphisms
ΦDgau († D` ),v

gl. to loc.
∼−→

Ψgau († D` )Rv

of topological monoids for each v ∈ V. We also have a functorial algorithm, withrespect to the D` -prime-strip † D` to construct a global formal evaluation isomorphism
eval∼

Denv († D` ) → Dgau († D` )of (pre-)Frobenioids, which is compatible with local evaluation isomorphisms of (4),with respect to the localisation isomorphisms for each v ∈ V and the bijections∼
Prime(−) → V.
Proof. Corollary follows from the definitions.
(F-theoretic Monoids associated to -Hodge Theatres, [IUTchII,

Corollary 11.21.
Corollary 4.6]) Let(†

HT  =

be a -Hodge theatre, and

†

†

±

Θ
ψ±

†

ell

Θ
ψ±

)

F ←− † FT −→ † D}±‡

F = {‡ Fv }v∈V

an F-prime-strip. We assume, for simplicity, that the D--Hodge theatre associated to†HT  is equal to † HT D- in Corollary 11.20, and that the D-prime-strip associatedto ‡ F is equal to ‡ D in Corollary 11.20. We write ‡ F` = {‡ Fv` }v∈V for the associatedF ` -prime-strip to ‡ F.
(1) (Constant Monoids) By Proposition 11.19 (1) for Vgood , and the same grouptheoretic algorithm for v ∈ Vbad , we have a functorial algorithm, with respect to theF-prime-strip ‡ F, to construct the assignment{} Gv (‡ Πv ) y Ψ‡v ∈ Vnon ,Fv‡‡Ψcns ( F) : V 3 v 7→ Ψcns ( F)v :=Ψ‡v ∈ Varc ,Fv

where Ψcns (‡ F)v is well-defined only up to a ‡ Πv -conjugacy indeterminacy for v ∈Vnon . By Proposition 11.14 (2) for v ∈ Vbad (where we take “† Cv ” to be ‡ Fv ) and297

a proof of the abc conjecture after Mochizuki

Proposition 11.19 (1) for v ∈ Vgood , we obtain a collection of Kummer isomorphismKum∼Ψcns (‡ F) → Ψcns (‡ D).
(2) (Mono-analytic Semi-simplifications) By Proposition 11.19 (2) for Vgood , andthe same group-theoretic algorithm for v ∈ Vbad , we have a functorial algorithm,
with respect to the F ` -prime-strip ‡ F` , to construct the assignment‡ `ss ‡ `ssΨss
cns ( F ) : V 3 v 7→ Ψcns ( F )v := Ψ‡ Fv`‡ `‡where Ψsscns ( F )v is well-defined only up to a Gv -conjugacy indeterminacy for v ∈non‡ `Vnon . Each Ψss, withcns ( F )v is equipped with its natural splitting, and for v ∈ V
ss
a distinguished element (Note that the distinguished element in Ψ† F ` for v ∈ Varcv

is not preserved by automorphism of † Fv` . cf. also the first table in Section 4.3 cf.
[IUTchII, Remark 4.6.1]). By Proposition 11.19 (2) for v ∈ Vgood and the samegroup-theoretic algorithm for v ∈ Vbad , we have a functorial algorithm, with respect
to F ` -prime-strip ‡ F` , to construct the collection of poly-isomorphisms (analoguesof Kummer isomorphism)‡ `Ψsscns ( F )

“Kum”
poly
∼

−→

Let
‡

‡ `
Ψss
cns ( D ).

∼F = (‡ C , Prime(‡ C ) → V, ‡ F` , {‡ ρv }v∈V )be the F -prime-strip associated to ‡ F. We also have a functorial algorithm, withrespect to F -prime-strip ‡ F , to construct an isomorphism
‡

C

“Kum”
∼

−→

D (‡ D` )(We can regard this isomorphism as an analogue of Kummer isomorphism), whereD (‡ D` ) is constructed in Corollary 11.20 (2), which is uniquely determined by∼
the condition that it is compatible with the respective bijections Prime(−) → V andthe localisation isomorphisms of topological monoids for each v ∈ V, with respect to‡ `the above collection of poly-isomorphisms Ψsscns ( F )
“Kum”
poly
∼

“Kum”
poly
∼

→

‡ `
Ψss
cns ( D ) (Note that,
“Kum”
∼

‡ `
‡ `
‡
→ Ψss
→ D (‡ D` ) in aif we reconstruct both Ψss
cns ( D ) and C
cns ( F )arc
compatible manner, then the distinguished elements in Ψss
can be† F ` at v ∈ V
vcomputed from the distinguished elements at v ∈ Vnon and the structure (e.g.. using
rational function monoids) of the global realified Frobenioids ‡ C , D (‡ D` ). cf.
[IUTchII, Remark 4.6.1]).

298

Go Yamashita(3) (Conjugate Synchronisation) For each t ∈ LabCusp± († D ), the collection of
isomorphisms in (1) determine a collection of compatible Kummer isomorphisms
Kum
∼

†

→ Ψcns († D )t ,

Ψcns ( F )twhere Ψcns († D )t is the labelled data constructed in Corollary 11.20 (3), and the†Πv -conjugacy indeterminacy at each v ∈ V is independent of t ∈ LabCusp± († D ),o±and Fo±l -symmerising isomorphisms induced by the various local Fl -actions inCorollary 11.17 (1) and Proposition 11.19 (3) between the data labelled by distinctt ∈ LabCusp± († D ). These Fo±l -symmetrising isomorphisms are compatible withello±the (doubly transitive) Fl -action on the index set T of the D-Θell -bridge † φΘ±with respect to † ζ in Corollary 11.20 (3), hence determine (diagonal submonoids
and) an isomorphismdiag∼Ψcns († F )0 → Ψcns († F )hF> ilconsisting of the local isomorphisms in Corollary 11.17 (3) and Proposition 11.19(3).
(4) (Local Theta and Gaussian Monoids) Let
†

†

Θ
ψ>

FJ −→ † D> 99K † HT Θ

be a Θ-bridge which is glued to the Θ± -bridge associate to the -Hodge theatre†HT  via the algorithm in Lemma 10.38 (Hence J = T > ). By Corollary 11.17 (2),(3) and Proposition 11.19 (4), we have a functorial algorithm, with respect to theabove Θ-bridge with its gluing to the Θ± -bridge associated to † HT  , to constructassignmentsΘ†(∞) ΨFenv ( HT ) : V 3 v 7→†(∞) ΨFenv ( HT

Θ

)v :=


{G († Π )}v ∈ Vnon ,(∞) Ψ† F Θv ∈ Varc ,vv

y (∞) Ψ† FvΘ
j∈F>
l

v

and
Θ
†
(∞) ΨFgau ( HT ) : V 3 v 7→†(∞) ΨFgau ( HT

Θ

)v :=


{Gv († Πv )} > y (∞) ΨF († F )
gau
j∈F
v
(∞) ΨF

gaul(† F v )v ∈ Vnonv ∈ Varc(Here the notation (−)(† HT Θ ) is slightly abuse of notation), where we write ∞ ΨFenv († HT Θ )v:= ΨFenv († HT Θ )v , and ∞ ΨFgau († HT Θ )v := ΨFgau († HT Θ )v for v ∈ Vgood , and299

a proof of the abc conjecture after MochizukiΘΘ††(∞) ΨFenv ( HT )v ’s, (∞) ΨFgau ( HT )v ’s are equipped with natural splittings, andcompatible evaluation isomorphisms†(∞) ΨFenv ( HT

Θ

Kum
∼

†

) →

(∞) Ψenv ( D )

eval
∼

→

†

(∞) Ψgau ( D )

Kum−1
∼

Θ
†
(∞) ΨFgau ( HT )→constructed by Corollary 11.17 (2) and Proposition 11.19 (4).
(5) (Global Realified Theta and Gaussian Monoids) By Proposition 11.19 (4)“Kum”∼→ D (‡ D` ) of
for labelled and non-labelled versions of the isomorphism ‡ C(2) to the global realified Frobenioids Denv († D` ), Dgau († D` ) constructed in Corollary 11.20 (5), we obtain a functorial algorithm, with respect to the above Θ-bridge,
to construct (pre-)Frobenioids
Cenv († HT Θ ), Cgau († HT Θ )(Here the notation (−)(† HT Θ ) is slightly abuse of notation. Note also that theconstruction of Cenv († HT Θ ) is similar to the one of Ctheta in Definition 10.5 (4))
with equipped with bijections∼

Prime(Cenv († HT Θ )) → V,∼

Prime(Cgau († HT Θ )) → V,

localisation isomorphisms
ΦCenv († HT Θ ),v

gl. to loc.
∼−→

ΨFenv († HT Θ )Rv,

ΦCgau († HT Θ ),v

gl. to loc.
∼−→

ΨFgau († HT Θ )Rv

of topological monoids for each v ∈ V, and evaluation isomrphisms†

Cenv ( HT )
Θ

“Kum”
∼

→

†

Denv ( D` )

eval
∼

→

†

Dgau ( D` )“Kum−1 ”∼→

Cgau († HT Θ )of (pre-)Frobenioids constructed by Proposition 11.19 (4) and Corollary 11.20 (5),which are compatible with local evaluation isomorphisms of (4), with respect to the∼
localisation isomorphisms for each v ∈ V and the bijections Prime(−) → V.
Proof. Corollary follows from the definitions.
Next, we consider -portion.
(Π-theoretic Monoids associated to D--Hodge Theatres, [IUTchII,

Corollary 11.22.
Corollary 4.7]) Let
†

HT

D-

†

=( D

}

†

φNF
> †

†

φΘ
> †

←− DJ −→ D> )be a D--Hodge theatre, which is glued to the D--Hodge theatre † HT D- of Corollary 11.20 via the algorithm in Lemma 10.38 (Hence J = T > ).

300

Go Yamashita(1) (Global Non-realified Structures) By Example 9.5, we have a functorial algorithm, with respect to the category † D} , to construct the morphism†

D} → † D~ ,the monoid/field/pseudo-monoid~

π1 († D~ ) y M~ († D} ), π1 († D~ ) y M († D} ), π1rat († D~ ) y M~(† D} )
∞κwith π1 († D~ )-/π1rat († D~ )-actions (Here, we use the notation π1 († D} ), π1 († D~ ) and
π1rat († D~ ), not † Π} , † Π~ , († Π~ )rat in Example 9.5, respectively, for making clear
the dependence of objects), which is well-defined up to π1 († D~ )-/π1rat († D~ )-conjugay
indeterminacies, the submooid/subfield/subset
~

~

† }
† }
~ † }
~ † }
† }
~ † }
M~
mod ( D ) ⊂ M ( D ), Mmod ( D ) ⊂ M ( D ), Mκ ( D ) ⊂ M∞ κ ( D ),of π1 († D~ )-/π1rat († D~ )-invariant parts, the Frobeniods~
Fmod(† D} ) ⊂ F ~ († D} ) ⊃ F } († D} )~~(Here, we write Fmod(† D} ), F } († D} ) for † Fmod
, † F } in Example 9.5, respectively)
with a natural bijection (by abuse of notation)∼~
Prime(Fmod(† D} )) → V,

and the natural realification functor~~R † }
Fmod(† D} ) → Fmod
( D ).

† }(2) (F>l -symmetry) By Definition 10.22, for j ∈ LabCusp( D ), we have a functorial
algorithm, with respect to the category † D} , to construct an F-prime-strip

F } († D} )|j ,which is only well-defined up to isomorphism, Moreover, the natural poly† }action of F>
induces isomorphisms between the labelled datal on D~† }† }F } († D} )|j , M~
mod ( D )j , Mmod ( D )j ,~~R † }{π1rat († D~ ) y M~(† D} )}j , Fmod(† D} )j → Fmod
( D )j∞κ

301

a proof of the abc conjecture after Mochizuki

for distinct j ∈ LabCusp(† D} ). We shall refer to these isomorphisms as F>l >symmetrising isomorphisms. These Fl -symmetrising isomorphisms are compatbile with the (simply transitive) F>l -action on the index set J of the D-NF-bridge
∼
∼
† NF
†
φ> with respect to ζ> : LabCusp(† D} ) → J(→ F>l ) in Proposition 10.19 (3),and determine diagonal objects
∏ ~
∏
~
~
† }
† }
† }
>
>
M
(
D
)
Mmod († D} )j .
M~
(
D
)
⊂
⊂
M
(
D
)
,
j
mod
mod
mod
hF i
hF i
l

l

j∈F>
l

j∈F>
l

We also write~~R † }F } († D} )|hF> i , {π1rat († D~ ) y M~(† D} )}hF> i , Fmod(† D} )hF> i → Fmod
( D )hF> i
∞κ
l

l

l

l

for a purely formal notational shorthand for the above F>l -symmetrising isomorphisms for the respective objects (cf. also Remark 11.22.1 below).
(3) (Localisations and Global Realified Structures) For simplicity, we write†
Dj = {† Dvj }v∈V (resp. † D`j = {† Dv`j }v∈V ) for the D-(resp. D` -)prime-strip

associated to the F-prime-strip F } († D} )|j (cf. Definition 10.22 (2)). By Definition 10.22 (2), Definition 9.6 (2), (3), and Definition 10.23 (3), we have a functorial
algorithm, with respect to the category † D} , to construct (1-)compatible collectionsof “localisation” functors/poly-morphisms~
Fmod(† D} )jgl. to loc.

−→

~R † }F } († D} )|j , Fmod
( D )jgl. to loc.

−→

(F } († D} )|j )R ,}{gl. to loc.
~†
rat † ~~ † }~†{π1 ( D ) y M∞ κ ( D )}j −→ M∞ κv ( Dvj ) ⊂ M∞ κ×v ( Dvj )v∈V

up to isomorphism, together with a natural isomorphismD†( D`j )

gl. real’d to gl. non-real’d⊗R∼−→~R † }
Fmod( D )jof global realified Frobenioids (global side), and a natural isomorphism†R≥0 ( D`j )v

localised (gl. real’d to gl. non-real’d⊗R)∼−→Ψ(F } († D} )|j )R ,v

of topological monoids for each v ∈ V (local side), which are compatible with the re∼spective bijections Prime(−) → V and the localisation isomorphisms {ΦD († D`j ),v

gl. to loc.
∼−→

gl. to loc.

~R † }R≥0 († D`j )v }v∈V constructed by Corollary 11.20 (2) and the above Fmod
( D )j−→} † }R(F ( D )|j ) . Finally, all of these structures are compatible with the respectiveF>l -symmetrising isomorphisms of (2).

302

Go Yamashita

Remark 11.22.1. ([IUTchII, Remark 4.7.2]) Recall that F>
l , in the context of>
Fl -symmetry, is a subquotient of Gal(K/F ) (cf. Definition 10.29), hence we cannot
perform the kind of conjugate synchronisations in Corollary 11.20 (3) for F>l -symmetry~ † }(for example, it nontrivially acts on the number field M ( D )). Therefore, we have towork with(1) F-prime-strips, instead of the corresponding ind-topological monoids with Galois
actions as in Corollary 11.20 (3),(2) the objects labelled by (−)mod (Note that the natural action of Galois group Gal(K/F )on them is trival, since they are in the Galois invariant parts), and(3) the objects labelled by (−)∞ κ ,because we can ignore the conjugacy indeterminacies for them (In the case of (2), there
is no conjugacy indeterminacy). cf. also Remark 9.6.2 (4) (in the second numeration).
Proof. Corollary follows from the definitions.
Corollary 11.23.
(F-theoretic Monoids associated to -Hodge Theatres, [IUTchII,
Corollary 4.8]) Let
(
)
† NF
† Θ
ψ>
ψ>

Θ
†
† ~
† }
†
†
†
HT = F
L99 F←−
FJ −→ F> 99K HT
be a -Hodge theatre, which lifts the D--Hodge theatre † HT D- of Corollary 11.22,
and is glued to the -Hodge theatre † HT  of Corollary 11.21 via the algorithm inLemma 10.38 (Hence J = T > ).
(1) (Global Non-realified Structures) By Definition 9.6 (1) (the Kummer isomorb × = {1} (Cyc. Rig. NF1)),
phism by the cyclotomic rigidity isomorphism via Q>0 ∩ Z
we have a functorial algorithm, with respect to the pre-Frobenioid † F } , to construct
Kummer isomorphism}} Kum{ rat † ~∼ {(† D } ) ,−→ π1rat († D~ ) y M~
π1 ( D ) y † M~
∞κ
∞κ

†

Kum
∼

~ † }
M~
κ −→ Mκ ( D )of pseudo-monoids with group actions, which is well-defined up to conjugacy indeterminacies, and by restricting Kummer classes (cf. Definition 9.6 (1)), natural
Kummer isomorphisms}{ rat † ~} Kum
∼ {π1 ( D ) y † M~ −→ π1rat († D~ ) y M~ († D} ) ,†{} Kum {}~~∼
π1rat († D~ ) y † M−→ π1rat († D~ ) y M († D} ) ,

†

Kum
∼

~
† }
M~
mod −→ Mmod ( D ),
~

Kum
∼

~

Mmod −→ Mmod († D} ).

303

a proof of the abc conjecture after Mochizuki

These isomorphisms can be interpreted as a compatible collection of isomorphisms
†

Kum
∼

Kum
∼

Kum
∼

Kum
∼

~
~
~R
~R † }
F } −→ F } († D} ), † F ~ −→ F ~ († D} ), † Fmod−→ Fmod(† D} ), † Fmod−→ Fmod( D )of (pre-)Frobenioids (cf. Definition 9.6 (1), and Example 9.5).
(2) (F>l -symmetry) The collection of isomorphisms of Corollary 11.21 (1) for thecapsule † FJ of the F-prime-strips and the isomorphism in (1) give us, for each∼j ∈ LabCusp(† D} )(→ J), a collection of Kummer isomorphisms

†

Kum
∼

∼

Fj → † F } |j → F } († D} )|j ,( M~mod )j†Kum∼{ rat † ~
} Kum
}
∼ {
π1 ( D ) y † M~
→ π1rat († D~ ) y M~(† D } ) j ,
∞κ j
∞κ

† }
M~
mod ( D )j ,→~( Mmod )j
†

Kum
∼

Kum
∼

~

→ Mmod († D} )j ,

Kum
∼

~
~
~R
~R † }(† Fmod)j → Fmod(† D} )j , († Fmod)j → Fmod( D )j ,† }and F>
l -symmetrising isomorphisms between the data indexed by distinct j ∈ LabCusp( D ),>† }induced by the natural poly-action of F>
l on F . These Fl -symmetrising isomorphisms are compatbile with the (simply transitive) F>l -action on the index set J of
∼
∼
† NF
†
the D-NF-bridge φ> with respect to ζ> : LabCusp(† D} ) → J(→ F>l ) in Proposition 10.19 (3), and determine various diagonal objects( † M~mod )hF> i ⊂
l

∏

~

†
(† M~
mod )j , ( Mmod )hF> i ⊂
l

j∈F>
l

∏

~

(† Mmod )j ,j∈F>l

and formal notational “diagonal objects” (cf. Corollary 11.22 (2))†~~RF } |hF> i , {π1rat († D~ ) y † M~
} > , († Fmod
)hF> i , († Fmod
)hF> i .
∞ κ hF i
l

l

l

l

(3) (Localisations and Global Realified Structures) By Definition 10.22 (2) andDefinition 10.23 (3), we have a functorial algorithm, with respect to the NF-bridge
†

†

NF
ψ>

FJ → † F } 99K † F ~ , to construct mutually (1-)compatible collections of localisation functors/poly-morphisms,~(† Fmod)j

gl. to loc. †−→~R
Fj , († Fmod)jgl. to loc. † R
−→
Fj ,

{{
}
} gl. to loc. †rat † ~† ~†π1 ( D ) y M∞ κ j −→M∞ κvj ⊂ M∞ κ×vj

v∈V

,

304

Go Yamashita

up to isomorphism, which is compatible with the collections of functors/poly-morphismsof Corollary 11.22 (3), with respect to the various Kummer isomorphisms of (1),(2), together with a natural isomorphism†

Cj

gl. real’d to gl. non-real’d⊗R
∼

−→

~R
)j
(† Fmodof global realified Frobenioids (global side), which is compatible with respective bijec∼
tions Prime(−) → V, and a natural isomorphismΨ† F`j ,vlocalised (gl. real’d to gl. non-real’d⊗R)∼−→Ψ† FjR ,vof topological monoids for each v ∈ V (local side), which are compatible with the∼respective bijections Prime(−) → V, the localisation isomorphisms {Φ† Cj ,v† ~R
ΨR† F` ,v }v∈V constructed by Corollary 11.20 (2) and the above ( Fmod )jjgl. to loc.
∼−→

gl. to loc.

−→

† R
Fj , the isomorphisms of Corollary 11.22 (3), and various (Kummer) isomorphismsof (1), (2). Finally, all of these structures are compatible with the respective F>l -symmetrising isomorphisms of (2).
Proof. Corollary follows from the definitions.
Write the results of this Chapter together, we obtain the following:
Corollary 11.24.
(Frobenius-picture of -Hodge Theatres, [IUTchII, Corol ‡†lary 4.10]) Let HT, HT  be -Hodge theatres with respect to the fixed initialΘ-data. We write † HT D- , ‡ HT D- for the assosiated D-  -Hodge theatres respectively.
(1) (Constant Prime-strips) Apply the constructions of Corollary 11.21 (1), (3)for the underlying -Hodge theatre of † HT  . Then the collection Ψcns († F )t of
data determines an F-prime-strip for each t ∈ LabCusp± († D ). We identify thecollections
Ψcns († F )0 , Ψcns († F )hF> i
l

diag
∼

of data, via the isomorphisms → in Corollary 11.21 (3), and we write†∼F∆ = († C∆ , Prime(† D∆ ) → V, † F`∆ , {† ρ∆,v }v∈V )(i.e., “∆ = { 0 , hF>
l i }”)

for the resulting F -prime-strip determined by the algorithm “F 7→ F ”. Note that∼
we have a natural isomorphism † F∆ → † Fmod of F -prime-strips, where † Fmod isthe data contained in the Θ-Hodge theatre of † HT  .

305

a proof of the abc conjecture after Mochizuki(2) (Theta and Gaussian Prime-strips) Apply Corollary 11.21 (4), (5) to the underlying Θ-bridge and -Hodge theatre of † HT  . Then the collection ΨFenv († HT Θ )of data, the global realified Frobenioid † Cenv := Cenv († HT Θ ), localisation isomorphisms Φ† Cenv ,v†gl. to loc.
∼−→

ΨFenv († HT Θ )Rv for v ∈ V give rise to an F -prime-strip∼

Fenv = († Cenv , Prime(† Denv ) → V, † F`env , {† ρenv,v }v∈V )(Note that † F`env is the F ` -prime-strip determined by ΨFenv († HT Θ )). Thus, there∼
is a natural identification isomorphism † Fenv → † Ftheta , where † Fenv is associatedto data in † HT Θ (cf. Definition 10.5 (4) for † Fenv ).
Similarly, the collection ΨFgau († HT Θ ) of data, the global realified Frobenioid † Cgau :=†Cgau ( HT ), localisation isomorphisms Φ† Cgau ,v
give rise to an F -prime-stripΘ†

gl. to loc.
∼−→

ΨFgau († HT Θ )Rv for v ∈ V∼

Fgau = († Cenv , Prime(† Dgau ) → V, † F`gau , {† ρgau,v }v∈V )(Note that † F`gau is the F ` -prime-strip determined by ΨFgau († HT Θ )). Finally, theevaluation isomorphisms of Corollary 11.21 (4), (5) determine an evaluation isomorphism
†

eval
∼

Fenv −→ † Fgau

of F -prime-strips.
(3) (Θ×µ - and Θ×µ
gau -Links) Let‡F∆I×µ (resp.

†

I×µ
Fenv
, resp.

†

I×µ
Fgau
)

donote F I×µ -prime-strip associated to the F -prime-strip ‡ F∆ (resp. † Fenv , resp.
†
Fgau ) (cf. Definition 10.12 (3) for F I×µ -prime-strips). Then the functriality ofthis algorithm induces maps
I×µ ‡IsomF († Fenv , ‡ F∆ ) → IsomF I×µ († Fenv
, F∆I×µ ),I×µ ‡, F∆I×µ ).
IsomF († Fgau , ‡ F∆ ) → IsomF I×µ († Fgau

Note that the second map is equal to the composition of the first map with theevalevaluation isomorphism † Fenv → † Fgau and the functorially obtained isomorphism
†

eval

I×µ
I×µ
Fenv
→ † Fgaufrom this isomorphism. We shall refer to the full poly-isomorphism

†

I×µ
Fenv

full poly∼‡−→F∆I×µ (resp.

†

I×µ
Fgau

full poly∼‡−→F∆I×µ )

306

Go Yamashita†as the Θ×µ -link (resp. Θ×µto ‡ HT  (cf. Definition 10.8),
gau -link) from HT
and we write it as
†

HT 

Θ×µ

‡

−→

HT  (resp.

†

HT 

Θ×µ
gau

−→

‡

HT  )

and we shall refer to this diagram as the Frobenius-picture of -Hodge theatres (This is an enhanced version of Definition 10.8). Note that the essential
meaning of the above link is
∼

∼

N
“ ΘN
−→ q N ” (resp. “ {q j }N1≤j≤l> −→ q ” )
v
2

v

v

v

for v ∈ Vbad .
(4) (Horizontally Coric F `×µ -Prime-strips) By the definition of the unit portionof the theta monoids and the Gaussian monoids, we have natural isomorphisms
∼

† `×µ
F∆

→

∼

† `×µ
Fenv

→

† `×µ
Fgau ,† `×µ † `×µ`×µ-prime-strips associated to the F ` -primewhere † F`×µ
∆ , Fenv , Fgau are the F
strips † F`∆ , † F`env , † F`gau , respectively. Then the composite
† `×µ
F∆

∼

→

† `×µ
Fenv

poly
∼

→

† `×µ
F∆

(resp.

† `×µ
F∆

∼

→

† `×µ
Fgau

with the poly-isomorphism induced by the full poly-isomorphism

poly
∼

→
†

† `×µ
F∆

I×µ
Fenv

)

full poly∼−→full poly
∼
I×µ
‡
†
I×µ
−→ ‡ F∆I×µ ) in the definition of Θ×µ -link (resp.
F∆(resp. Fgau
`×µΘ×µgau link) is equal to the full poly-isomorphism of F
-prime-strips. This means that(−) `×µF∆ is preserved (or “shared”) under both the Θ×µ -link and Θ×µ
gau -link (This isan enhanced version of Remark 10.8.1 (2)). Note that the value group portion is notshared under the Θ×µ -link and the Θ×µgau -link. Finally, this full poly-isomorphism
induces the full poly-isomorphism†

D`∆full poly
∼

‡

−→

D`∆

of the associated D` -prime-strips. We shall refer to this as the D-  -link from†
HT D- to ‡ HT D- , and we write it as†

HT D-D−→‡

HT D- .

This means that (−) D`∆ is preserved (or “shared”) under both the Θ×µ -link andΘ×µgau -link (This is an enhanced version of Remark 10.8.1 (1), Definition 10.21 and307

a proof of the abc conjecture after Mochizuki

Definition 10.35). Note that the holomorphic base “HT D- ” is not shared under×µ
the Θ×µ -link and the Θ×µ-link and Θ×µ
gau -link (i.e., Θ
gau -link share the underlying
mono-analytic base structures, but not the arithmetically holomorphic base structures).
(5) (Horizontally Coric Global Realified Frobenioids) The full poly-isomorphism†

D`∆full poly∼‡−→D`∆ in (4) induces an isomorphism∼∼∼(D († D`∆ ), Prime(D († D`∆ )) → V, {† ρD ,v }v∈V ) → (D (‡ D`∆ ), Prime(D (‡ D`∆ )) → V, {‡ ρD ,v }v∈V )of triples. This isomorphism is compatible with the R>0 -orbits
∼

“Kum”
poly
∼

∼

“Kum”
poly
∼

(† C∆ , Prime(† C∆ ) → V, {† ρ∆,v }v∈V )→∼(D († D`∆ ), Prime(D († D`∆ )) → V, {† ρD ,v }v∈V )and(‡ C∆ , Prime(‡ C∆ ) → V, {‡ ρ∆,v }v∈V )→∼(D (‡ D`∆ ), Prime(D (‡ D`∆ )) → V, {‡ ρD ,v }v∈V )of isomorphisms of triples obtained by the functorial algorithm in Corollary 11.21(2), with respect to the Θ×µ -link and the Θ×µgau -link. Here, the R>0 -orbits are naturally defined by the diagonal (with respect to Prime(−)) R>0 -action on the divisor
monoids.
Proof. Corollary follows from the definitions.
Remark 11.24.1. (Étale picture of D-  -Hodge Theatres, [IUTchII, Corollary
4.11]) We can visualise the “shared” and “non-shared” relation in Corollary 11.24 as
follows:†

HT D- − − > † D`∆ ∼= ‡ D`∆ > − − ‡ HT D-We shall refer to this diagram as the étale-picture of -Hodge theatres (Thisis an enhanced version of Remark 10.8.1, Remark 10.21.1 and Remark 10.35.1). Note
that, there is the notion of the order in the Frobenius-picture (i.e., † (−) is on the left,and ‡ (−) is on the right), on the other hand, there is no such an order and it has apermutation symmetry in the étale-picture (cf. also the last table in Section 4.3). Notethat these constructions are compatible, in an obvious sense, with Definition 10.21 and∼Definition 10.35, with respect to the natural identification (−) D`∆ → (−) D`> .

308

Go Yamashita§ 12.

Log-links — An Arithmetic Analogue of Analytic Continuation.
§ 12.1.

Log-links and Log-theta-lattices.

Definition 12.1.
([IUTchIII, Definition 1.1]) Let † F = {† Fv }v∈V be an Fprime-strip with the associated F ` -prime-strip (resp. F `×µ -prime-strip, resp. D-primestrip) † F` = {† Fv` }v∈V (resp. † F`×µ = {† Fv`×µ }v∈V , resp. † D = {† Dv }v∈V ).
(1) Let v ∈ Vnon . We writepf(Ψ† Fv ⊃ Ψ× ) Ψ∼:= (Ψ×
†F
†F
†F )
v
v
v

for the perfection of Ψ×(cf. Section 5.1). By the Kummer isomorphism of Re†Fv
gp

mark 3.19.2, we can construct an ind-topological field structure on Ψ† Fv , which is
an isomorph of Kv (cf. Section 5.1 for the notation (−)gp ). Then we can define thegp∼∼→ Ψ† Fv
pv -adic logarithm on Ψ∼† F , and this gives us an isomorphism logv : Ψ† Fv
v
of ind-topological groups. Thus, we can transport the ind-topological field structuregp
of Ψ† Fv into Ψ∼† F . Hence we can consider the multiplicative monoid “O ” of nonv
gp∼
zero integers of Ψ∼† F , and we write Ψlog(† Fv ) for it. Note that Ψlog(† F ) = Ψ† F .
v
vv

The pair † Πv y Ψlog(† Fv ) determines a pre-Frobenioidlog(† Fv ).
The resulting † Πv -equivariant diagramgp

Ψ† Fv ⊃ Ψ× Ψ∼
= Ψlog(† Fv )
†F
†F
v
v

(Log-link v ∈ Vnon )is called the tautological log-link associated to † Fv (This is a review, in our
setting, of constructions of the diagram (Log-link (non-Arch)) in Section 5.1), andwe write it aslog†
Fv −→ log(† Fv ).
(poly)∼‡For any (poly-)isomorphism (resp. the full poly-isomorphism) log(† Fv ) →(resp. log(† Fv )†full poly∼‡→Fv ) of pre-Frobenioids, we shall refer to the composite(poly)
∼
‡

log

Fv

Fv −→ log(† Fv ) → Fv as a log-link (resp. the full log-link) from † Fv to‡
Fv and we write it as
†

Fv

log

−→

‡

Fv (resp. † Fv

full log

−→

‡

Fv ).

Finally, put
I† Fv :=

)
(
1
gp
∼
∼
Gv († Πv )→Ψ⊂ΨIm (Ψ×
)
=
Ψ
,
†
†F
†F
F
log(† Fv )
v
v
v
2pv

309

a proof of the abc conjecture after Mochizuki

and we shall refer to this as the Frobenius-like holomorphic log-shell associated to † Fv (This is a review of Definition 5.1 in our setting). By the recongp
structible ind-topological field structure on Ψ† Fv∼ = Ψlog(† Fv ) , we can regard I† Fv
log

as an object associated to the codomain of any log-link † Fv −→ ‡ Fv .
(2) Let v ∈ Varc . Recall that † Fv = († Cv , † Dv , † κv ) is a triple of a pre-Frobenioid † Cv ,an Aut-holomorphic space † Uv := † Dv , and a Kummer structure † κv : Ψ† Fv :=†O († Cv ) ,→ A Dv , which is isomorphic to the model triple (Cv , Dv , κv ) of Defini⊂ Ψgp
for the subgroup oftion 10.2 (3). For N ≥ 1, we write Ψµ† FNv ⊂ Ψ×
†F
†F
v
v
gp
N -th roots of unity, and Ψ∼ Ψ† Fv for the universal covering of the topological
†F
v
exp

group Ψgp
(Recall that Ψ∼ Ψgp
is an isomorph of “C −→ C× ”). Then the
†F
†F
†F
v
v
v
composite
µN
Ψ∼
 Ψgp
 Ψgp
†F
†F
† F /Ψ† F
v
v
v
v
µN
is also a universal covering of Ψgp
We can regard Ψ∼as constructed†F† F /Ψ† F .
v
v
v
gp
µN
from Ψ† Fv /Ψ† Fv (cf. also Remark 10.12.1, Proposition 12.2, (4) in this definition,

Proposition 13.7, and Proposition 13.11). By the Kummer structure † κv , we cangp
construct a topological field structure on Ψ† Fv . Then we can define the Archimedean∼gp→ Ψ† Fv of
logarithm on Ψ∼: Ψ∼† F , and this gives us an isomorphism logv
†F
v
v
gp

topological groups. Thus, we can transport the topological field structure of Ψ† Fv†

Dv
into Ψ∼into a Kummer structure† F , and the Kummer structure Ψ† Fv ,→ Av†

† ∼κv : Ψ∼†Fv,→ A Dv . Hence we can consider the multiplicative monoid “O ” ofnon-zero elements of absolute values ≤ 1 of Ψ∼† F , and we write Ψlog(† Fv ) for it.
v
gp

Note that Ψlog(† Fv ) = Ψ∼
† F . The triple of topological monoid Ψlog(† Fv ) , the Autv
holomorphic space † Uv , and the Kummer structure † κ∼v determines a triplelog(† Fv ).
The resulting co-holomorphicisation-compatible-diagramgp

Ψ† Fv ⊂ Ψgp
 Ψ∼= Ψlog(† Fv )
†F
†F
v
v

(Log-link v ∈ Varc )is called the tautological log-link associated to † Fv (This is a review, in our
setting, of constructions of the diagram (Log-link (Arch)) in Section 5.2), and wewrite it aslog†
Fv −→ log(† Fv ).
†(poly)∼‡For any (poly-)isomorphism (resp. the full poly-isomorphism) log( Fv ) →(resp. log(† Fv )full poly
∼
‡

→

Fv

log

Fv ) of triples, we shall refer to the composite † Fv −→310

Go Yamashita†(poly)∼‡

log( Fv ) →we write it as

Fv as a log-link (resp. the full log-link) from † Fv to ‡ Fv and
†

Fv

log

−→

‡

Fv (resp. † Fv

full log

‡

−→

Fv ).

Finally, we write
I † Fvfor the Ψ×-orbit of the uniquely determined closed line segment of Ψ∼† F whichlog(† Fv )v
is preserved by multiplication by ±1 and whose endpoints diﬀer by a generator of thekernel of the natural surjection Ψ∼ Ψgp(i.e., “the line segment [−π, +π]”), or
†F
†F
v
v

µN
×
(when we regard Ψ∼as constructed from Ψgp†F† F /Ψ† F ) equivalently, the Ψlog(† F ) vvvv
orbit of the result of multiplication by N of the uniquely determined closed line
segment of Ψ∼which is preserved by multiplication by ±1 and whose endpoints†FvµN
diﬀer by a generator of the kernel of the natural surjection Ψ∼ Ψgp†F† F /Ψ† Fv
v
v
ππ(i.e., “the line segment N [− N
, +N
] = [−π, +π]”), and we shall refer to this as the
Frobenius-like holomorphic log-shell associated to † Fv (This is a review ofDefinition 5.3 in our setting). By the reconstructible topological field structure on
gpΨ† Fv∼ = Ψlog(† Fv ) , we can regard I† Fv as an object associated to the codomain oflog

any log-link † Fv −→ ‡ Fv .
(3) We put{}log(† Fv ) := Ψ∼†Fvlog(† F) :=v∈Vfor the collection of ind-topological modules (i.e., we forget the field structure onΨ∼† F ), where the group structure arises from the additive portion of the field strucv

nontures on Ψ∼, we regard Ψ∼as equipped with natural Gv († Πv )† F . For v ∈ V†Fvv
action. Write alsolog(† F) := {log(† Fv )}v∈V

for the Fv -prime-strip determined by log(† Fv )’s, and we write†log

F −→ log(† F)log

for the collection {† Fv −→ log(† Fv )}v∈V of diagrams, and we shall refer to this as
the tautological log-link associated to † F. For any (poly-)isomorphism (resp. the†(poly)∼ ‡full poly-isomorphism) log( F) →F (resp. log(† F)log(poly)∼ ‡we shall refer to the composite † F −→ log(† F) →log-link) from † F to ‡ F and we write it as
†

log

F −→

‡

F (resp. † F→F) of F-prime-strips,

F as a log-link (resp. the full

full log−→

full poly
∼
‡

‡

F ).

311

a proof of the abc conjecture after Mochizuki

Finally, we put
I† F := {I† Fv }v∈V ,and we shall refer to this as the Frobenius-like holomorphic log-shell associated to † F. We also write
I† F ⊂ log(† F)
for {I† Fv ⊂ log(† Fv )}v∈V . We can regard I† F as an object associated to thelog

codomain of any log-link † F −→ ‡ F.
(4) For v ∈ Vnon (resp. v ∈ Varc ), the ind-topological modules with Gv († Π)-action(resp. the topological module and the closed subspace) I† Fv ⊂ log(† Fv ) can beconstructed only from the v-component † Fv`×µ of the associated F `×µ -prime-strip,by the ×µ-Kummer structure, since these constructions only use the perfection(−)pf of the units and are unaﬀected by taking the quotient by Oµ (−) (cf. (Step2) of Proposition 5.2) (resp. only from the v-component † Fv` of the associatedF ` -prime-strip, by (Step 3) of Proposition 5.4, hence only from the v-component† `×µ
Fvof the associated F `×µ -prime-strip, by regarding this functorial algorithm
as an algorithm which only makes us of the quotien of this unit portion by µN for
N ≥ 1 with a universal covering of this quotient). We writeI† Fv`×µ ⊂ log(† Fv`×µ )for the resulting ind-topological modules with Gv († Πv )-action (resp. the resulting
topological module and a closed subspace). We shall refer to this as the Frobeniuslike mono-analytic log-shell associated to † Fv`×µ . Finally, we put
I† F`×µ := {I† Fv`×µ }v∈V ⊂ log(† F`×µ ) := {log(† Fv`×µ )}v∈V
for the collections constructed from the F `×µ -prime-strip † F`×µ (not from † F). We
shall refer to this as the Frobenius-like mono-analytic log-shell associated to
† `×µ
F
.
Proposition 12.2.
(log-Links Between F-Prime-strips, [IUTchIII, Proposition††‡1.2]) Let F = { Fv }v∈V , F = {‡ Fv }v∈V be F-prime-strips with associated F `×µ -primestrips (resp. D-prime-strips, resp. D` -prime-strips) † F`×µ = {† Fv`×µ }v∈V , † F`×µ ={‡ Fv`×µ }v∈V (resp. † D = {† Dv }v∈V , ‡ D = {‡ Dv }v∈V , resp. † D` = {† Dv` }v∈V , ‡ D` =log{‡ Dv` }v∈V ), respectively, and † F −→ ‡ F a log-link from † F to ‡ F. We recall the log-link
diagrams(lognon )gp

 log(† Fv ) = Ψlog(† Fv )Ψ† F v ⊃ Ψ×†Fv

(poly)∼→

gpΨ‡ F v ,312

Go YamashitaΨ† F v ⊂(logarc )

Ψgp
†F
v

†

 log( Fv ) =(poly)∼

gp
Ψlog(† Fv )→gp

Ψ‡ F v .

for v ∈ Vnon and v ∈ Varc , respectively.
log(1) (Vertically Coric D-Prime-strips) The log-link † F −→ ‡ F induces (poly-)isomorphisms†

D(poly)
∼
‡

→

D,

†

D

`

(poly)∼‡→

D`of D-prime-strips and D` -prime-strips, respectively. In particular, the (poly-)isomorphism†(poly)∼ ‡D →D induces a (poly-)isomorphism†Ψcns ( D)(poly)∼→

Ψcns (‡ D).

(2) (Compatibility with Log-volumes) For v ∈ Vnon (resp. v ∈ Varc ), the diagram lognon (resp. the diagram logarc ) is compatible with the natural pv -adic log†
gp †gp
volumes on (Ψ† Fv ) Πv , and (Ψlog(† Fv ) ) Πv (resp. the natural angular log-volume on
gp

Ψ×
and the natural radial log-volume on Ψlog(† Fv ) ) in the sense of the formula†Fv(5.1) of Proposition 5.2 (resp. in the sense of the formula (5.2) of Proposition 5.4).
µN
When we regard Ψ∼as constructed from Ψgp(cf. Definition 12.1 (2)),†F† F /Ψ† F
v
v
v
gp
µN
then we equip Ψ† Fv /Ψ† Fv the metric obtained by descending the metric of Ψgp
†F ;
v
µN
µN
however, we regard the object Ψgp(or Ψ×† F /Ψ† F ) as being equipped with
† F /Ψ† F
v
v
v
v

µN
a “weight N ”, that is, the log-volume of Ψ×
is equal to the log-volume of† F /Ψ† Fv
v
gpΨ† Fv ([IUTchIII, Remark1.2.1 (i)]) (cf. also Remark 10.12.1, Definition 12.1 (2),(4), Proposition 13.7, and Proposition 13.11).

(3) ((Frobenius-like) Holomorphic Log-shells) For v ∈ Vnon (resp. v ∈ Varc ), we
have
(
)
†
(
)
†
Πv
×
Πv
†
Ψlog(† Fv ) , Im (Ψ† Fv )→ log( Fv ) ⊂ I† Fv ⊂ log(† Fv )(cf. the inclusions (Upper Semi-Compat. (non-Arch)) Ok× , log(Ok× ) ⊂ Ik in Section 5.1) (resp.
(
)
(
)
gp
×
†
Ψlog(† Fv ) ⊂ I† Fv ⊂ log( Fv ) , Ψ† Fv ⊂ Im I† Fv → Ψ† Fv(cf. the inclusions (Upper Semi-Compat. (Arch)) Ok∼ ⊂ Ik , Ok× ⊂ expk (Ik ) in
Section 5.2) ).
(4) ((Frobenius-like and Étale-like) Mono-analytic Log-shells) For v ∈ Vnon
(resp. v ∈ Varc ), by Proposition 5.2 (resp. Proposition 5.4), we have a functorial
algorithm, with respect to the category † Dv` (= B(† Gv )0 ) (resp. the split monoid

313

a proof of the abc conjecture after Mochizuki†

Dv` ), to construct an ind-topological module equipped with a continuous † Gv -action(resp. a topological module){}log(† Dv` ) := † Gv y k ∼ († Gv ) (resp. log(† Dv` ) := k ∼ († Gv ) )and a topological submodule (resp. a topological subspace)I† Dv` := I(† Gv ) ⊂ k ∼ († Gv )(which is called the étale-like mono-analytic log-shell associated to † Dv` )equipped with a pv -adic log-volume (resp. an angular log-volume and a radial logvolume). Moreover, we have a natural functorial algorithm, with respect to thesplit-×µ-Kummer pre-Frobenioid † Fv`×µ (resp. the triple † Fv`×µ ), to construct an
Isomet-orbit (resp. {±1} × {±1}-orbit arising from the independent {±1}actions on each of the direct factors “k ∼ (G) = C ∼ × C ∼ ” in the notation of Proposition 5.4)log(† Fv`×µ )

“Kum”
poly
∼

→

log(† Dv` )of isomorphisms of ind-topological modules (resp. topological modules) (cf. the poly“Kum”
poly∼ss ‡ `‡ `isomorphism Ψss
cns ( F ) −→ Ψcns ( D ) of Corollary 11.21 (2)). We also havea natural functorial algorithm, with respect to the pv -adic Frobenioid † Fv (resp.
the triple † Fv ), to construct isomorphisms (resp. poly-isomorphisms of the {±1} ×{±1}-orbit arising from the independent {±1}-actions on each of the direct factors“k ∼ (G) = C ∼ × C ∼ ” in the notation of Proposition 5.4)
(poly)

gp∼=(Ψ‡ Fv) log(† Fv )(poly)∼=

gp(resp. (Ψ‡ Fv

tauto
∼

) log(† Fv )→log(† Fv`×µ )

tauto
∼

→

induced by Kum∼log(† Fv`×µ )−→log(† Dv` )induced by Kum
poly, {±1}×{±1}∼−→log(† Dv` ) )of isomorphisms of ind-topological modules (resp. topological modules) (cf. the iso∼ss ‡ ` ×morphism Ψcns (‡ D)×
v → Ψcns ( D )v of Corollary 11.20 (2) and the Kummer iso‡Kum∼morphism Ψcns ( F) → Ψcns (‡ D) of Corollary 11.21), which is compatible with therespective † Gv and Gv († Πv )-actions, the respective log-shells, and the respective logvolumes on these log-shells (resp. compatible with the respective log-shells, and therespective angular and radial log-volumes on these log-shells).
The above (poly-)isomorphisms induce collections of (poly-)isomorphismslog(† F`×µ ) := {log(† Fv`×µ )}v∈V

“Kum”
poly
∼

→

log(† D` ) := {log(† Dv` )}v∈V ,314

Go Yamashita

I† F`×µ := {I† Fv`×µ }v∈V

gp

gp(Ψcns (‡ F) := {Ψ‡ Fv }v∈V

“Kum”
poly
∼

→

I† D` := {I† Dv` }v∈V ,

tauto
∼

(poly)

induced by Kum
poly
∼
† `×µ
†

∼
= ) log(† F) := {log(† Fv )}v∈V → log( FI† F := {I† Fv }v∈V

tauto
∼

→

) → log( D` ),induced by Kum
poly∼I† F`×µ → I† D`

gp(Here, we regard each Ψ‡ Fv as equipped with Gv (‡ Πv )-action in the definition ofgp

Ψcns (‡ F)).
(5) ((Étale-like) Holomorphic Vertically Coric Log-shells) Let ∗ D be a D-primestrip with associated D` -prime-strip ∗ D` . We writeF(∗ D)
for the F-prime-strip determined by Ψcns (∗ D). Assume that † F = ‡ F = F(∗ D),
full log

and that the given log-link is the full log-link † F −→ ‡ F = F(∗ D). We have afunctorial algorithm, with respect to the D-prime-strip ∗ D, to construct a collectionof topological subspaces
I∗ D := I† F
(which is called a collection of vertically coric étale-like holomorphic loggp
gp
shell associated to ∗ D) of the collection Ψcns (∗ D) = Ψcns (∗ F),and a collection of
isomorphisms∼

I ∗ D → I ∗ D`∼

ss ‡ ` ×(cf. the isomorphism Ψcns (‡ D)×
v → Ψcns ( D )v of Corollary 11.20 (2)).

Remark 12.2.1.
(Kummer Theory, [IUTchIII, Proposition 1.2 (iv)]) Note that
the Kummer isomorphisms
Kum
∼

Kum
∼

Ψcns († F) → Ψcns († D), Ψcns (‡ F) → Ψcns (‡ D)(poly)∼of Corollary 11.21 (1) are not compatible with the (poly-)isomorphism Ψcns († D) →Ψcns (‡ D) of (1), with respect to the diagrams (lognon ) and (logarc ).

315

a proof of the abc conjecture after Mochizuki

Remark 12.2.2. (Frobenius-picture, [IUTchIII, Proposition 1.2 (x)]) Let {n F}n∈Z
be a collection of F-prime-strips indexed by Z with associated collection of D-primestrips (resp. D` -prime-strips) {n D}n∈Z (resp. {n D` }n∈Z ). Then the chain of full
log-links
full log (n−1)· · · −→

F

full log n−→full log (n+1)

−→

F

F

full log−→···of F-prime-strips (which is called the Frobenius-picture of log-links for F-primestrips) induces chains of full poly-isomorphisms······

full poly
∼
(n−1)

→

full poly
∼
(n−1)

→

D

D`

full poly∼n→full poly
∼
n

→

D

full poly
∼
(n+1)

→

D

full poly
∼
(n+1)

D`

→

full poly
∼

D`

→

··· ,

full poly
∼

→

···

of D-prime-strips and D` -prime-strips respectively. We identify (−) D’s by these fullpoly-isomorphisms, then we obtain a diagram
full log
full log/ Ψcns (n F) full log / Ψcns ((n+1) F) full log/· · · VVVV / Ψcns ((n−1) F)
hhh · · ·
VVVV
h
OOO
h
o
h
h
o
VVVV
hh
KumoooVVVV OOKumOOO Kum
hhhh
o
h
h
o
h
V
V
h
O
o
Kum
Kum
h
VVVV OO
V* '
wooothhhhh
Ψcns ((−) D).

This diagram expresses the vertical coricity of Ψcns ((−) D). Note that Remark 12.2.1
says that this diagram is not commutative.
Proof. Proposition follows from the definitions.
Definition 12.3.
tion 1.3 (i)]) Let(log-Links Between -Hodge Theatres, [IUTchIII, Proposi†HT  ,‡HT 

be -Hodge theatres with associated D-  -Hodge theatres † HT D- , ‡ HT D-
respectively. We write † F> , † F , † Fj (in † FJ ), † Ft (in † FT ) (resp. ‡ F> , ‡ F , ‡ Fj (in‡FJ ), ‡ Ft (in ‡ FT )) for F-prime-strips in the -Hodge theatre † HT  (resp. ‡ HT  ).
For an isomorphism∼Ξ : † HT D- → ‡ HT D-of D-  -Hodge theatres, the poly-isomorphisms determined by Ξ between the Dprime-strips associated to † F> , ‡ F> (resp. † F , ‡ F , resp. † Fj , ‡ Fj , resp. † Ft , ‡ Ft )
poly
∼

poly
∼

uniquely determines a poly-isomorphism log(† F> ) → ‡ F> (resp. log(† F ) → ‡ F ,

316

Go Yamashita
poly
∼ ‡

†

resp. log( Fj ) →
†

poly
∼ ‡

†

Fj , resp. log( Ft ) →

log
∼ ‡

log
∼ ‡

log
∼ ‡

†

Ft ), hence a log-link F> → F> (resp.

log
∼ ‡

F → F , resp. † Fj → Fj , resp. † Ft → Ft ), by Lemma 10.10 (2). We write†HT  −→
log

‡

HT 

∼

for the collection of data Ξ : † HT D- → ‡ HT D- , † F> → ‡ F> , † F → ‡ F , {† Fj →
‡

log

log

log

Fj }j∈J , and {† Ft → ‡ Ft }t∈T , and we shall refer to it as a log-link from † HT 

to

log

‡

HT



†

. When Ξ is replaced by a poly-isomorphism HT

D-

poly
∼ ‡

→

HT D-

full poly∼(resp. the full poly-isomorphism † HT D- → ‡ HT D- ), then we shall refer tothe resulting collection of log-links constructed from each constituent isomorphism of thepoly-isomorphism (resp. full poly-isomorphism) as a log-link (resp. the full log-linkfrom † HT  to ‡ HT  , and we also write it†

HT  −→log‡

HT  (resp. † HT 

full log−→‡

HT  ).

Note that we have to carry out the construction of the log-link first for single Ξ for the
purpose of maintaining the compatibility with the crucial global {±1}-synchronisationin the -Hodge theatre ([IUTchIII, Remark 1.3.1]) (cf. Proposition 10.33 and Corollary 11.20 (3)) (For a given poly-isomorphism of -Hodge theatres, if we considered the uniquely determined poly-isomorphisms on F-prime-strips induced by thepoly-isomorphisms on D-prime-strips by the given poly-isomorphism of -Hodge theatres, not the “constituent-isomorphism-wise” manner, then the crucial global {±1}synchronisation would collapse (cf. [IUTchI, Remark 6.12.4 (iii)], [IUTchII, Remark
4.5.3 (iii)])).
Remark 12.3.1. (Frobenius-picture and Vertical Coricity of D-  -Hodge theatres, [IUTchIII, Proposition 1.3 (ii), (iv)]) Let {n HT  }n∈Z be a collection of Hodge theatres indexed by Z with associated collection of D-  -Hodge theatres{n HT D- }n∈Z . Then the chain of full log-links
full log (n−1)· · · −→HT 

full log n

−→

HT 

full log (n+1)

−→

HT 

full log−→···of -Hodge theatres (which is called the Frobenius-picture of log-links for Hodge theatres) induces chains of full poly-isomorphisms···

full poly
∼
(n−1)

→

HT

D-

full poly
∼
n

→

HT

D-

full poly
∼
(n+1)

→

HT

D-

full poly
∼

→

··· ,

of D-  -Hodge theatres. We identify (−) HT D- ’s by these full poly-isomorphisms,

317

a proof of the abc conjecture after Mochizuki

then we obtain a diagram
full log
full log n/ HT  full log / (n+1) HT  full log/· · · VVVV / (n−1) HT O
hhh · · ·
OOO
o
VVVV
h
h
o
h
VVVV
hhh
KumooooOOO Kum
VVVV OOKum
hhhh
o
h
o
h
h
V
O
o
Kum
Kum
h
VVVV OOwoooshhhhh
V+ '(−)
HT D- ,where Kum expresses the Kummer isomorphisms in Remark 12.2.1. This diagram expresses the vertical coricity of (−) HT D- . Note that Remark 12.2.1 says that thisdiagram is not commutative.
Definition 12.4.
([IUTchIII, Definition 1.4]) Let {n,m HT  }n,m∈Z be a collection of -Hodge theatres indexed by pairs of integers. We shall refer to either of the
diagrams
..
..
.O
.O
full log···

Θ×µ / n,m+1


HT
O

full log
Θ×µ / n+1,m+1


O HT

full log···

Θ×µ/ n,m HT OΘ×µ/ n+1,m HT O..
.Ofull log/ n,m+1 HT Ofull log
Θ×µgau/ n+1,m+1 HT Ofull log···/ ··· ,

..
.
..
.O

Θ×µ
gau

Θ×µ

full log

..
.

···

/ ···

full log

full log

Θ×µ
gau

Θ×µ

/ n,m HT 
O

/ ···

full log
Θ×µgau/ n+1,m HT Ofull log

..
.

Θ×µ
gau

Θ×µ
gau

/ ···

full log..
.

as the log-theta-lattice. We shall refer to the former diagram (resp. the latter diagram) as non-Gaussian (resp. Gaussian).
Remark 12.4.1.
For the proof of the main Theorem 0.1, we need only two adjacentcolumns in the (final update version of) log-theta-lattice. In the analogy with p-adic

318

Go Yamashita

Teichmüller theory, this means that we need only “lifting to modulo p2 ” (cf. the lasttable in Section 3.5).
Theorem 12.5.
Fix an initial T h-data(Bi-Cores of the Log-Theta-Lattice, [IUTchIII, Theorem 1.5])(F /F, XF , l, C K , V, Vbad
mod , ).
For any Gaussian log-theta-lattice corresponding to this initial Θ-data, we write n,m D(resp. n,m D> ) for the D-prime-strip labelled “” (resp. “>”) of the -Hodge theatre.
(1) (Vertical Coricity) The vertical arrows of the Gaussian log-theta-lattice induce
the full poly-isomorphisms between the associated D-  -Hodge theatres···

full poly
∼
n,m

→

HT

D-

full poly
∼
n,m+1

→

HT

D-

full poly
∼

→

··· ,

where n is fixed (cf. Remark 12.3.1).
(2) (Horizontal Coricity) The horizontal arrows of the Gaussian log-theta-lattice induce the full poly-isomorphisms between the associated F `×µ -prime-strips
full polyfull polyfull poly∼∼∼n,m `×µ
n+1,m `×µ
··· →
F∆
→
F∆
→ ···

where m is fixed (cf. Corollary 11.24 (4)).
(3) (Bi-coric F `×µ -Prime-strips) Let n,m D`∆ for the D` -prime-strip associated to
the F ` -prime-strip n,m F`∆ of Corollary 11.24 (1) for the -Hodge theatre n,m HT  .
We idenfity the collections Ψcns (n,m D )0 , Ψcns (n,m D )hF> i of data via the isomorl

diag∼phism Ψcns (n,m D )0 → Ψcns (n,m D )hF> i constructed in Corollary 11.20 (3), andl
we writeF`∆ (n,m D )for the resulting F ` -prime-strip (Recall that “∆ = {0, hF>l i}”) Note also we have
∼
` n,m
` n,m
a natural identification isomorphism F∆ ( D ) → F> ( D> ), where we writeF`> (n,m D> ) for the F ` -prime-strip determined by Ψcns (n,m D> ) (Recall that “>={0, }”. cf. Lemma 10.38). We write
n,m
n,m
F`×
D ), F`×µ
D )
∆ (
∆ (

for the associated F `× -prime-strip and F `×µ -prime-strip to F`∆ (n,m D ), respec∼
ss ‡ ` ×tively. By the isomorphism “Ψcns (‡D)×
v → Ψcns ( D )v ” of Corollary 11.20 (2), we
have a functorial algorithm, with respect to the D` -prime-strip n,m D`∆ , to construct319

a proof of the abc conjecture after Mochizukin,m `an F `× -prime-strip F`×D∆ ). We also have a functorial algorithm, with respect∆ (n,mto the D-prime-strip
D , to construct an isomorphism
n,m
F`×
D )
∆ (

tauto
∼

→

n,m `
F`×
D∆ ),
∆ (

by definitions. Then the poly-isomorphisms of (1) and (2) induce poly-isomorphisms
poly
∼

··· →

poly
∼

··· →

n,m
F`×µ
D )
∆ (
n,m `
F`×µ
D∆ )
∆ (

poly
∼

→

poly
∼

→

n,m+1
F`×µ
D )
∆ (

poly
∼

n+1,m `
F`×µ
D∆ )
∆ (

→ ··· ,

poly
∼

→ ···

of F `×µ -prime-strips, respectively. Note that the poly-isomorphisms (as sets of isomorphisms) of F `×µ -prime-strips in the first line is strictly smaller than the polyisomorphisms (as sets of isomorphisms) of F `×µ -prime-strips in the second line
tauto
∼

n,m
n,m `
in general, with respect to the above isomorphism F`×D ) → F`×
D∆ ),
∆ (
∆ (
by the existence of non-scheme-theoretic automorphisms of absolute Galois groups
of MLF’s (cf. the inclusion (nonGC for MLF) in Section 3.5), and that the polymorphisms in the second line are not full by Remark 8.5.1. In particular, by composing these isomorphisms, we obtain poly-isomorphisms
n,m `
F`×µ
D∆ )
∆ (

poly
∼

0

0

n ,m
→ F`×µD`∆ )∆ (of F `×µ -prime-strips for any n0 , m0 ∈ Z. This means that the F `×µ -prime-strip`×µ n,m `F∆( D∆ ) is coric both horizontally and vertically, i.e., it is bi-coric. Finally,Kum∼

the Kummer isomorphism “Ψcns (‡ F) → Ψcns (‡ D)” of Corollary 11.21 (1) determines Kummer isomorphismn,m `×µF∆

induced by Kum
∼

−→

n,m `
F`×µ
D∆ )
∆ (

which is compatible with the poly-isomorphisms of (2), and the ×µ-Kummer structures at v ∈ Vnon and a similar compatibility for v ∈ Varc (cf. Definition10.12(1)).
(4) (Bi-coric Mono-analytic Log-shells) The poly-isomorphisms in the bi-coricity
in (3) induce poly-isomorphisms{

In,m D`∆ ⊂ log(

n,m

D`∆ )

} poly {}00∼→In0 ,m0 D` ⊂ log(n ,m D`∆ ) ,∆{} poly {}∼`×µ n0 ,m0 `
n,m `
0
0
IF`×µ (n,m D` ) ⊂ log(F`×µ
(
D
))
→
I
⊂
log(F
(
D
))
`×µ
∆
∆
∆
∆
F
(n ,m D` )
∆

∆

∆

∆

320

Go Yamashita

for any n, m, n0 , m, ∈ Z, which are compatible with the natural poly-isomorphisms
“Kum”
poly
∼

{
}
n,m `
IF`×µ (n,m D` ) ⊂ log(F`×µ
(
D
))
∆
∆

−→

∆

∆

{
}
In,m D`∆ ⊂ log(n,m D`∆ )of Proposition 12.2 (4). On the other hand, by Definition 12.1 (1) for “Ψcns († F )0 ”
and “Ψcns († F )hF> i ” in Corollary 11.24 (1) (which construct n,m F`∆ ), we obtainlIn,m F∆ ⊂ log(n,m F∆ )(This is a slight abuse of notation since no F-prime-strip “n,m F∆ ” has been defined).
Then we have natural poly-isomorphisms
induced by Kum{} poly {}{} tauto∼∼n,mn,m `×µIn,m F∆ ⊂ log( F∆ ) → In,m F`×µ ⊂ log( F∆ ) → In,m D`∆ ⊂ log(n,m D`∆ )∆(cf. Proposition 12.2 (4)), where the last poly-isomorphism is compatible with thepoly-isomorphisms induced by the poly-isomorphisms of (2).
(5) (Bi-coric Mono-analytic Global Realified Frobenioids) The poly-isomorphisms
D`∆

poly
∼ n0 ,m0→D`∆ of D` -prime-strips induced by the full poly-isomorphisms of (1)and (2) for n, m, n0 , m0 induce an isomorphismn,m∼(D (n,m (D`∆ ), Prime(D (n,m (D`∆ )) → V, {n,m ρD ,v }v∈V )
∼

0

0

0

0

∼

0

0

→ (D (n ,m (D`∆ ), Prime(D (n ,m (D`∆ )) → V, {n ,m ρD ,v }v∈V )of triples (cf. Corollary 11.20 (2), and Corollary 11.24 (5)). Moreover, this isomorphism of triples is compatible, with respect to the horizontal arrows of the Gaussian
log-theta-lattice, with the R>0 -orbits of the isomorphisms∼(n,m C∆ , Prime(n,m C∆ ) → V, {n,m ρ∆,v }v∈V )
“Kum”
∼

→

∼

(D (n,m D`∆ ), Prime(D (n,m D`∆ )) → V, {n,m ρD ,v }v∈V )of triples, obtained by the functorial algorithm in Corollary 11.21 (2) (cf. also Corollary 11.24 (1), (5)).
Proof. Theorem follows from the definitions.
§ 12.2.

Kummer Compatible Multiradial Theta Monoids.

In this subsection, we globalise the multiradiality of local theta monoids (Proposition 11.7, and Proposition 11.15) to cover the theta monoids and the global realified

321

a proof of the abc conjecture after Mochizuki

theta monoids in Corollary 11.20 (4), (5) Corollary 11.21 (4), (5), in the setting of
log-theta-lattice.
In this subsection, let † HT  be a -Hodge theatre with respect to the fixed
initial Θ-data, and n,m HT  a collection of -Hodge theatres arising from a Gaussian
log-theta-lattice.
Proposition 12.6.
(Vertical Coricity and Kummer Theory of Theta Monoids,[IUTchIII, Proposition 2.1]) We summarise the theta monoids and their Kummer theory
as follows:(1) (Vertically Coric Theta Monoids) By Corollary 11.20 (4) (resp. Corollary 11.20(5)), each isomorphism of the full poly-isomorphism induced by a vertical arrow of
the Gaussian log-theta-lattice induces a compatible collection∼n,mD> ) → (∞) Ψenv (n,m+1 D> )(∞) Ψenv (∼(resp. Denv (n,m D`> ) → Denv (n,m+1 D`> ) )of isomorphisms, where the last isomorphism is compatible with the respective bijec∼
tion Prime(−) → V, and localisation isomorphisms.
(2) (Kummer Isomorphisms) By Corollary 11.21 (4) (resp. Corollary 11.21 (5)),
we have a functorial algorithm, with respect to the -Hodge theatre † HT  , toconstruct the Kummer isomorphism†(∞) ΨFenv ( HT

Θ

Kum
∼

†

†

) → (∞) Ψenv ( D> ) (resp. Cenv ( HT )
Θ

“Kum”
∼

→

Denv († D`> ) ).
∼Here, the resp’d isomorphism is compatible with the respective Prime(−) → V andthe respective localisation isomorphisms. Note that the collection Ψenv († D> ) of
data gives us an F ` -prime-strip F`env († D> ), and an F -prime-strip Fenv († D> ) =∼(Denv († D`> ), Prime(Denv († D`> )) → V, F`env († D> ), {ρDenv ,v }v∈V ) and that the nonresp’d (resp. the resp’d) Kummer isomorphism in the above can be interpreted as
an isomorphism
† `
Fenv

induced by Kum∼→

F`env († D> ) (resp.

†

Fenv

“Kum”
∼

→

Fenv († D> ) )of F ` -prime-strips (resp. F -prime-strips).
(3) (Compatibility with Constant Monoids) By the definition of the unit portionof the theta monoids (cf. Corollary 11.24 (4)), we have natural isomorphisms† `× ∼ † `×F∆ → Fenv ,

∼

† `
`× †
F`×
∆ ( D∆ ) → Fenv ( D> ),induced by Kum∼† `which are compatible with the Kummer isomorphisms Fenv→
F`env († D> ),induced by Kum
∼
† `×µ
† `
F∆
→
F`×µ
∆ ( D∆ ) of (2) and Theorem 12.5 (3).

322

Go Yamashita

Proof. Proposition follows from the definitions.
Theorem 12.7.
(Kummer-Compatible Multiradiality of Theta Monoids, [IUTchIII,
Theorem 2.2]) Fix an initial T h-data(F /F, XF , l, C K , V, Vbad
mod , ).
Let † HT  be a -Hodge theatre with respect to the fixed initial Θ-data.
(1) The natural functors which send an F -prime-strip to the associated F I×µ - andF `×µ -prime-strips and composing with the natural isomorphisms of Proposition 12.6
(3) give us natural homomorphisms
I×µ †
† `
AutF (Fenv († D> )) → AutF I×µ (Fenv( D> ))  AutF `×µ (F`×µ∆ ( D∆ )),I×µ)  AutF `×µ († F`×µ
AutF († Fenv ) → AutF I×µ († Fenv
∆ )(Note that the second homomorphisms in each line are surjective), which are compat†

“Kum”
∼

Fenv ( D> ), † F`×µ∆†ible with the Kummer isomorphisms Fenv →`×µ † `
F∆
( D∆ ) of Proposition 12.6 (2), and Theorem 12.5 (3)induced by Kum∼→(2) (Kummer Aspects of Multiradiality at Bad Primes) For v ∈ Vbad , we write⊥ †∞ Ψenv ( D> )v⊂ ∞ Ψenv († D> )v ,Θ⊥†∞ ΨFenv ( HT )v⊂ ∞ ΨFenv († HT Θ )v ,for the submonoids corresponding to the respective splittings (i.e., the submonoids
generated by “∞θιenv (MΘ∗ )” and the respective torsion subgroups). We have a commutative diagram
⊥
†
Θ
†
Θ µ
†
Θ ×
†
Θ ×µ
∞ ΨFenv ( HT )v ⊃ ∞ ΨFenv ( HT )v ⊂ ∞ ΨFenv ( HT )v  ∞ ΨFenv ( HT )v
Kum ↓∼=⊥†∞ Ψenv ( D> )v

Kum ↓∼=⊃µ†∞ Ψenv ( D> )v

Kum ↓∼=⊂†×∞ Ψenv ( D> )v

Kum ↓∼=
×µ†∞ Ψenv ( D> )vpoly∼† ` ×µ→ Ψss
cns ( F∆ )v
poly“Kum” ↓ ∼
=
poly
∼
×µ
ss
†
`
→ Ψcns ( D∆ )v ,where † D`∆ and † F`∆ are as in Theorem 12.5 (3), and Corollary 11.24 (1), respectively, the most right vertical arrow is the poly-isomorphism of Corollary 11.21 (2),the most right lower horizontal arrow is the poly-isomorphism obtained by composing∼`× † `†
the inverse of the isomorphism F`×env ( D> ) ←− F∆ ( D∆ ) of Proposition 12.6 (3)† ` ×µand the poly-automorphism of Ψss
induced by the full poly-automorphism
cns ( D∆ )v`† `of the D -prime-strip D∆ , and the most right upper horizontal arrow is the polyisomorphism defined such a manner that the diagram is commutative. This commutative diagram is compatible with the various group actions with respect to thediagram
temp
Θ †
Θ †
Θ †
Θ †
Π
(M∗ ( D>,v ))  Gv (M∗ ( D>,v )) = Gv (M∗ ( D>,v )) = Gv (M∗ ( D>,v ))X

full poly∼Θ †→Gv (M∗ ( D>,v )).

† ` ×µFinally, each of the various composite ∞ Ψenv († D> )µv → Ψss
is equal to thecns ( F∆ )vzero map, hence the identity automorphism on the following objects is compatible

323

a proof of the abc conjecture after Mochizuki(with respect to the various natural morphisms) with the collection of automorphisms† ` ×µof Ψssinduced by any automorphism in AutF `×µ († F`×µ
cns ( F∆ )v∆ ):(⊥, µ)étv(µZb )étv(M)étv(spl)étv††µ
the submonoid and the subgroup ∞ Ψ⊥env ( D> )v ⊃ ∞ Ψenv ( D> )v ,†the cyclotome µZb (MΘ∗ ( D>,v )) ⊗ Q/Z with respect to the natural isomorphism∼††µµZb (MΘ∗ ( D>,v )) ⊗ Q/Z → ∞ Ψenv ( D> )v†
the projective system MΘ
∗ ( D>,v ) of mono-theta environments††µthe splittings ∞ Ψ⊥env ( D> )v  ∞ Ψenv ( D> )v by the restriction to the zerolabelled evaluation points (cf. Corollary 11.11 (3) and Definition 11.12 (1)).

Proof. Theorem follows from the definitions.
Remark 12.7.1.

([IUTchIII, Remark 2.2.2 (iii)]) Note that the Galois evaluation(O× ∞ θ ⊃)∞ θ 7→ 1 ∈ O×µ

by which we obtain the multiradial algorithm of constructing the splittings of the thetamonoid O× ∞ θ is compatible with the respective Kummer theories of O× (⊂ O× ∞ θ)and the coric O×µ .
If we replace the Θ-link by a naive correspondence q 7→ q λ with λ ∈ Z>0 , then theanaloguous map
q λ 7→ 1 ∈ O×µ
is not compatible with the Kummer theories, since the Kummer class of q λ in a cohomology group of Π/∆ is never sent to the trivial element of the related cohomology
poly∼group of G via the poly-isomorphism Π/∆ → G.
Corollary 12.8.
([IUTchIII, Étale Picture of Multiradial Theta Monoids, Coroln,mlary 2.3]) Let { HT}n,m∈Z be a collection of -Hodge theatres arising from a
Gaussian log-theta-lattice, with associated D-  -Hodge theatres n,m HT D- . We
consider the following radial environment. We define a radial datum
†

`×µ †
† `
R = († HT D- , Fenv († D> ), † Rbad , F`×µ
∆ ( D∆ ), Fenv ( D> )to be a quintuple of(HT D )étR(F )étR

a D-  -Hodge theatre † HT D- ,the F -prime-strip Fenv († D> ) associated to † HT D- ,full poly
∼

→

† `
F`×µ
∆ ( D∆ ))324(bad)ét
R

Go Yamashitaétét
the quadruple † Rbad = ((⊥, µ)vét , (µZb )étv , (M)v , (spl)v ) of Theorem 12.7 (2) forv ∈ Vbad ,(F `×µ )ét
R`×µ † `
the F `×µ -prime-strip F∆( D∆ ) associated to † HT D- , and(env∆)ét
R†
the full poly-isomorphim F`×µ
env ( D> )full poly
∼

→

† `
F`×µ
∆ ( D∆ ).

We define a morphism from a radial datum † R to another radial datum ‡ R to be aquintuple of(HT D )ét
MorR∼

an isomorphism † HT D- → ‡ HT D- of D-  -Hodge theatres,∼(F )ét
MorR

the isomorphism Fenv († D> ) → Fenv (‡ D> ) of F -prime-strips induced by the isoétmorphism (HT D )Mor,(bad)ét
MorR

the isomorphism † Rbad → ‡ Rbad of quadruples induced by the isomorphism(HT D )ét
Mor , and(F `×µ )ét
MorR

∼

∼

`×µ ‡ `
† ``×µ
an isomorphism F`×µ-prime-strips∆ ( D∆ ) → F∆ ( D∆ ) of Fét(Note that the isomorphisms of (F )Morand (F `×µ )ét
Mor are automatically compatibleétwith (env∆) ).
We define a coric datum†

C = († D` , F`×µ († D` ))to be a pair of(D` )`ét
C(F`×µ )`ét
Ca D` -prime-strip † D` , andthe F `×µ -prime-strip F`×µ († D` ) associated to † D` .

We define a morphism from a coric datum † C to another coric datum ‡ C to be a pairof(D` )`ét
MorC(F`×µ )`ét
MorC∼

an isomorphism † D` → ‡ D` of D` -prime-strips, and∼an isomorphism F`×µ († D` ) → F`×µ (‡ D` ) of F `×µ -prime-strips which induces`
the isomorphism (D` )`ét
MorC on the associated D -prime-strips.

325

a proof of the abc conjecture after Mochizuki

We define the radial algorithm to be the assignment
†

† `
`×µ †
R = († HT D- , Fenv († D> ), † Rbad , F`×µ
∆ ( D∆ ), Fenv ( D> )full poly
∼

→

† `
F`×µ
∆ ( D∆ ))† `7→ † C = († D`∆ , F`×µ
∆ ( D∆ ))ét
and the assignment on morphisms determined by the data (F `×µ )Mor.
R

(1) (Multiradiality) The functor defined by the above radial algorithm is full andessentially surjective, hence the above radial environment is multiradial.
(2) (Étale Picture) For each D-  -Hodge theatre n,m HT D- with n, m ∈ Z, we
can associate a radial datum n,m R. The poly-isomorphisms induced by the vertical
poly
∼ n,m

arrows of the Gaussian log-theta-lattice induce poly-isomorphisms · · · →
n,m+1

poly
∼

R →

poly
∼

R → · · · of radial data by Theorem 12.5 (1). We writen,◦R

for the radial datum obtained by idenfitying n,m R for m ∈ Z via these poly-isomorphisms,
and we writen,◦Cfor the coric datum obtained by applying the radial algorithm to n,◦ R. Similarly, thepoly-isomorphisms induced by the horizontal arrows of the Gaussian log-theta-latticefull poly∼n,minduce full poly-isomorphisms · · · →D` -prime-strips Theorem 12.5 (2). We write
◦,◦

D`∆

full poly
∼
n+1,m

→

D`∆

full poly∼→· · · ofC

for the coric datum obtained by idenfitying n,◦ C for n ∈ Z via these full polyisomorphisms. We can visualise the “shared” and “non-shared” relation in Corollary 12.8 (2) as follows:
0

0

◦,◦ `
Fenv (n,◦ D> ) + n,◦ Rbad + · · · −− > F`×µD∆ ) >−− Fenv (n ,◦ D> ) + n ,◦ Rbad + · · ·∆ (We shall refer to this diagram as the étale-picture of multiradial theta monoids.
Note that it has a permutation symmetry in the étale-picture (cf. also the last table
in Section 4.3). Note also that these constructions are compatible, in an obvious
sense, with Definition 11.24.1.

326

Go Yamashita(3) (Kummer Compatibility of Θ×µ
gau -Link, env → ∆) The (poly-)isomorphisms`×µét`×µ étof F-prime-strips of/induced by (env∆)R, (F )ét)MorR are comMorR , and (Fpatible with the poly-isomorphisms n,m F`×µ∆full poly∼→ n+1,m F`×µ
of Theorem 12.5 (2)∆arising from the horizontal arrows of Gaussian log-theta-lattice, with respect to the
induced by Kum∼

induced by Kum∼n,m `Kummer isomorphisms n,m F`×µ→F`×µD∆ ), n,m F`env
→
∆
∆ (
`
n,m
Fenv ( D> ) of Theorem 12.5 (3) and Proposition 12.6 (2). In particular, we havea commutative diagram
full poly∼n,m `×µF∆/ n+1,m F`×µ∆

induced by Kum & “∆7→env” ∼=∼= induced by Kum & “∆7→env”


full poly∼ / `×µ n+1,◦ `
n,◦ `
F`×µ
(
D
)
F
(
D> ).
env
>
env

∼

n,m
D> ) →(4) (Kummer Compatibility of Θ×µ
gau -Link, ⊥ & ) The isomorphisms Fenv (
∼
ét
n+1,m
n,m bad
n+1,m
bad
ét
Fenv (
D> ),R→
R
of (F )MorR , (bad)MorR are compatible

with the poly-isomorphisms n,m F`×µ∆full poly∼→ n+1,m F`×µ
of Theorem 12.5 (2) arising∆from the horizontal arrows of Gaussian log-theta-lattice, with respect to the Kummer
isomorphisms n,m Fenv

“Kum”
∼

→

∼

Fenv (n,m D> ), n,m F`×µ∆

induced by Kum
∼

→

n,m `
F`×µ
D∆ ),
∆ (

“Kum”
∼

and ( C∆ , Prime( C∆ ) → V, { ρ∆,v }v∈V ) → (D (n,m D`∆ ), Prime (D (n,m D`∆ ))∼→ V, {n,m ρD ,v }v∈V ) of Proposition 12.6 (2), Theorem 12.5 (3), (5) and theirn+1,m(−)-labelled versions, and the full poly-isomorphism of projective system of
n,m

n,m

n,m

†
mono-theta environments “MΘ∗ ( D>,v )full poly
∼

→

†
MΘ
∗ ( F v )” of Proposition 11.15.

Proof. Corollary follows from the definitions.
Remark 12.8.1. ([IUTchIII, Remark 2.3.3]) In this remark, we explain similarities and diﬀerences between theta evaluations and NF evaluations. Similarities are as
follows: For the theta case, the theta functions are multiradial in two-dimensional geometric containers, where we use the cyclotomic rigidity of mono-theta environmentsin the Kummer theory, which uses only µ-portion (unlike the cyclotomic rigidity via
LCFT), and the evaluated theta values (in the evaluation, which depends on a holomorphic structure, the elliptic cuspidalisation is used), in log-Kummer correspondence
later (cf. Proposition 13.7 (2)), has a crucial non-interference property by the constantmultiple rigidity (cf. Proposition 13.7 (2c)). For the NF case, the κ-coric functions are
multiradial in two-dimensional geometric containers, where we use the cyclotomic rigidb × = {1} in the Kummer theory, which uses only {1}-portion (unlike
ity of via Q>0 ∩ Z

327

a proof of the abc conjecture after Mochizuki

the cyclotomic rigidity via LCFT), and the evaluated number fields (in the evaluation,
which depends on a holomorphic structure, the Beyli cuspidalisation is used), in logKummer correspondence later (cf. Proposition 13.11 (2)), has a crucial non-interference∏××∩ v≤∞ Ov = µ(Fmod) (cf. Proposition 13.11 (2)). cf. also the followproperty by Fmod
ing table:

multirad. geom. container
theta

theta fct.

eval

in mono-an. container
2

theta values q j (ell. cusp’n)cycl. rig.

log-Kummer

mono-thetano interf. by(depends on labels&hol. str.)NF∞ κ-coric fct.

eval×
NF Fmod(up to {±1})(Belyi cusp’n)const. mul. rig.
b×
via Q>0 ∩ Z= {1}(indep. of labels, dep. on hol. str.)

no interf. by
×
Fmod
∩

∏

v≤∞ Ov = µ

The diﬀerences are as follows: The output theta values q j depend on the labels j ∈ F>l
(Recall that the labels depend on a holomorphic structure), and the evaluation is com×patible with the labels, on the other hand, the output number field Fmod
(up to {±1})>does not depend on the labels j ∈ Fl (Note also that, in the final multiradial algorithm,
we also use global realified monoids, and these are of mono-analytic nature (since units
are killed) and do not depend of holomorphic structure). We continue to explain thediﬀerences of the theta case and the NF case. The theta function is transcendental
and of local nature, and the cyclotomic rigidity of mono-theta environments, which iscompatible with profinite topology (cf. Remark 9.6.2), comes from the fact that the orderof zero at each cusp is equal to one (Such “only one valuation” phenomenon corresponds precisely to the notion of “local”). Note that such a function only exists as atranscendental function. (Note also that the theta functions and theta values do nothave Fo±l -symmetry; however, the cyclotomic rigidity of mono-theta enrionments haveo±Fl -symmetry. cf. Remark 11.17.1). On the other hand, the rational functions used in
Belyi cuspidalisation are algebraic and of global nature, and the cyclotomic rigidity via
b × = {1}, which is obtained by sacrificing the compatibility with profinite topolQ>0 ∩ Z
ogy (cf. Remark 9.6.2). Algebraic rational function never satisfy the property like “theorder of zero at each cusp is equal to one” (Such “many valuations” phenomenon corresponds precisely to the notion of “global”). cf. also the following table (cf. [IUTchIII,
Fig. 2.7]):2

328

Go Yamashitatheta (0 is permuted)

transcendental

local

compat. w/prof. top.

“one valuation”

NF (0 is isolated)algebraic

globalincompat. w/prof. top.

“many valuations”

We also explain the “vicious circles” in Kummer theory. In the mono-anabelian∗reconstruction algorithm, we use various cyclotomes µét
arising from cuspidal inertia
subgroups (cf. Theorem 3.17), these are naturally identified by the cyclotomic rigidity
isomorphism for inertia subgroups (cf. Proposition 3.14 and Remark 3.14.1). We writeµ∀ét for the cyclotome resulting from the natural identifications. In the context of logKummer correspondence, the Frobenius-like cyclotomes µFr ’s are related to µ∀ét , via
cyclotomic rigidity isomorphisms:• † µO FrF
FF
FFKum
log
FF
F#
Kum /
‡
◦ µ∀
• µO Fr:u étuuuu
loguuu Kumuu∀
If we consider these various Frobenius-like µFr ’s and the vertically coric étale-like µét
as distinct labelled objects, then the diagram does not result in any “vicious circles” or“loops”. On the other hand, ultimately in Theorem 13.12, we will construct algorithms
to describe objects of one holomorphic structure on one side of Θ-link, in terms ofanother alien arithmetic holomorphic structure on another side of Θ-link by means ofmultiradial containers. These multiradial containers arise from étale-like versions ofobjects, but are ultimately applied as containers for Frobenius-like versions of objects.
Hence we need to contend with the consequences of identifying the Frobenius-like µFr ’s
and the étale-like µ∀ét , which gives us possible “vicious circles” or “loops”. We consider
the indeterminacies arising from possible “vicious circles”. The cyclotome µ∀ét is subject
to indeterminacies with respect to multiplication by elements of the submonoid

Iord ⊂ N≥1 × {±1}generated by the orders of the zeroes of poles of the rational functions appearing the cyclotomic rigidity isomorphism under consideration (Recall that constructing cyclotomic
rigidity isomorphisms associated to rational functions via the Kummer-theoretic approach of Definition 9.6 amounts to identifying various µ∗ét ’s with various sub-cyclotomesof µFr ’s via morphisms which diﬀer from the usual natural identification precisely bya proof of the abc conjecture after Mochizuki

329

multiplication by the order ∈ Z at a cusp “∗” of the zeroes/poles of the rational function). In the theta case, we have
Iord = {1}as a consequence of the fact that the order of the zeros/poles of the theta function at
any cusp is equal to 1. On the other hand, for the NF case, such a phenomenon never
happens for algebraic rational functions, and we haveIm(Iord → N≥1 ) = {1}b × = {1}. Note also that the indeterminacy arising from Im(Iord →by the fact Q>0 ∩ Z{±1}) (⊂ {±1}) is avoided in Definition 9.6, by the fact that the inverse of a nonconstand κ-coric rational function is never κ-coric, and that this thechnique is incompat∀
ible with the identification of µFr and µét
discussed above. Hence in the final multiradialordalgorithm, a possible Im(I → {±1}) (⊂ {±1})-indeterminacy arises. However, the to×
tality Fmodof the non-zero elements is invariant under {±1}, and this indeterminacy is2harmless (Note that, in the theta case, the theta values q j have no {±1}-invariance).
§ 13.

Multiradial Representation Algorithms.

In this section, we construct the main multiradial algorithm to describe objects of
one holomorphic structure on one side of Θ-link, in terms of another alien arithmetic
holomorphic structure on another side of Θ-link by means of multiradial containers. We
briefly explain the ideas. We want to “see” (=multiradiality) the alien ring structureon the left-hand side of Θ-link (more precisely, Θ×µ
LGP -link) from the right-hand side of
Θ-link:^ (eye) Θ-link /••As explained in Section 4.3, after constructing link (or wall) by using Frobenius-like objects, we relate Frobenius-like objects to étale-like objects via Kummer theory (Kummer detachment). Then étale-like objects can penetrate the wall (étale transport)(cf. Remark 9.6.1). We also have another step to go from holomorphic structure to theunderlying mono-analytic structure for the purpose of using the horizontally coric (i.e.,shared) objects in the final multiradial algorithm. This is a fundamental strategy:arith.-holomorphic Frobenius-like obj’s data assoc. to F-prime-strips↓ Kummer theory
arith.-holomorphic étale-like obj’s
data assoc. to D-prime-strips↓ forget arith.-hol. str.
mono-analytic étale-like obj’s
data assoc. to D` -prime-strips.

330

Go Yamashita

We look more. The Θ-link only concerns the multiplicative structure (), hence it seems
diﬃcult to see the additive structure () on the left-hand side, from the right-hand side.
First, we try to overcome this diﬃculty by using a log-link (Note that Fo±l -symmetrisingisomorphisms are compatible with log-links, hence we can pull back Ψgau via log-link
to construct ΨLGP ):^ (eye)O Θ-link /•O•

log(O×µ )O

log-link



O×µ

•

However, the square^ (eye)O Θ-link /•O•O

log(O×µ )
O

log



O×µ

•

log

Θ

/•

is non-commutative (cf. log(aN ) 6= (log a)N ), hence we cannot describe the left verticalarrow in terms of the right vertical arrow. We overcome this diﬃculty by considering
the infinite chain of log-links:..
.O^ (eye)

log


•O

Θ

/•

log

•O
log

..
.
Then the infinite chain of log-links is invariant under the vertical shift, and we candescribe the infinite chain of log-links on the left-hand side, in terms of the infinite
chain of log-links on the right-hand side. This is a rough explanation of the idea.
§ 13.1.

Local and Global Packets.

Here, we introduce a notion of processions.

331

a proof of the abc conjecture after Mochizuki

Definition 13.1.
([IUTchI, Definition 4.10]) Let C be a category. A n-processionof C is a diagram of the form
P1

all capsule-full poly,→P2

all capsule-full poly

,→

···

all capsule-full poly

,→

Pn ,

where Pj is a j-capsule of Ob(C) for 1 ≤ j ≤ n, and each ,→ is the set of all capsule-fullpoly-morphisms. A morphism from an n-procession of C to an m-procession of C()()
all capsule-full poly
all capsule-full poly
all capsule-full poly
all capsule-full poly
P1
,→
···
,→
Pn → Q1
,→
···
,→
Qm
consists of an order-preserving injection ι : {1, . . . , n} ,→ {1, . . . , m} together with acapsule-full poly-morphism Pj

capsule-full poly,→Qι(j) for 1 ≤ j ≤ n.

Ultimately, l> -processions of D` -prime-strips corresponding to the subsets {1} ⊂{1, 2} ⊂ · · · ⊂ F>l will be important.
Remark 13.1.1. As already seen, the labels (LabCusp(−)) depend on the arithmetically holomorphic structures (cf. also Section 3.5), i.e., ∆(−) ’s or Π(−) ’s (Recall thatΠ(−) for hyperbolic curves of strictly Belyi type over an MLF has the information of the
field structure of the base field, and can be considered as arithmetically holomorphic,on the other hand, the Galois group of the base field (Π(−) )G(−) has no informationof the field structure of the base field, and can be considered as mono-analytic). In
inter-universal Teichmüller theory, we will reconstruct an alien ring structure on one
side of (the updated version of) Θ-link from the other side of (the updated version of)
Θ-link (cf. also the primitive form of Θ-link shares the mono-analytic structure † Dv` ,but not the arithmetically holomorphic structures † Dv , ‡ Dv (Remark 10.8.1)), and wecannot send arithmetically holomorphic structures from one side to the other side of(the updated version of) Θ-link. In particular, we cannot send the labels (LabCusp(−))
from one side to the other side of (the updated version of ) Θ-link, i.e., we cannot see
the labels on one side from the other side:
1, 2, . . . , l> 7−→?, ?, . . . , ?.

>Then we have (l> )l -indeterminacies in total. However, we can send processions:{1} ,→ {1, 2} ,→ {1, 2, 3} ,→ · · · ,→ {1, 2, . . . , l> } 7−→ {?} ,→ {?, ?} ,→ · · · ,→ {?, ?, . . . , ?}.
>In this case, we can reduce the indeterminacies from (l> )l to (l> )!. If we did not
use this reduction of indeterminacies, then the final inequality of height function would
be weaker (More precisely, it would be ht . (2 + )(log-diﬀ + log-cond), not ht .
(1 + )(log-diﬀ + log-cond)). More concretely, in the calculations of Lemma 1.10, if we∑>
did not use the processions, then the calculation l1> 1≤j≤l> (j + 1) = l 2+1 + 1 would∑
be changed into l1> 1≤j≤l> (l> + 1) = l> + 1, whose coeﬃcient of l would be twice.

332

Go Yamashita

For j = 1, . . . , l± (Recall that l± = l> + 1 = l+1
2 (cf. Section 0.2)), we writeS>j := {1, . . . , j},S±j := {0, . . . , j − 1}.

Note that we have>>>S>
1 ⊂ S2 ⊂ · · · ⊂ Sl> = Fl ,±±S±
1 ⊂ S2 ⊂ · · · ⊂ Sl± = |Fl |.

±We also consider S>
j as a subset of Sj+1 .

([IUTchI, Proposition 4.11, Proposition 6.9]) For a D-Θ-bridge

Definition 13.2.
†

†

φΘ
>

†

±

†

†

±

φΘ
±

DJ −→ D> (resp. D-Θ -bridge DT −→ † D ), we write
Proc(† DJ ) (resp. Proc(† DT ) )for the l> -processin (resp. l± -procession) of D-prime-strips determined by the sub>>>capsules of † DJ (resp. † DT ) corresponding to the subsets S>
1 ⊂ S2 ⊂ . . . ⊂ Sl> = Fl
∼
>
±
±
†
(resp. S±
1 ⊂ S2 ⊂ . . . ⊂ Sl± = |Fl |), with respect to the bijection χ : J → Fl∼of Proposition 10.19 (1) (resp. the bijection |T | → |Fl | determined by the F±
l -group† `† ``structure of T ). For the capsule DJ (resp. DT ) of D -prime-strips associated to † DJ
(resp. † DT ), we similarly define the l> -processin (resp. l± -procession)Proc(† D`J ) (resp. Proc(† D`T ) )±±† Θof D` -prime-strips. If the D-Θ-bridge † φΘ
> (resp. the D-Θ -bridge φ± ) arises froma capsule Θ-bridge (resp. Θ± -bridge), we similarly define the l> -processin (resp. l± procession)Proc(† FJ ) (resp. Proc(† FT ) )of F-prime-strips.
Proposition 13.3.
3.1]) Let

(Local Holomorphic Tensor Packets, [IUTchIII, Proposition{α F}α∈S± =j{ α}{ Fv }v∈V α∈S±jbe a j-capsule of F-prime-strips with index set S±
j . For V 3 v( | vQ ∈ VQ := V(Q)),α
we regard log( Fv ) as an inductive limit of finite-dimensional topological modules over
QvQ , by log(α Fv ) = limJ⊂α Π : open (log(α Fv ))J . We shall refer to the assignment−→v
VQ 3 vQ 7→ log(α FvQ ) :=⊕log(α Fv )V3v|vQas the 1-tensor packet associated to the F -prime-strip α F, and the assignment⊗±log(α FvQ )VQ 3 vQ 7→ log(Sj FvQ ) :=α∈S±ja proof of the abc conjecture after Mochizuki

333

the j-tensor packet associated to the collection {α F}α∈S± of F -prime-strips,jwhere the tensor product is taken as a tensor product of ind-topological modules.
(1) (Ring Structures) The ind-topological field structures on log(α Fv ) for α ∈ S±j±determine an ind-topological ring structure on log(Sj FvQ ) as an inductive limit of
direct sums of ind-topological fields. Such decompositions are compatible with thenatural action of the topological group α Πv on the direct summand with subscript vof the factor labelled α.
(2) (Integral Structures) Fix α ∈ S±
j+1 , v ∈ V, vQ ∈ VQ with v | vQ . Write






⊗
±
,α
S±
α
β
j+1
log(
Fv ) := log( Fv ) ⊗log( FvQ )⊂ log(Sj+1 FvQ ).
β∈S± \{α}
j+1±Then the ind-topological submodule log(Sj+1 ,α Fv ) forms a direct summand of the±±ind-topological ring log(Sj+1 FvQ ). Note that log(Sj+1 ,α Fv ) is also an inductive limitof direct sums of ind-topological fields. Moreover, by forming the tensor productwith 1’s in the factors labelled by β ∈ S±
j+1 \ {α}, we obtain a natural injective
homomorphism±
log(α Fv ) ,→ log(Sj+1 ,α Fv )of ind-topological rings, which, for suitable (cofinal) choices of objects in the inductive limit descriptions for the domain and codomain, induces an isomorphism ofsuch an object in the domain onto each of the direct summand ind-topological fieldsof the object in the codomain. In particular, the integral structureΨlog(α Fv ) := Ψlog(α Fv ) ∪ {0} ⊂ log(α Fv )determines integral structures on each of the direct summand ind-topological fields±±
appearing in the inductive limit descriptions of log(Sj+1 ,α Fv ), log(Sj+1 FvQ ).
×

Note that log(α Fv ) is an isomorph of log(Kv ) ∼= Kv , the integral structure Ψlog(α Fv )
⊗
⊕
∼
S±
,α
is an isomorph of OKv , and log( j+1 Fv ) is an isomorph of
Kv → lim Kv .
−→Proof. Proposition follows from the definitions.
Remark 13.3.1. ([IUTchIII, Remark 3.1.1 (ii)]) From the point of view of “an∼alytic section” Vmod → V(⊂ V(K)) of Spec K  Spec Fmod , we need to consider the
log-volumes on the portion of log(α Fv ) corresponding to Kv relative to the weight1,[Kv : (Fmod )v ]334

Go Yamashita∼

where we write v ∈ Vmod for the valuation corresponding to v via the bijection Vmod →⊕V (cf. also Definition 10.4). When we consider V3v|vQ as in case of log(α FvQ ), we use
the normalised weight(∑[Kv : (Fmod )v ] ·

1

)
[(F
)
:
Q
]
mod w
vQ
Vmod 3w|vQ

so that the multiplication by pvQ aﬀects log-volumes as + log(pvQ ) (resp. by − log(pvQ ))nonfor vQ ∈ Varc
Q (resp. vQ ∈ VQ ) (cf. also Section 1.2). Similarly, when we consider±

log-volumes on the portion of log(Sj+1 FvQ ) corresponding to the tensor product of Kvi
with V 3 v i | vQ for 0 ≤ i ≤ j, we have to consider these log-volumes relative to theweight1∏,0≤i≤j [Kv i : (Fmod )vi ]where vi ∈ Vmod corresponds to v i . Moreover, when we consider direct sums over all
possible choices for the data {v i }i∈S± , we use the normalised weight
j+1

(∏

1
)}
(∏
) {∑
0≤i≤j [(Fmod )wi : QvQ ]0≤i≤j [Kv i : (Fmod )vi ] ·{wi }0≤i≤j ∈((Vmod )v )j+1
Q

(cf. also Section 1.2) so that the multiplication by pvQ aﬀects log-volumes as + log(pvQ )non(resp. by − log(pvQ )) for vQ ∈ Varc
Q (resp. vQ ∈ VQ ) (cf. Section 0.2 for the notation(Vmod )vQ ).
Proposition 13.4.
tion 3.2]) Let(Local Mono-analytic Tensor Packets, [IUTchIII, Proposi{α D` }α∈S± =j{}{α Dv` }v∈Vα∈S±jbe a j-capsule of D` -prime-strips with index set S±
j . We shall refer to the assignment⊕log(α Dv` )VQ 3 vQ 7→ log(α Dv`Q ) :=V3v|vQas the 1-tensor packet associated to the D ` -prime-strip α D, and the assignment⊗±log(α Dv`Q )VQ 3 vQ 7→ log(Sj Dv`Q ) :=α∈S±j

the j-tensor packet associated to the collection {α D` }α∈S± of D ` -prime-strips,jwhere the tensor product is taken as a tensor product of ind-topological modules. For
α ∈ S±
j+1 , v ∈ V, vQ ∈ VQ with v | vQ , put






⊗
±
S±
,α
`
α
`
β
`
j+1
log(
Dv ) := log( Dv ) ⊗log( DvQ )⊂ log(Sj+1 Dv`Q ).
β∈S± \{α}

j+1

335

a proof of the abc conjecture after Mochizuki

If {α D` }α∈S± arises from a j-capsulej{α F`×µ }α∈S± =j{}{α Fv`×µ }v∈Vα∈S±jof F `×µ -prime-strips, then we put
±

±

±

±

log(α Fv`×µ) := log(α Dv`Q ), log(Sj Fv`×µ) := log(Sj Dv`Q ), log(Sj+1 ,α Fv`×µ ) := log(Sj+1 ,α Dv` ),Q
Qand we shall refer to the first two of them as the 1-tensor packetassociated to theF `×µ -prime-strip α F`×µ , and the j-tensor packet associated to the collection{α F`×µ }α∈S± of F `×µ -prime-strips, respectively.
j(1) (Mono-analytic/Holomorphic Compatibility) Assume that {α D` }α∈S± arisesjfrom a j-capsule}{{α F}α∈S± = {α Fv }v∈V α∈S±
j

j

α `×µ

of F-prime-strips. We write { F}α∈S± for the j-capsule of F `×µ -prime-strips
j

tauto
∼

associated to {α F}α∈S± . Then the (poly-)isomorphisms log(† Fv ) → log(† Fv`×µ )

“Kum”
poly
∼

→

j

log(† Dv` ) of Proposition 12.2 (4) induce natural poly-isomorphisms
tauto
∼

)
log(α FvQ ) → log(α Fv`×µ
Q

“Kum”
poly
∼

→

S±
j+1 ,α

log(

±

tauto
∼

±

log(α Dv`Q ), log(Sj FvQ ) → log(Sj Fv`×µ
)
Q
tauto
∼

S±
j+1 ,αFv ) → log(

Fv`×µ )

“Kum”
poly
∼

→

“Kum”
poly
∼

→

±

log(Sj Dv`Q ),±log(Sj+1 ,α Dv` )of ind-topological modules.
(2) (Integral Structures) For V 3 v | vQ ∈ Vnon
the étale-like mono-analytic logQ
shells “I† Dv` ” of Proposition 12.2 (4) determine topological submodules
±

±

±

±

I(α Dv`Q ) ⊂ log(α Dv`Q ), I(Sj Dv`Q ) ⊂ log(Sj Dv`Q ), I(Sj+1 ,α Dv` ) ⊂ log(Sj+1 ,α Dv` ),which can be regarded as integral structures on the Q-spans of these submodules.
For V 3 v | vQ ∈ Varc
Q by regarding the étale-like mono-analytic log-shells “I† Dv` ” ofProposition 12.2 (4) as the “closed unit ball” of a Hermitian metric on “log(† Dv` )”,and putting the induced direct sum Hermitian metric on log(α Dv`Q ), and the induced±

tensor product Hermitian metric on log(Sj Dv`Q ), we obtain Hemitian metrics on±±log(α Dv`Q ), log(Sj Dv`Q ), and log(Sj+1 ,α Dv`Q ), whose associated closed unit balls
±

±

±

±

I(α Dv`Q ) ⊂ log(α Dv`Q ), I(Sj Dv`Q ) ⊂ log(Sj Dv`Q ), I(Sj+1 ,α Dv` ) ⊂ log(Sj+1 ,α Dv` ),336

Go Yamashita±±

can be regarded as integral structures on log(α Dv`Q ), log(Sj Dv`Q ), and log(Sj+1 ,α Dv`Q ),respectively. For any V 3 v | vQ ∈ VQ , we put±±±

I Q (α Dv`Q ) := Q-span of I(α Dv`Q ) ⊂ log(α Dv`Q ), I Q (Sj Dv`Q ) := Q-span of I(Sj Dv`Q ) ⊂ log(Sj Dv`Q ),±±±I Q (Sj+1 ,α Dv` ) := Q-span of I(Sj+1 ,α Dv` ) ⊂ log(Sj+1 ,α Dv` ).
If {α D` }α∈S± arises from a j-capsule {α F}α∈S± of F-prime-strips then, the objects
j

I(

α

j

±
±
±
±
Dv`Q ), I Q (α Dv`Q ), I(Sj Dv`Q ), I Q (Sj Dv`Q ), I(Sj+1 ,α Dv` ), I Q (Sj+1 ,α Dv` ) determine
±

±

±

±

I(α FvQ ), I Q (α FvQ ), I(Sj FvQ ), I Q (Sj FvQ ), I(Sj+1 ,α Fv ), I Q (Sj+1 ,α Fv ),
and
±

±

±

±

I(α Fv`×µ), I Q (α Fv`×µ), I(Sj Fv`×µ), I Q (Sj Fv`×µ), I(Sj+1 ,α Fv`×µ ), I Q (Sj+1 ,α Fv`×µ )
Q
Q
Q
Q
tauto
∼

)
via the above natural poly-isomorphisms log(α FvQ ) → log(α Fv`×µ
Q
±

tauto
∼

±

log(Sj FvQ ) → log(Sj Fv`×µ)Q
S±
j+1 ,α

log(

“Kum”
poly
∼

→

±

±

tauto
∼

“Kum”
poly
∼

→

log(α Dv`Q ),±log(Sj Dv`Q ), log(Sj+1 ,α Fv ) → log(Sj+1 ,α Fv`×µ )Dv` ) of ind-topological modules.

Proof. Proposition follows from the definitions.
Proposition 13.5.

(Global Tensor Packets, [IUTchIII, Proposition 3.3]) Let
†

HT 

be a -Hodge theatre with associated - and -Hodge theatres † HT  , † HT  respectively. Let {α F}α∈S> be a j-capsule of F-prime-strips. We consider S>
j as a subset ofj∼the index set J appearing the -Hodge theatre † HT  via the isomorphism † χ : J → F>lof Proposition 10.19 (1). We assume that for each α ∈ S>,
alog-linkjαlog

F −→ † Fα

poly∼(i.e., a poly-morphism log(α F) → † Fα of F-prime-strips) is given. Recall that we have~~a labelled version († Mmod )j of the field † Mmod (cf. Corollary 11.23 (1), (2)). We call⊗~~(† Mmod )S> :=(† Mmod )α
j

α∈S>
j


†
the global j-tensor packet associated to S>.
j and the -Hodge theatre HT

“Kum”
poly
∼

→

337

a proof of the abc conjecture after Mochizuki~(1) (Ring Structures) The field structures on († Mmod )α for α ∈ S>
j determine a~ring structure on († Mmod )S> , which decomposes uniquely as a direct sum of numj
ber fields. Moreover, by composing with the given log-links, the various localisation~
functors “(† Fmod)j → † Fj ” of Corollary 11.23 (3) give us a natural injective localisation ring homomorphism~(† Mmod )S>jgl. to loc.

,→±

log(Sj+1 FVQ ) :=∏±log(Sj+1 FvQ )vQ ∈VQto the product of the local holomorphic tensor packets of Proposition 13.3, where we
S±
±
j+1 F
consider S>
vQ ) ofj as a subset of Sj+1 , and the component labelled by 0 in log(
the localisation homomorphism is defined to be 1.
(2) (Integral Structures) For α ∈ S>
j , by taking the tensor product with 1’s in the>factors labelled by β ∈ Sj \ {α}, we obtain a natural injective ring homomorphism~~(† Mmod )α ,→ († Mmod )S>jwhich induces an isomorphism of the domain onto a subfield of each of the direct summand number fields of the codomain. For each vQ ∈ VQ , this homomorphism is compatible, in the obvious sense, with the natural injective homomorphism±
log(α Fv ) ,→ log(Sj+1 ,α Fv ) of ind-topological rings of Proposition 13.3 (2), with respect to the localisation homomorphisms of (1). Moreover, for each vQ ∈ Vnon
(resp.
Qarc
vQ ∈ VQ ), the composite~~(† Mmod )α ,→ († Mmod )S>t

gl. to loc.

,→

±

±

log(Sj+1 FVQ )  log(Sj+1 FvQ )of the above displayed homomorphism with the vQ -component of the localisation homomorphism of (1) sends the ring of integers (resp. the set of elements of absolute~value ≤ 1 for all Archimedean primes) of the number field († Mmod )α into the submodule (resp. the direct product of subsets) constituted by the integral structures on±±log(Sj+1 FvQ ) (resp. on various direct summand ind-topological fields of log(Sj+1 FvQ ))of Proposition 13.3 (2).
Proof. Proposition follows from the definitions.
§ 13.2.

Log-Kummer Correspondences and Multiradial Representation
Algorithms.

Proposition 13.6.
tion 3.4])(Local Packet-Theoretic Frobenioids, [IUTchIII, Proposi-

338

Go Yamashita(1) (Single Packet Monoids) In the situation of Proposition 13.3, for α ∈ S±
j+1 , v ∈V, vQ ∈ VQ with v | vQ , the image of the monoid Ψlog(α Fv ) , its submonoid Ψ×log(α Fv )αof units, and realification ΨRlog(α Fv ) , via the natural homomorphism log( Fv ) ,→±log(Sj+1 ,α Fv ) of Proposition 13.3 (2), determines monoids±
R S± ,α,αS,αΨlog(S±
Ψ×
Ψlog(
j+1
j+1
log( j+1
Fv ),F v ),
Fv )which are equipped with Gv (α Πv )-actions when v ∈ Vnon , and for the first monoid,
with a pair of an Aut-holomorphic orbispace and a Kummer structure when v ∈ Varc .
±We regard these monoids as (possibly realified) subquotients of log(Sj+1 ,α Fv ) which±

act on appropriate (possibly realified) subquotients of log(Sj+1 ,α Fv ). (For the purpose±of equipping Ψlog(α Fv ) etc. with the action on subquotients of log(Sj+1 ,α Fv ), in thealgorithmical outputs, we define Ψ S± ,αetc. by using the image of the naturallog( j+1

homomorphism log( Fv ) ,→ log(αS±
j+1 ,αFv )Fv )).

(2) (Local Logarithmic Gaussian Procession Monoids) Let‡

HT  −→ † HT 
log

be a log-link of -Hodge theatres. Consider the F-prime-strip processions Proc(† FT ).
Recall that the Frobenius-like Gaussian monoid (∞) ΨFgau († HT Θ )v of Corollary 11.21∏(4) is defined by the submonoids in the product j∈F> (Ψ† F )j (cf. Corollary 11.17lv(2), Proposition 11.19 (4)). Consider the following diagram:
∏

∏

poly
∼ ∏

(Ψ† F )j ←−
j∈F>
l
v

j∈F>
l

log(j;‡ Fv )

⊂

∏

∪

Ψlog(j;‡ Fv )j∈F>lby (1)
∼ ∏

→

±

j∈F>
l

log(Sj+1 j;‡ Fv )∪±Ψlog( j+1
j∈F>
l
S

,j;‡

Fv )

,

∪
ΨFgau († F v )where, in the last line, we write ΨFgau († F v ), by abuse of notation, for ΨFξ († F v )for a value profie ξ in the case of v ∈ Vbad . We take the pull-backs of ΨFgau († F v )via the poly-isomorphism given by log-link ‡ HT  −→ † HT  , and send them
∏
∼ ∏
,j;‡
to the isomorphism j∈F> Ψlog(j;‡ Fv ) → j∈F> Ψlog(S±j+1
Fv ) constructed in (1).
ll
By this construction, we obtain a functorial algorithm, with respect to the log-linklog‡
HT  −→ † HT  of -Hodge theatres, to construct collections of monoids
log

log

V 3 v 7→ ΨFLGP ((‡−→)† HT  )v ,log(‡−→)†HT  )v ,∞ ΨFLGP (a proof of the abc conjecture after Mochizuki

339

equipped with splittings up to torsion when v ∈ Vbad (resp. splittings when v ∈Vgood ). We shall refer to them as Frobenius-like local LGP-monoids or Frobeniuslike local logarithmic Gaussian procession monoids. Note that we are able
to perform this construction, thanks to the compatibility of log-link with theFo±l -symmetrising isomorphisms.
Note that, for v ∈ Vbad , we have
(
)
‡
log
±
 Gv ( Πv )(‡−→)†
j-labelled component of ΨFLGP (⊂ I Q (Sj+1 ,j;‡ Fv )HT)v
fv ⊃) O× ·q j 2 ⊂ Q log(O× )”), where we write (−)Gv (‡ Πv ) for the invariant(i.e., “(KKv
Kvv

part, and the above j-labelled component of Galois invariant part acts multiplica±
tively on I Q (Sj+1 ,j;‡ Fv ). For any v ∈ V, we also have
(
)
log
±
 × Gv (‡ Πv )(‡−→)†⊂ I Q (Sj+1 ,j;‡ Fv )j-labelled component of (ΨFLGP (HT)v )fv ⊃) O× ⊂ Q log(O× )” for v ∈ Vgood ), where ‡ Πv = {1} for v ∈ Varc ,(i.e., “(KKv
Kv
and the above j-labelled component of Galois invariant part of the unit portion acts±
multiplicatively on I Q (Sj+1 ,j;‡ Fv ).
Proof. Proposition follows from the definitions.
Proposition 13.7.
(Kummer Theory and Upper Semi-Compatibility for Vertically Coric Local LGP-Monoids, [IUTchIII, Proposition 3.5]) Let {n,m HT p }n,m∈Z be
a collection of -Hodge theatres arising from a Gaussian log-theta-lattice. For each ninZ, we writen,◦HT D-
for the D-  -Hodge theatre determined, up to isomorphism, by n,m HT p for m ∈ Z,
via the vertical coricity of Theorem 12.5 (1).
(1) (Vertically Coric Local LGP-Monoids and Associated Kummer Theory)
We writeF(n,◦ D )t
for the F-prime-strip associated to the labelled collection of monoids “Ψcns (n,◦ D )t ”
of Corollary 11.20 (3). Then by applying the constructions of Proposition 13.6 (2)to the full log-links associated these (étale-like) F-prime-strips (cf. Proposition 12.2(5)), we obtain a functorial algorithm, with respect to the D-  -Hodge theatren,◦
HT D- , to construct collections of monoids
V ∈ v 7→ ΨLGP (n,◦ HT D- )v ,n,◦
HT D- )v∞ ΨLGP (340

Go Yamashita

equipped with splittings up to torsion when v ∈ Vbad (resp. splittings when v ∈Vgood ). We shall refer to them as vertically coric étale-like local LGP-monoids
or vertically coric étale-like local logarithmic Gaussian procession monoids.
Note again that we are able to perform this construction, thanks to the compatibility of log-link with the Fo±l -symmetrising isomorphisms. For each n, m ∈ Z,
this functorial algorithm is compatible, in the obvious sende, with the functorial
alogrithm of Proposition 13.6 (2) for † (−) = n,m (−), and ‡ (−) = n,m−1 (−), withrespect to the Kummer isomorphism
0Ψcns (n,m F )tKum∼→ Ψcns (n,◦ D )t0of labelled data of Corollary 11.21 (3) and the identification of n,m Ft with the F0
prime-strip associated to Ψcns (n,m F )t for m0 = m − 1, m. In particular, for eachn, m ∈ Z, we obtain Kummer isomorphismslog(∞) ΨFLGP ((n,m−1−→)n,m

HT



)v

Kum
∼

→

(∞) ΨLGP (n,◦

HT D- )v

for local LGP-monoids for v ∈ V.
(2) (Upper Semi-Compatibility) The Kummer isomorphisms of the above (1) are
upper semi-compatible with the log-links n,m−1 HT  −→ n,m HT  of Hodge theatres in the Gaussian log-theta-lattice in the following sense:log(a) (non-Archimedean Primes) For vQ ∈ Vnon
Q , (and n ∈ Z) by Proposition 13.6
(2), we obtain a vertically coric topological module±

I(Sj+1 F(n,◦ D )vQ ).
Then for any j = 0, . . . , l> , m ∈ Z, v | vQ , and m0 ≥ 0, we have(()n,m Πv )
⊗
±
m0
×
n,m
Kum ◦ log
Ψcns ( F )|t|⊂ I(Sj+1 F(n,◦ D )vQ ),|t|∈S±
j+1

0

where we write Kum for the Kummer isomorphism of (1), and logm for the m0 th iteration of pv -adic logarithm part of the log-link (Here we consider the m0 -th
iteration only for the elements whose (m0 − 1)-iteration lies in the unit group).
cf. also the inclusion (Upper Semi-Compat. (non-Arch)) in Section 5.1.
(b) (Archimedean Primes) For vQ ∈ Varc
Q , (and n ∈ Z) by Proposition 13.6 (2),
we obtain a vertically coric closed unit ball±I(Sj+1 F(n,◦ D )vQ ).

341

a proof of the abc conjecture after Mochizuki

Then for any j = 0, . . . , l> , m ∈ Z, v | vQ , we have()⊗
S±
j+1 F(n,◦ D )Kum Ψcns (n,m F )×
 vQ ),|t| ⊂ I(
|t|∈S±
j+1⊗(
Kum closed ball of radius π inside Ψcns (|t|∈S±
j+1

n,m

gp
F )|t|)±⊂ I(Sj+1 F(n,◦ D )vQ ),and, for m0 ≥ 1,
(

0
)
logm
0
gp
closed ball of radius π inside Ψcns (n,m F )|t| ⊃ (a subset)  Ψcns (n,m−m F )×|t| ,0

where we write Kum for the Kummer isomorphism of (1), and logm for them0 -th iteration of the Archimedean exponential part of the log-link (Here we
consider the m0 -th iteration only for the elements whose (m0 − 1)-iteration liesin the unit group). cf. also the inclusion (Upper Semi-Compat. (Arch)) in
Section 5.2.
log(c) (Bad Primes) Let v ∈ Vbad , and j 6= 0. Recall that the monoids (∞) ΨFLGP ((‡−→)† HT  )v ,and (∞) ΨLGP (n,◦ HT D- )v are equipped with natural splitting up to torsion
in the case of ∞ Ψ(−), and up to 2l-torsion in the case of Ψ(−). We write
log⊥(n,m−1−→)n,m
HT  )v(∞) ΨFLGP (⊥n,◦
HT D- )v(∞) ΨLGP (log⊂ (∞) ΨFLGP ((n,m−1−→)n,m HT  )v ,⊂ (∞) ΨLGP (n,◦ HT D- )v

for the submonoids defined by these splittings. Then the actions of the monoids
log(n,m−1−→)n,m
Ψ⊥HT  )v
FLGP ((m ∈ Z)on the ind-topological modules±±

I Q (Sj+1 ,j F(n,◦ D )v ) ⊂ log(Sj+1 ,j F(n,◦ D )v )(j = 1, . . . , l> ),via the Kummer isomorphisms of (1) is mutually compatible, with respect tothe log-links of the n-th column of the Gaussian log-theta-lattice, in the following sense: The only portions of these actions which are possibly related to eachother via these log-links are the indeterminacies with respect to multiplication
by roots of unity in the domains of the log-links (since Ψ⊥ (−) ∩ Ψ× (−) = µ2l ).
Then the pv -adic logarithm portion of the log-link sends the indeterminacies atm (i.e., multiplication by µ2l ) to addition by zero, i.e., no indeterminacy! atm + 1 (cf. also Remark 10.12.1, Definition 12.1 (2), (4), and Proposition 12.2
(2) for the discussion on quotients by Ψµ† FNv for v ∈ Varc ).

342

Go Yamashita

Now, we consider the groups
Gv (((Ψcns (n,m F )|t| )×
v)

n,m

Πv )

G (n,m−1 Πv )log

ΨFLGP ((n,m−1−→)n,m HT  )v v,

of units for v ∈ V, and the splitting monoids
log(n,m−1−→)n,m
Ψ⊥HT  )v
FLGP (

for v ∈ Vbad as acting on the modules±

I Q (Sj+1 F(n,◦ D )vQ )not via a single Kummer isomorphism of (1), which fails to be compatible with the loglinks, but rather via the totality of the pre-composites of Kummer isomorphisms with
iterates of the pv -adic logarithmic part/Archimedean exponential part of log-links as in
the above (2). In this way, we obtain a local log-Kummer correspondence between
the totality of the various groups of units and splitting monoids for m ∈ Z, and theiractions on the “I Q (−)” labelled by “n, ◦”0{ Kum ◦ logm (groups of units, splitting monoids at (n, m)) y I Q (n,◦ (−)) }m∈Z, m0 ≥0 ,which is invariant with respect to the translation symmetries m 7→ m + 1 of the n-th
column of the Gaussian log-theta-latice.
Proof. Proposition follows from the definitions.
Proposition 13.8.
tion 3.7])(Global Packet-Theoretic Frobenioids, [IUTchIII, Proposi-(1) (Single Packet Global non-Realified Frobenioid, -Line Bundle Version)
In the situation of Proposition 13.5, for each α ∈ S>
j , by the construction of Definition 9.7 (1), we have a functorial algorithm, from the image
(
)
±
~
~
~
(† MMOD )α := Im († Mmod )α ,→ († Mmod )S> ,→ log(Sj+1 FVQ )jof the number field, via the homomorphisms of Proposition 13.5 (1), (2) to constructa (pre-)Frobenioid~(† FMOD)αwith a natural isomorphism∼~~(† Fmod)α → († FMOD)α~of (pre-)Frobenioids (cf. Corollary 11.23 (2) for († Fmod)α ), which induces the tau∼ † ~† ~tological isomorphism ( Mmod )α → ( MMOD )α on the associated rational function~~
monoids. We ofthen identify († Fmod
)α with († FMOD
)α , via the above isomorphism.
† ~R† ~We write ( FMOD )α for the realification of ( FMOD )α .

343

a proof of the abc conjecture after Mochizuki(2) (Single Packet Global non-Realified Frobenioid, -Line Bundle Version)
For each α ∈ S>
j , by the construction of Definition 9.7 (2), we have a functorial~~

algorithm, from the number field († Mmod )α := († MMOD )α and the Galois invariant
local monoids
Gv (α Πv ),α(Ψlog(S±j+1
Fv ))of Proposition 13.6 (1) for v ∈ V, to construct a (pre-)Frobenioid~(† Fmod)α(Note that, for v ∈ Vnon (resp. v ∈ Varc ), the corresponding local fractional ideal Jv±of Definition 9.7 (2) is a submodule (resp. subset) of I Q (Sj+1 ,α Fv ) whose Q-span is±equal to I Q (Sj+1 ,α Fv )) with natural isomorphisms
∼

∼

~
~
~
~
(† Fmod)α → († Fmod
)α , († Fmod
)α → († FMOD
)α
~

~

∼

of (pre-)Frobenioids, which induces the tautological isomorphisms († Mmod )α → († Mmod )α ,~~∼(† Mmod )α → († MMOD )α on the associated rational function monoids, respectively.
~R~
We write († Fmod
)α for the realification of († Fmod)α .
(3) (Global Realified Logarithmic Gaussian Procession Frobenioids, -LineBundle Version) Let ‡ HT  −→ † HT  a log-link. In this case, in the construc±tion of the above (1), (2), the target log(Sj+1 FVQ ) of the injection is ‡-labbeled oblog

~

±

~

~
ject log(Sj+1 ,j;‡ FVQ ), thus, we write ((‡→)† MMOD )α , ((‡→)† Mmod )α , ((‡→)† FMOD
)α ,
~

~

~
~
~
((‡→)† Fmod
)α for († MMOD )α , († Mmod )α , († FMOD
)α , († Fmod
)α , respectively, in orderto specify the dependence. Consider the diagram

∏

j∈F>
l
†

†

∪
Cgau

Cj

gl. real’d to gl. non-real’d⊗R
∏
∼

−→

∼ ∏
~R
~R
(† Fmod)j → j∈F> ((‡→)† FMOD
)j ,
j∈F>
l
l

where the isomorphisms in the upper line are Corollary 11.23 (3) and the realification of the isomorphism in (1). Then by sending the global realified portion†
Cgau of the F -prime-strip † Fgau of Corolllary 11.24 (2) via the isomorphismsof the upper line, we obtain a functorial algorithm, with respect to the log-linklog‡
HT  −→ † HT  of Proposition 13.6 (2), to construct a (pre-)Frobenioid‡ log †

CLGP (( −→) HT  ).
‡ log †We shall refer to (‡→)† CLGP := CLGP (( −→) HT  ) as a Frobenius-like global
realified LGP-monoid or Frobenius-like global realified -logarithmic Gaus‡ log †sian procession monoids. The combination of it with the collection ΨFLGP (( −→) HT  )344

Go Yamashita

of data constructed by Proposition 13.6 (2) gives rise to an F -prime-strip
(‡→)†

∼

FLGP = ((‡→)† CLGP , Prime((‡→)† CLGP ) → V, (‡→)† F`LGP , {(‡→)† ρLGP,v }v∈V )with a natural isomorphism†∼

Fgau → (‡→)† FLGPof F -prime-strips.
(4) (Global Realified Logarithmic Gaussian Procession Frobenioids, -LineBundle Version) Write‡ log †‡ log †

ΨFlgp (( −→) HT  ) := ΨFLGP (( −→) HT  ),(‡→)† `
Flgp := (‡→)† F`LGP .
∼

∼

~R
~R
~R
In the construction of (3), by replacing († Fmod)j → († FMOD
)j by († Fmod)j →~R(† Fmod)j , we obtain a functorial algorithm, with respect to the log-link ‡ HT  −→†HT  of Proposition 13.6 (2), to construct a (pre-)Frobenioid
log

(‡→)†

‡ log †

Clgp := Clgp (( −→) HT  ).

and an F -prime-strip
(‡→)†

∼

Flgp = ((‡→)† Clgp , Prime((‡→)† Clgp ) → V, (‡→)† F`lgp , {(‡→)† ρlgp,v }v∈V )with tautological isomorphisms†∼∼

Fgau → (‡→)† FLGP → (‡→)† Flgp
‡ log †of F -prime-strips. We shall refer to (‡→)† Clgp := Clgp (( −→) HT  ) as a Frobeniuslike global realified lgp-monoid or Frobenius-like global realified -logarithmic
Gaussian procession monoids.
(5) (Global Realified to Global non-Realified⊗R) By the constructions of global‡ log †‡ log †

realified Frobenioids CLGP (( −→) HT  ) and Clgp (( −→) HT  ) of (3), (4), we
have a commutative diagram‡ log †

CLGP (( −→) HT  )


/

∏

~R
(† FMOD
)j
j∈F>
l

∼
=


log

(‡ −→)†

Clgp (HT∼=  

)

/∏



~R
j∈Fl ( Fmod )j .
†
>

345

a proof of the abc conjecture after Mochizuki~
In particular, by the definition of († Fmod)j in terms of local fractional ideals, and∏∏~R~)j , we ob)j → j∈F> († Fmod
the product of the realification functors j∈F> († Fmodlltain an algorithm, which is compatible, in the obvious sense, with the localisation
isomorphisms {† ρlgp,v }v∈V and {† ρLGP,v }v∈V , to construct objects of the (global)
log

log

categories Clgp ((‡−→)† HT  ), CLGP ((‡−→)† HT  ), from the local fractional ideals
log

generated by elements of the monoid ΨFlgp ((‡−→)† HT  )v for v ∈ Vbad .
Proof. Proposition follows from the definitions.
Definition 13.9.

([IUTchIII, Definition 3.8])

log

log

(‡−→)†(1) Write Ψ⊥
HT  )v := ΨFlgp ((‡−→)† HT  )v for v ∈ Vbad . When we reFlgp (gard the object of∏~(† Fmod)jj∈F>l

and its realification determined by any collection, indexed by v ∈ Vbad , of generatorslog(‡−→)†
up to µ2l of the monoids Ψ⊥
HT  )v , as an object of the global realified
Flgp (‡ log †‡ log †

Frobenioid (‡→)† CLGP = CLGP (( −→) HT  ) or (‡→)† Clgp = Clgp (( −→) HT  ),then we shall refer to it as a Θ-pilot object.
We shall refer to the object of the global realified Frobenioid † C∆ of Corollary 11.24(1) determined by any collection, indexed by v ∈ Vbad , of generators up to torsion`of the splitting monoid associated to the split Frobenioid † F∆,v
in the v-component`† `of the F -prime-strip F∆ of Corollary 11.24 (1), as a q-pilot object.
(2) Let ‡ HT  −→ † HT  be a log-link of -Hodge theatres, and
log

∗

HT 

a -Hodge theatre. Let
∗

I×µ
I×µ
, resp. (‡→)† Flgp)
F∆I×µ (resp. (‡→)† FLGP

be the F I×µ -prime-strip associated to the F -prime strip ∗ F∆ of Corollary 11.24(1) (resp. (‡→)† FLGP , resp. (‡→)† FLGP ). We shall refer to the full poly-isomorphism
(‡→)†

I×µ
FLGP

full poly
∼
∗

−→

F∆I×µ

(resp.

(‡→)†

I×µ
Flgp

full poly∼∗−→F∆I×µ )×µ†as the Θ×µ
to ∗ HT  , relative to the log-link
LGP -link (resp. Θlgp -link) from HT‡

HT  −→ † HT  , and we write it as
log

†

HT 

Θ×µ
LGP

−→

∗

HT  (resp. † HT 

Θ×µ
lgp

−→

∗

HT  ).

346

Go Yamashita(3) Let {n,m HT  }n,m∈Z be a collection of -Hodge theatres indexed by pairs ofintegers. We shall refer to the diagram
..
.O

..
.O
full log

···

Θ×µ
LGP

/ n,m+1 HT Ofull log
Θ×µLGP

full log

···

Θ×µ
LGP

/ n,m HT 
O

Θ×µ
LGP

/ n+1,m+1 HT 
O

/ ···

full log
Θ×µLGP/ n+1,m HT Ofull log

Θ×µLGP/ ··· ,

full log

..
.

..
.

(resp.
..
.O..
.O
full log

···

Θ×µ
lgp

/ n,m+1 HT Ofull log
Θ×µlgp

full log

···

Θ×µ
lgp

/ n,m HT 
O

Θ×µ
lgp

/ n+1,m+1 HT 
O

/ ···

full log
Θ×µlgp/ n+1,m HT 
O

Θ×µ
lgp

/ ···

full log

full log

..
.

..
.

) as the LGP-Gaussian log-theta-lattice (resp. lgp-Gaussian log-theta-lattice),×µn,mwhere the Θ×µ
HT  to n+1,m HT  is taken relLGP -link (resp. Θlgp -link) fromative to the full log-link n,m−1 HT  −→ n,m HT  . Note that both Θ×µ
LGP -link×µand Θlgp -link send Θ-pilot objects to q-pilot objects.
full log

Proposition 13.10.
sition 3.9])(Log-volume for Packets and Processions, [IUTchIII, Propo-(1) (Local Holomorphic Packets) In the situation of Proposition 13.4 (1), (2), for±V 3 v | vQ ∈ Vnon
(resp. V 3 v | vQ ∈ Varc
QQ ), α ∈ Sj+1 , the pvQ -adic log-volume(resp. the radial log-volume) on each of the direct summand pvQ -adic fields (resp.
±±

complex Archimedean fields) of I Q (α FvQ ), I Q (Sj+1 FvQ ), and I Q (Sj+1 ,j FvQ ) with thenormalised weights of Remark 13.3.1 determines log-volumes
Q αµlogα,vQ : M(I ( FvQ )) → R,±µlog
: M(I Q (Sj+1 FvQ )) → R,
S± ,v
j+1

Q

347

a proof of the abc conjecture after Mochizuki±µlog: M(I Q (Sj+1 ,α Fv )) → R,
S± ,α,v
j+1

where we write M(−) for the set of compact open non-empty subsets of (−) (resp.
the set of compact closures of open subsets of (−)), such that the log-volume of eachof the local holomorphic integral structures
Oα FvQ ⊂ I Q (α FvQ ), OS±

j+1 F

±

vQ

⊂ I Q (Sj+1 FvQ ), OS± ,α
j+1

±

Fv

⊂ I Q (Sj+1 ,α Fv ),given by the integral structures of Proposition 13.3 (2) on each of the direct summand, is equal to zero. Here, we assume that these log-volumes are normalised insuch a manner that multiplication by pv corresponds to − log(pv ) (resp. + log(pv ))on the log-volume (cf. Remark 13.3.1) (cf. Section 0.2 for pv with Archimedeanv). We shall refer to this normalisation as the packet-normalisation. Note that“µlog
” is invaariant by permutations of S±
j+1 . When we are working with colS± ,v
j+1

Qlections of capsules in a procession, we normalise log-volumes on the products of“M(−)” associated to the various capsules by taking the average over the various
capsules. We shall refer to this normalisation as the procession-normalisation.
(2) (Mono-analytic Compatibility) In the situation of Proposition 13.4 (1), (2),±(resp. V 3 v | vQ ∈ Varc
for V 3 v | vQ ∈ Vnon
QQ ), α ∈ Sj+1 , by applying the pvQ -adic
log-volume (resp. the radial log-volume) on the mono-analytic log-shells “I† Dv` ”
of Proposition 12.2 (4), and adjusting appropriately the discrepancy between thelocal holomorphic integral structures of Proposition 13.3 (2) and the mono-analytic
integral structures of Proposition 13.4 (2), we obtain log-volumes
Q α `
µlog
α,vQ : M(I ( DvQ )) → R,±µlog
: M(I Q (Sj+1 Dv`Q )) → R,
S± ,v
j+1

Q

±

µlog
: M(I Q (Sj+1 ,α Dv` )) → R,
S± ,α,v
j+1

where we write M(−) for the set of compact open non-empty subsets of (−) (resp.
the set of compact closures of open subsets of (−)), which are compatible with thelog-volumes of (1), with respect to the natural poly-isomorphisms of Proposition 13.4
(1). In particular, these log-volumes can be constructed via a functorial alogrithm
from the D` -prime-strips. If we consider the mono-analyticisation of an F-primestrip procession as in Proposition 13.6 (2), then taking the average of the packetnormalised log-volumes gives rise to procession-normalised log-volumes, which arecompatible with the procession-normalised log-volumes of (1), with respect to thenatural poly-isomorphisms of Proposition 13.4 (1). By replacing “D` ” by F `×µ ,348

Go Yamashita

we obtain a similar theory of log-volumes for the various objects associated to themono-analytic log-shells “I† Fv`×µ ”
Q α `×µµlog
α,vQ : M(I ( FvQ )) → R,±µlog
: M(I Q (Sj+1 Fv`×µ)) → R,
Q
S± ,v
j+1

Q

±

: M(I Q (Sj+1 ,α Fv`×µ )) → R,
µlog
S± ,α,v
j+1

which is compatible with the “D` ”-version, with respect to the natural poly-isomorphismsof Proposition 13.4 (1).
(3) (Global Compatibility) In the situation of Proposition 13.8 (1), (2), write
∏
∏
±
±
±
±
log(Sj+1 FvQ )I Q (Sj+1 FVQ ) :=I Q (Sj+1 FvQ ) ⊂ log(Sj+1 FVQ ) =vQ ∈VQvQ ∈VQand we write±

M(I Q (Sj+1 FVQ )) ⊂∏±M(I Q (Sj+1 FvQ ))vQ ∈VQfor the subset of elements whose components have zero log-volume for all but finitely
many vQ ∈ VQ . Then by adding the log-volumes of (1) for vQ ∈ VQ , we obtain a
global log-volume±: M(I Q (Sj+1 FVQ )) → R
µlog
S± ,V
j+1

Q

which is invariant by multiplication by elements of±† ~Q Sj+1(† M~
FVQ )mod )α = ( MMOD )α ⊂ I (log(product formula), and permutations of S±
j+1 . The global log-volume µS± ,V ({Jv }v∈V )j+1~( Fmod)α†Qof an object {Jv }v∈V of(cf. Definition 9.7 (2)) is equal to the degreeof the arithmetic line bundle determined by {Jv }v∈V (cf. the natural isomorphism∼~~(† Fmod)α → († Fmod
)α of Proposition 13.8 (2)), with respect to a suitable normalisation.
(4) (log-Link Compatibility) Let {n,m HT  }n,m∈Z be a collection of -Hodge theatres arising from an LGP-Gaussian log-theta-lattice.
(a) For n, m ∈ Z, the log-volumes of the above (1), (2), (3) determine log-volumes
on the various “I Q (−)” appearing in the construction of the local/global LGP/lgp-monoids/Frobenioids in the F -prime-strips n,m FLGP , n,m Flgp of Proposition 13.8 (3), (4), relative to the log-link n,m−1 HT  −→full log n,mHT  .

349

a proof of the abc conjecture after Mochizuki(b) At the level of the Q-spans of log-shells “I Q (−)” arising from the various Fprime-strips involved, the log-volumes of (a) indexed by (n, m) are compatible,
in the sense of Proposition 12.2 (2) (i.e., in the sense of the formula (5.1) of
Proposition 5.2 and the formula (5.2) of Proposition 5.4), with the log-volumes
indexed by (n, m − 1) with respect to the log-link n,m−1 HT  −→ n,m HT (This means that we do not need to be worried about how many times loglinks are applied in the log-Kummer correspondence, when we take valuesof the log-volumes).
full log

Proof. Proposition follows from the definitions.
Proposition 13.11.
(Global Kummer Theory and Non-Interference with Local
Integers, [IUTchIII, Proposition 3.10]) Let {n,m HT p }n,m∈Z be a collection of Hodge theatres arising from an LGP-Gaussian log-theta-lattice. For each ninZ, we writen,◦HT D-
for the D-  -Hodge theatre determined, up to isomorphism, by n,m HT p for m ∈ Z,
via the vertical coricity of Theorem 12.5 (1).
(1) (Vertically Coric Global LGP- lgp-Frobenioids and Assosiated Kummer
Theory) By applying the construcions of Proposition 13.8 to the (étale-like) Fprime-strips “F(n,◦ D )t ” and to the full log-links associated to these (étale-like) Fprime-strips (cf. Proposition 12.2 (5)), we obtain functorial algorithms, with respect
to the D-  -Hodge theatre n,◦ HT D- , to construct vertically coric étale-likenumber fields, monoids, and (pre-)Frobenioids equipped with natural
isomorphisms
~

~

n,◦
n,◦
Mmod (n,◦ HT D- )α = MMOD (n,◦ HT D- )α ⊃ M~
HT D- )α = M~
HT D- )α ,
mod (
MOD (

~

n,◦
Mmod (n,◦ HT D- )α ⊃ M~
HT D- )α ,
mod (
∼

∼

~
~
~
Fmod
(n,◦ HT D- )α → Fmod(n,◦ HT D- )α → FMOD(n,◦ HT D- )αvia † χ⊂ J, and vertically coric étale-like F -prime-strips equipped
for α ∈ S>
jwith natural isomorphisms∼∼

F (n,◦ HT D- )gau → F (n,◦ HT D- )LGP → F (n,◦ HT D- )lgp .
Note again that we are able to perform this construction, thanks to the compatibility of log-link with the Fo±l -symmetrising isomorphisms. For each

350

Go Yamashitan, m ∈ Z, these functorial algorithms are compatible, in the obvious sense, withthe (non-vertically coric Frobenius-like) functorial algorithms of Proposition 13.8
for † (−) = n,m (−), and ‡ (−) = n,m−1 (−), with respect to the Kummer isomorphisms
n,m0Ψcns (
n,m0(

M~mod )j

Kum
∼

→

F )t

Kum
∼

0

→ Ψcns (n,m D )t ,n,m0 }M~D )j ,
mod (

(

n,m0

~
Mmod )j

Kum
∼

~

0

→ Mmod (n,m D} )jof labelled data (cf. Corollary 11.21 (3), and Corollary 11.23 (2)), and the evident
00
identification of n,m Ft with the F-primes-strip associated to Ψcns (n,m F )t for m0 =m − 1, m. In particular, for each n, m ∈ Z, we obtain Kummer isomorphisms
Kum
∼

~

~

~

Kum
∼

~

(n,m Mmod )α → Mmod (n,◦ HT D- )α , ((n,m−1→)n,m MMOD/mod )α → MMOD/mod (n,◦ HT D- )α ,
Kum
∼

Kum
∼

~
~
n,◦
n,◦
(n,m M~HT D- )α , ((n,m−1→)n,m M~
HT D- )α ,mod )α → Mmod (MOD/mod )α → MMOD/mod (
Kum
∼

Kum
∼

~
~
~
~
(n,m Fmod)α → Fmod(n,◦ HT D- )α , ((n,m−1→)n,m FMOD/mod)α → FMOD/mod(n,◦ HT D- )α ,
n,m

Kum
∼

Kum
∼

Fgau → F (n,◦ HT D- )gau , ((n,m−1→)n,m FLGP/lgp ) → F (n,◦ HT D- )LGP/lgp ,(Here (−)MOD/mod is the shorthand for “(−)MOD (resp. (−)mod )”, and (−)LGP/lgp
is the shorthand for “(−)LGP (resp. (−)lgp )”) of fields, monoids, Frobenioids, andF -prime-strips, which are compatible with the above various equalities, natural
inclusions, and natural isomorphisms.
(2) (Non-Interference with Local Integers) In the notation of Proposition 13.4(2), Proposition 13.6 (1), Proposition 13.8 (1), (2), and Proposition 13.10 (3), we
have


∏
∏
∏
±
±
±
† ~
⊂
,α
I Q (Sj+1 FvQ ) = I Q (Sj+1 FVQ )
(† M~
Ψlog(S±I Q (Sj+1 ,α Fv ) =j+1
Fv ) = µ(( MMOD )α )MOD )α ∩
v∈V

v∈V

×
(i.e., “Fmod∩±∏vQ ∈VQ±∏×
Q Sj+1 ,αFv )v≤∞ O(Fmod )v = µ(Fmod )”) (Here, we identify
V3v|vQ I (with I Q (Sj+1 FvQ )). Now, we consider the multiplicative groups((n,m−1→)n,m M~
MOD )jof non-zero elements of number fields as acting on the modules±

I Q (Sj+1 F(n,◦ D )VQ )351

a proof of the abc conjecture after Mochizuki

not via a single Kummer isomorphism of (1), which fails to be compatible withthe log-links, but rather via the totality of the pre-composites of Kummer isomorphisms with iterates of the pv -adic logarithmic part/Archimedean exponential partof log-links, where we observe that these actions are mutually compatible, withrespect to the log-links of the n-th column of the LGP-Gaussian log-theta-lattice, inthe following sense: The only portions of these actions which are possibly related toeach other via these log-links are the indeterminacies with respect to multiplication
by roots of unity in the domains of the log-links (by the above displayed equality). Then the pv -adic logarithm portion of the log-link sends the indeterminacies
at m (i.e., multiplication by µ((n,m−1→)n,m M~
MOD )j ) to addition by zero, i.e., noindeterminacy! at m + 1 (cf. also Remark 10.12.1, Definition 12.1 (2), (4), andProposition 12.2 (2) for the discussion on quotients by Ψµ† FNv for v ∈ Varc ). In thisway, we obtain a global log-Kummer correspondence between the totality ofthe various multiplicative groups of non-zero elements of number fields for m ∈ Z,
and their actions on the “I Q (−)” labelled by “n, ◦”0

Q n,◦{ Kum ◦ logm (((n,m−1→)n,m M~(−)) }m∈Z, m0 ≥0 ,MOD )j ) y I (which is invariant with respect to the translation symmetries m 7→ m + 1 of the n-th
column of the LGP-Gaussian log-theta-latice.
(3) (Frobenioid-theoretic log-Kummer Correspondences) The Kummer isomorphisms of (1) induce, via the log-Kummer correspondence of (2), isomorphisms of(pre-)Frobenioids((n,m−1→)n,m

~
FMOD
)α

Kum
∼

→

~
FMOD
(n,◦ HT D- )α ,((n,m−1→)n,m

~R
FMOD
)α

Kum
∼

~R
→ FMOD(n,◦ HT D- )αwhich are mutually compatible with the log-links of the LGP-Gaussian log-thetalattice, as m rus over the elements of Z. These compatible isomorphisms of (pre)Frobenioids with the Kummer isomorphisms of (1) induce, via the global logKummer correspondence of (2) and the splitting monoid portion of the the local
log-Kummer correspondence of Proposition 13.7 (2), a Kummer isomorphism(n,m−1→)n,m

⊥
FLGP

Kum
∼

→ F ⊥ (n,◦ HT D- )LGP

of associated F ⊥ -prime-strips, which are mutually compatible with the log-linksof the LGP-Gaussian log-theta-lattice, as m rus over the elements of Z.
Note that we use only MOD-/LGP-labelled objects in (2) and (3), since these are defined
only in terms of multiplicative operations (), and that the compatibility of Kummer
isomorphisms with log-links does not hold for mod-/lgp-labelled objects, since these are352

Go Yamashita

defined in terms of both multiplicative and additive operaions ( and ), where we only
expect only a upper semi-compatibility (cf. Definition 9.7, and Proposition 13.7 (2)).
Proof. Proposition follows from the definitions.
The following the Main Theorem of inter-universal Teichmüller theory:
Theorem 13.12.
(Multiradial Algorithms via LGP-Monoids/Frobenioids, [IUTchIII,
Theorem 3.11]) Fix an initial Θ-data(F /F, XF , l, C K , V, Vbad
mod , ).
Let{n,m HT  }n,m∈Z
be a collection of -Hodge theatres, with respect to the fixed initial Θ-date, arisingfrom an LGP-Gaussian log-theta-lattice. For each n ∈ Z, we writen,◦HT D-

for the D-  -Hodge theatre determined, up to isomorphism, by n,m HT  for m ∈ Z,
via the vertical coricity of Theorem 12.5 (1).
(1) (Multiradial Representation) Consider the procession of D` -prime-strips Proc(n,◦ D`T ){n,◦ D`0 } ,→ {n,◦ D`0 , n,◦ D`1 } ,→ . . . ,→ {n,◦ D`0 , n,◦ D`1 , . . . , n,◦ D`l> }.
Consider also the following data:(Shells) (Unit portion — Mono-anaytic Containers) For V 3 v | vQ , j ∈ |Fl |, thetopological modules and mono-analytic integral structures
±

±

±

±

I(Sj+1 ;n,◦ Dv`Q ) ⊂ I Q (Sj+1 ;n,◦ Dv`Q ), I(Sj+1 ,j;n,◦ Dv` ) ⊂ I Q (Sj+1 ,j;n,◦ Dv` ),which we regard as equipped with the procession-normalised mono-analytic logvolumes of Proposition 13.10 (2),
(ThVals) (Value Group Portion — Theta Values) For v ∈ Vbad , the splitting monoidn,◦
HT D- )v
Ψ⊥
LGP (

of Proposition 13.7 (2c), which we regard as a subset of∏±
I Q (Sj+1 ,j;n,◦ Dv` ),
j∈F>
l

353

a proof of the abc conjecture after Mochizuki

equipped with a multiplicative action on
poly-isomorphisms

Q S±
j+1 ,j;n,◦

I (

Dv` )

“Kum”−1
poly
∼

→

Q S±
j+1 ,j;n,◦

I (

F

∏

Q Sj+1 ,j;n,◦`×µtauto−1
∼

I
j∈F>
l

(

(D )v )±→Dv` ), via the natural±

I Q (Sj+1 ,j;n,◦ F(D` )v )of Proposition 13.4 (2), and(NFs) (Global Portion — Number Fields) For j ∈ F>
l , the number field
∏
±
±
~
~
MMOD (n,◦ HT D- )j = Mmod (n,◦ HT D- )j ⊂ I Q (Sj+1 ;n,◦ DV`Q ) :=I Q (Sj+1 ;n,◦ Dv`Q )vQ ∈VQ

with natural isomorphisms
∼

∼

~
~
~R
~R n,◦
FMOD(n,◦ HT D- )j → Fmod(n,◦ HT D- )j , FMOD(n,◦ HT D- )j → Fmod
( HT D- )j(cf. Proposition 13.11 (1)) between the associated global non-realified/realified
Frobenioids, whose associated global degrees can be computed by means of thelog-volumes of (a).
We writen,◦

RLGP

for the collection of data (Shells), (ThVals), (NFs) regarded up to indeterminaciesof the following two types:(Indet xy) the indeterminacies induced by the automorphisms of the procession of D` prime-strip Proc(n,◦ D`T ), andarc(Indet →) for each vQ ∈ Vnon
Q (resp. vQ ∈ VQ ), the indeterminacies induced by the actionof independent copies of Isomet (resp. copies of {±1}×{±1}-orbit arising fromthe independent {±1}-actions on each of the direct factors “k ∼ (G) = C ∼ ×C ∼ ”
of Proposition 12.2 (4)) on each of the direct summands of the j + 1 factors±
appearing in the tensor product used to define I Q (Sj+1 ;n,◦ Dv`Q )Then we have a functorial algorithm, with respect to Proc(n,◦ D`T ), to constructn,◦ LGP
R(from the given initial Θ-data). For n, n0 ∈ Z, the permutation symmetriesof the étale picture of Corollary 12.8 (2) induce compatible poly-isomorphisms
poly
∼

0

Proc(n,◦ D`T ) → Proc(n ,◦ D`T ),

n,◦

poly
∼
n0 ,◦

RLGP →RLGP

354

Go Yamashita

poly
∼

D`0

poly
∼ n0 ,◦

which are, moreover, compatible with the poly-isomorphisms→
D`0 induced by the bi-coricity of the poly-isomorphisms of Theorem 12.5 (3). We shall ren,◦0

fer to the switching poly-isomorphism n,◦ RLGP → n ,◦ RLGP as an étale-transportpoly-isomorphism (cf. also Remark 11.1.1), and we also shall refer to (Indet xy)as the étale-transport indeterminacies.
(2) (log-Kummer Correspondence) For n, m ∈ Z, the Kummer isomorphisms
Ψcns (

n,m

F )t

Kum
∼

→ Ψcns (n,◦D )t , ({π1rat (n,m D~ ) y n,m M~
}
∞κ j

n,m

~
Mmod )jKum∼~→ Mmod (n,◦ D} )j ,Kum∼

→ {π1rat (n,◦ D~ ) y M~(n,◦ D} )}j∞κ(where t ∈ LabCusp± (n,◦ D )) of labelled data of Corollary 11.21 (3), Corollary 11.23(1), (2) (cf. Proposition 13.7 (1), Proposition 13.11 (1)) induce isomorphisms between the vertically coric étale-like data (Shells), (ThVals), and (NFs) of (1), andthe corresponding Frobenius-like data arising from each -Hodge theatre n,m HT  :(a) for V 3 v | vQ , j ∈ |Fl |, isomorphisms
I (Q) (

S±
j+1 ;n,m

FvQ )S±
j+1 ,j;n,m

I (Q) (

tauto
∼

Fv )

→

tauto
∼

→

S±
j+1 ;n,m

I (Q) (

)
Fv`×µ
Q

S±
j+1 ,j;n,m

I (Q) (

Fv`×µ )

“Kum”
poly
∼

→

“Kum”
poly
∼

→

±

I (Q) (Sj+1 ;n,◦ DvQ ),±I (Q) (Sj+1 ,j;n,◦ Dv )of local mono-analytic tensor packets and their Q-spans (cf. Proposition 13.4(2)), all of which are compatible with the respective log-volumes by
Proposition 13.10 (2) (Here, I (Q) (−) is a shorthand for “I(−) (resp. I Q (−))”),(b) for Vbad 3 v, isomorphisms(n,m−1→)n,m
Ψ⊥HT  )v
FLGP (Kum∼n,◦
HT D- )v
→ Ψ⊥
LGP (

of splitting monoids (cf. Proposition 13.7 (1)),(c) for j ∈ F>
l , isomorphisms(n,m−1→)n,m(~
MMOD/mod )j(n,m−1→)n,m(~
FMOD/mod)j~R((n,m−1→)n,m FMOD/mod
)j

Kum
∼

~

→ MMOD/mod (n,◦ HT D- )j ,

Kum
∼

~
→ FMOD/mod(n,◦ HT D- )j ,

Kum
∼

~R
→ FMOD/mod(n,◦ HT D- )j ,

355

a proof of the abc conjecture after Mochizuki

of number fields and global non-realified/realified Frobenioids (cf. Proposition 13.11(1)), which are compatible with the respective natural isomorphisms between“(−)MOD ” and “(−)mod ” (Here, (−)MOD/mod is a shorthand for “(−)MOD(resp. (−)mod )”), here, the last isomorphisms induce isomorphisms(n,m−1→)n,m

CLGP/lgp

Kum∼→ CLGP/lgp (n,◦ HT D- )(Here, (−)LGP/lgp is a shorthand for “(−)LGP (resp. (−)lgp )”) of the global realified Frobenioid portions of the F -prime-strips (n,m−1→)n,m FLGP , F (n,◦ HT D- )LGP ,(n,m−1→)n,m
Flgp , and F (n,◦ HT D- )lgp (cf. Proposition 13.11 (1)).
Kum∼(n,m−1→)n,mn,◦
HT  )v → Ψ⊥HT D- )v ’s,
Moreover, the various isomorphisms Ψ⊥
FLGP (LGP (~MMOD/mod )j

Kum
∼

~

and (
→ MMOD/mod (n,◦ HT D- )j ’s in (b), (c) are mutually compatible with each other, as m runs over Z, with respect to the log-linksof the n-th column of the LGP-Gaussian log-theta-lattice, in the sense that the
only portions of the domains of these isomorphisms which are possibly related toeach other via the log-links consist of µ in the domains of the log-links at (n, m),and these indeterminacies at (n, m) (i.e., multiplication by µ) are sent to addition by zero, i.e., no indeterminacy! at (n, m + 1) (cf. Proposition 13.7 (2c),(n,m−1→)n,m

~

Kum
∼

Proposition 13.11 (2)). This mutual compatibility of ((n,m−1→)n,m MMOD/mod )j →

Kum
~
∼
D-
n,◦
(n,m−1→)n,m ~MMOD/mod ( HT)j ’s implies mutual compatibilities of (FMOD )j →
Kum
∼
~
~
~
FMOD
(n,◦ HT D- )j ’s, and ((n,m−1→)n,m FMOD
)j → FMOD(n,◦ HT D- )j ’s (Notethat the mutual compatibility does not hold for (−)mod -labelled objects, since theseare defined in terms of both multiplicative and additive operaions ( and ), where
we only expect only a upper semi-compatibility (cf. Definition 9.7, Proposition 13.7(2), and Proposition 13.11 (3)). On the other hand, the isomorphisms of (a) are
subject to the following indeterminacy:(Indet ↑) the isomorphisms of (a) are upper semi-compatible, with respect to the loglinks of the n-th column of the LGP-Gaussian log-theta-lattice, as m runs overZ, in a sense of Proposition 13.7 (2a), (2b).
(We shall refer to (Indet →) and (Indet ↑) as the Kummer detachment indeterminacies.) Finally, the isomorphisms of (a) are compatible with therespective log-volumes, with respect to the log-links of the n-th column of the
LGP-Gaussian log-theta-lattice, as m runs over Z (This means that we do not need356

Go Yamashita

to be worried about how many times log-links are applied in the log-Kummer
correspondence, when we take values of the log-volumes).
(3) (Θ×µ
LGP -Link Compatibility) The various Kummer isomorphisms of (2) are compatible with the Θ×µ
LGP -links in the following sense:(a) (Kummer on ∆) By applying the Fo±l -symmetry of the -Hodge theatre
n,m

Kum
∼

HT  , the Kummer isomorphism Ψcns (n,m F )t → Ψcns (n,◦ D )t induces

a Kummer isomorphism n,m F`×µ∆

induced by Kum
∼

n,◦ `
→
F`×µ
D∆ ) (cf. Theo∆ (rem 12.5 (3)). Then we have a commutative diagramn,m `×µF∆
induced by Kum ∼=full poly∼/ n+1,m F`×µ∆∼
= induced by Kum



n,◦ `
F`×µ
D∆ )
∆ (


/ F`×µ (n+1,◦ D` ),full poly∼∆∆

where the upper horizontal arrow is induced (cf. Theorem 12.5 (2)) by theΘ×µ
LGP -link between (n, m) and (n + 1, m) by Theorem 12.5 (3).
(b) (∆ → env) The F -prime-strips n,m Fenv , Fenv (n,◦ D> ) appearing implicitly inthe construction of the F -prime-strips (n,m−1→)n,m FLGP , F (n,◦ HT D- )LGP ,∼(n,m−1→)n,m
Flgp , F (n,◦ HT D- )lgp , admit natural isomorphisms n,m F`×µ→∆∼n,m `×µn,◦ `n,◦ `Fenv , F`×µD∆ ) → F`×µD> ) of associated F `×µ -prime-strips (cf.
env (
∆ (
Proposition 12.6 (3)). Then we have a commutative diagramn,m `×µF∆
induced by Kum & “∆7→env” ∼=full poly∼/ n+1,m F`×µ∆∼
= induced by Kum & “∆7→env”


full poly∼ / `×µ n+1,◦ `
n,◦ `
F`×µ
(
D
)
F
(
D> ),
env
>
env

where the upper horizontal arrow is induced (cf. Theorem 12.5 (2)) by theΘ×µ
LGP -link between (n, m) and (n + 1, m) by Corollary 12.8 (3).
(c) (env → gau) Recall that the (vertically coric étale-like) data “n,◦ R” i.e.,
(
[
n,◦
n,◦
n,◦
n,◦
HT D- , Fenv (n,◦ D> ), ∞ Ψ⊥D> )v ⊃ ∞ Ψenv (n,◦ D> )µv , µZb (MΘD>,v )) ⊗ Q/Z, MΘ
D>,v ),
env (
∗(
∗(
⊥ n,◦
D> )v  ∞ Ψenv (n,◦ D> )µv∞ Ψenv (]v∈Vbadn,◦ `n,◦, F`×µD∆ ), F`×µ
D> )
env (
∆ (

full poly
∼

→

)
n,◦ `
F`×µ
D∆ )
∆ (

of Corollary 12.8 (2) implicitly appears in the construction of the F -primestrips (n,m−1→)n,m FLGP , F (n,◦ HT D- )LGP , (n,m−1→)n,m Flgp , F (n,◦ HT D- )lgp .

357

a proof of the abc conjecture after Mochizuki

This (vertically coric étale-like) data arising from n,◦ HT D- is related tocorresponding (Frobenius-like) data arising from the projective system of themono-theta environments associated to the tempered Frobenioids of the Hodge theatre n,m HT  at v ∈ Vbad via the Kummer isomorphisms andpoly-isomorphisms of projective systems of mono-theta environments of Proposition 12.6 (2), (3) and Theorem 12.5 (3). With respect to these Kummer
isomorphisms and poly-isomorphisms of projective systems of mono-theta environments, the poly-isomorphism
n,◦

poly
∼
n+1,◦

R →

R
n,◦

induced by the permutation symmetry of the étale picturen+1,◦
HT D- is compatible with the full poly-isomorphismn,m `×µ
F∆

HT

D-

full poly∼→full poly∼n+1,m `×µ→F∆

of F `×µ -prime-strips induced by Θ×µ
LGP -link between (n, m) and (n + 1, m) andso on. Finally, the above two displayed poly-isomorphisms and the various
related Kummer isomorphisms are compatible with the various evaluation
map implicit in the portion of the log-Kummer correspondence of (2b), up toindeterminacies (Indet xy), (Indet →), (Indet ↑) of (1), (2).
(d) (κ-coric → NF) With respect to the Kummer isomorphisms of (2) and thegluing of Corollary 11.21, the poly-isomorphism[{]} gl. to loc.
n,◦ }
n,◦
n,◦
(
D
)
→
M
(
D
π1rat (n,◦ D~ ) y M~
)
⊂
M
(
D
)
vj
vj
∞ κv
∞ κ×v
∞κ
j
v∈V

poly []{ rat n+1,◦ ~} gl. to loc.
∼n+1,◦ }
n+1,◦
n+1,◦
→
π1 (
D ) y M~
(
D
)
→
M
(
D
)
⊂
M
(
D
)
vj
vj
∞ κv
∞ κ×v
∞κ
j

(cf. Corollary 11.22 (3)) induced by the permutation symmetry of the étale
picture n,◦ HT D-
isomorphism

full poly
∼
n+1,◦

→

HT D- is compatible with the full poly-n,m `×µF∆full poly∼n+1,m `×µ→F∆

of F `×µ -prime-strips induced by Θ×µ
LGP -link between (n, m) and (n + 1, m). Finally, the above two displayed poly-isomorphisms and the various related Kummer isomorphisms are compatible with the various evaluation map implicit in
the portion of the log-Kummer correspondence of (2b), up to indeterminacies(Indet xy), (Indet →), (Indet ↑) of (1), (2).

v∈V

358

Go Yamashita

Proof. Theorem follows from the definitions.

A rough picture of the final multiradial representation is as follows:

×
(Fmod
)1
y

{I0Q } ⊂ {I0Q ,

···

×
(Fmod
)l>
y

I1Q }
O

Q
⊂B · · ·⊂ {I0Q , . . . ,k5 Il> }
k
k

kkk

kkk

k
k
2
kk
q2

kkk

k
12
k

k
q
kk
 acts
kkk q(l> )2
k

k
k
k

kkk
 ?  1 (  kkkkk
Ψ⊥
LGP ,

×
where the multiplicative group (Fmod)j of non-zero elements of a j-labelled number field
2
Q
Q⊥
acts on Ij , and ΨLGP acts on Ij in the (j + 1)-capsule by multiplication by q j . Note
Q
Q
that Ψ⊥LGP does not act on other components I0 , . . . , Ij−1 of the (j + 1)-capsule. Notealso that the 0-labelled objects (together with the diagonal labelled objects) are used to×⊥form horizontally coric objects (Recall that “∆ = {0, hF>l i}”), and (Fmod )j ’s or ΨLGP
do not act on 0-labelled (Q-span of) log-shell I0Q .
The following table is a summary of Theorem 13.12 and related topics:

359

a proof of the abc conjecture after Mochizuki

×µ(temp. conj. vs. prof. conj. → Fo±l -conj. synchro.→ diag.→hor. core→ ΘLGP -link↓)Fo±l -sym.

(1) (Objects)(2) (log-Kummer)(3) (Comp’ty with Θ×µ
LGP -link)I (inv. after admitting

inv. after admitting

units)

(Indet ↑)(Indet →) (b × -indet.)ZFo±l -sym.

Ψ⊥
LGP val. gp.

no interf. by const. mult. rig.

b × -indet.
protected from Z(←compat. of log-link(ell. cusp’n←pro-p anab.

by mono-theta cycl. rig.

w/ Fo±
l -sym.)+hidden. endom.)(←quad. str. of Heis. gp.)

F>l -sym.

Mmod NF

no interf.

b × -indet.
protected from ZBelyi cusp’n(←pro-p anab.

×by Fmod∩∏v≤∞ Ov = µb × = {1}by Q>0 ∩ Z+hidden endom.)others: (compat. of log.-vol. w/ log-links), (Arch. theory:Aut-hol. space (ell. cusp’n is used))(disc. rig. of mono-theta), (étale pic.: permutable after admitting (Indet xy) (autom. of proc. incl.))Corollary 13.13.
lary 3.12]) We write(Log-volume Estimates for Θ-Pilot Objects, [IUTchIII, Corol−| log(Θ)| ∈ R ∪ {+∞}

for the procession-normalised mono-analytic log-volume (where the average is taken overj ∈ F>l ) of the holomorphic hull (cf. the definiton after Lemma 1.6) of the unionof the possible image of a Θ-pilog object, with respect to the relevant Kummer
isomorphisms in the multiradial representation of Theorem 13.13 (1), which we regard
as subject to the indeterminacies (Indet ↑), (Indet →), and (Indet xy) of Theorem 13.13(1), (2). We write−| log(q)| ∈ R
for the procession-normalised mono-analytic log-volume of the image of a q-pilot object,with respect to the relevant Kummer isomorphisms in the multiradial representation of360

Go Yamashita

Theorem 13.13 (1), which we do not regard as subject to the indeterminacies (Indet ↑),(Indet →), and (Indet xy) of Theorem 13.13 (1), (2) (Note that we have | log(q)| > 0).
Then we obtain−| log(q)| ≤ −| log(Θ)|(i.e., “0 . −(large number) + (mild indeterminacies)”. cf. also § A.4). Note also thatthe explicit computations of the indeterminacies in Proposition 1.12, in fact, shows that−| log(Θ)| < ∞.
Θ×µ

LGP 1,00,0Proof. The Θ×µ
HT  −→HT  induces the full poly-isomorphism
LGP -link

I×µ
FLGP

full poly∼
1,0→F∆I×µ of F I×µ -prime-strips, which sends Θ-pilot objects to aq-pilot objects. By the Kummer isomorphisms, the (0, 0)-labelled Frobenius-like objects corresponding to the objects in the multiradial representaion of Theorem 13.12(1) are isomorphically related to the (0, ◦)-labelled vertically coric étale-like objects(i.e., mono-analytic containers with actions by theta values, and number fields) in themultiradial representaion of Theorem 13.12 (1). After admitting the indeterminacies(Indet xy), (Indet →), and (Indet ↑), these (0, ◦)-labelled vertically coric étale-like objects are isomorphic (cf. Remark 11.1.1) to the (1, ◦)-labelled vertically coric étale-likeobjects. Then Corollary follows by comparing the log-volumes (Note that log-volumes
are invariant under (Indet xy), (Indet →), and also compatible with log-Kummer correspondence of Theorem 13.12 (2)) of (1, 0)-labelled q-pilot objects (by the compatibility
with Θ×µ
LGP -link of Theorem 13.12 (3)) and (1, ◦)-labelled Θ-pilot objects, since, in the
mono-analytic containers (i.e., Q-spans of log-shells), the holomorphic hull of the unionof possible images of Θ-pilot objects subject to indeterminacies (Indet xy), (Indet →),(Indet ↑) contains a region which is isomorphic (not equal) to the region determined by
the q-pilot objects (This means that “very small region with indeterminacies” contains“almost unit region”).
0,0

Then Theorem 0.1 (hence Corollary 0.2 as well) is proved, by combining Proposition 1.2, Proposition 1.15, and Corollary 13.13.
Remark 13.13.1. By admitting (Indet xy), (Indet →), and (Indet ↑), we obtain×µ
objects which are invariant under the ΘLGP -link. On the other hand, the Θ×µ
LGP -link can
be considered as “absolute Frobenius” over Z, since it relates (non-ring-theoretically)2
q to {q j }1≤j≤l> . Therefore, we can considervv(Indet xy) the permutation indeterminacy in the étale transport:

•



/◦o



•

“† Gv ∼
= ‡ Gv ” (and autom’s of processions)361

a proof of the abc conjecture after Mochizuki(Indet →) the horizontal indeterminacy in the Kummer detachment:†Θ• −→ •O×µ ∼= ‡ O×µ with integral structures,

and(Indet ↑) the vertical indeterminacy in the Kummer detachment:log(O× ) 
O

•O

log

log

O×

•



/ 1 log(O× )
2p
q8
q
q
q
qqq
+ qqq

as “descent data from Z to F1 ”.
Remark 13.13.2. The following diagram (cf. [IUTchIII, Fig. 3.8]) expresses thetautological two ways of computations of log-volumes of q-pilot objects in the
proof of Corollary 13.13:))((
étale transport
-line bdls.1≤j≤l> assoc. to{0,◦ q j 2 }v∈Vv

up to Indet.’s

O

Kummer detach.
via log-Kummer corr.

(_

-line bdls.1≤j≤l> assoc. tosuited to Fmod{1,◦ q j 2 }v∈V up to Indet.’s

hRRR
RRR
RRR
RRR
RRR
RRR
RRR
compatibility with Θ×µ
LGP -link
RRR
RRR
R
) Θ×µ
(
-link
LGP

-line bdl. assoc. to{0,0 q j 2 }v∈V
v

∼
=

∼
=

suited to FMODv

O

compare log-vol.’s)  ()
-line bdl. assoc. to-linebdl.
assoc.
to∼.
={1,0 q }v∈V
{1,0 q }v∈Vv

v

These tautological two ways of computations of log-volumes of q-pilot objects can beconsidered as computations of self-intersection numbers “∆.∆” of the diagonal “∆ ⊂Z ⊗F1 Z” from point view of Remark 13.13.1. This observation is compatible withthe analogy with p-adic Teichmüller theory (cf. last table in Section 3.5), where thecomputation of the global degree of line bundles arising from the derivative of thecanonical Frobenius lifting (↔ Θ-link) gives us an inequality (1 − p)(2g − 2) ≤ 0 (Recall
that self-intersection numbers give us Euler numbers). This inequality (1−p)(2g−2) ≤ 0
essentially means the hyperbolicity of hyperbolic curves. Analogously, the inequality| log(Θ)| ≤ | log(q)| ; 0means the hyperbolicity of number fields.
cf. also the following table (cf. [IUTchIII, Fig. 3.2]):

362

Go Yamashita

-line bundles, MOD/LGP-labelled objects

-line bundles, mod/lgp-labelled objects

defined only in terms of 

defined in terms of both  and 

value group/non-coric portion

unit group/coric portion“(−) I ” of Θ×µ
LGP -link×µ“(−)`×µ ” of Θ×µ
LGP /Θlgp -link

precise log-Kummer corr.

only upper semi-compatible log-Kummer corr.

ill-suited to log-vol. computation

suited to log-vol. computation
subject to mild indeterminacies

Remark 13.13.3. In this remark, we consider the following natural questions:
How about the following variants of Θ-links?
(1)
2{q j }1≤j≤l> 7→ q λvv(λ ∈ R>0 ),(2)
2{(q j )N }1≤j≤l> 7→ qvv

(N > 1), and

(3)
q

v

7→ q λv(λ ∈ R>0 ).

From conclusions, (1) works, and either of (2) or (3) does not work.
(1) ([IUTchIII, Remark 3.12.1 (ii)]) We explain the variant (1). Recall that we havel ≈ ht >> |deg(q )| ; 0. Then the resulting inequalty from “the generalised Θ×µ
LGP vlink” isλ · 0 . −(ht) + (indet.)for λ << l, which gives us the almost same inequality of Corollary 13.13, and weakerinequality for λ > l than the inequality of Corollary 13.13 (since deg(q ) < 0).
v(2) ([EtTh, Introduction, Remark 2.19.2, Remark 5.12.5], [IUTchII, Remark 1.12.4,
Remark 3.6.4], [IUTchIII, Remark 2.1.1]) We explain the variant (2). There areseveral reasons that the variant (2) does not work (cf. also the principle of Galois
evaluation of Remark 11.10.1):

a proof of the abc conjecture after Mochizuki

363(a) If we replace Θ by ΘN (N > 1), then the crucial cyclotomic rigidity of monotheta environments (Theorem 7.23 (1)) does not hold, since the constructionof the cyclotomic rigidity of mono-theta environments uses the quadraticity of
the commutator [ , ] structure of the theta group (i.e., Heisenberg group) (cf.
also Remark 7.23.2). If we do not have the cyclotomic rigidity of mono-theta
environments, then we have no Kummer compatibility of theta monoids (cf.
Theorem 12.7).
(b) If we replace Θ by ΘN (N > 1), then the crucial constant multiple rigidityof mono-theta environments (Theorem 7.23 (3)) does not hold either, since, if
we consider N -th power version of mono-theta environments by relating the
1-st power version of mono-theta environments (for the purpose of maintaining
the cyclotomic rigidity of mono-theta environments) via N -th power map, thensuch N -th power map gives rise to mutually non-isomorphic line bundles, hence
a constant multiple indeterminacy under inner automorphisms arising fromautomorphisms of corresponding tempered Frobenioid (cf. [IUTchIII, Remark
2.1.1 (ii)], [EtTh, Corollary 5.12 (iii)]).
(c) If we replace Θ by ΘN (N > 1), then, the order of zero of ΘN at cusps isequal to N > 1, hence in the log-Kummer correspondence, one loop among thevarious Kummer isomorphisms between Frobenius-like cyclotomes in a columnof log-theta-lattice and the vertically coric étale-like cyclotome gives us the N power map before the loop, therefore, the log-Kummer correspondence totallycollapses. cf. also Remark 12.8.1 (“vicious circles”).
If it worked, then we would have
0 . −N (ht) + (indet.),which gives us an inequalityht .

1(1 + )(log-diﬀ + log-cond)N

for N > 1. This contradicts Masser’s lower bound in analytic number theory([Mass2]).
(3) ([IUTchIII, Remark 2.2.2]) We explain the variant (3). In the theta function case,
we have Kummer compatible splittings arisen from zero-labelled evaluation points(cf. Theorem 12.7):() −→.
0-labelled
ev.
pt.
id y O× · ∞ θ x Π←Π/∆ .. Aut(G), Isomet y (G y O×µ )
−→

∞θ

7→

1 ∈ O×µ .

364

Go Yamashita

Here, the crucial Kummer compatibility comes from the fact that the evaluation
map relates the Kummer theory of O× -portion of O× ·∞ θ on the left to the coric O×µon the right, via the evaluation ∞ θ 7→ 1 ∈ O×µ . On the other hand, in the case of
the variants (3) under consideration, the corresponding arrow maps q λ 7→ 1 ∈ O×µ ,hence this is incompatible with passage to Kummer classes, since the Kummer classof q λ in a suitable cohomology group of Π/∆ is never sent to the trivial element of
the relavant cohomology group of G, via the full poly-isomorphism Π/∆

Appendix A.

full poly∼→G.

Motivation of the Definition of the Θ-Link.

In this section, we explain a motivation of Θ-link from a historical point of view,i.e., in the order of classical de Rham’s comparison theorem, p-adic Hodge comparison
theorem, Hodge-Arakelov comparison theorem, and a motivation of Θ-link. This section
is an explanatory section, and we do not give proofs, or sometimes rigorous statements.
cf. also [Pano, §1].
§ A.1.

The Classical de Rham Comparison Theorem.

The classical de Rham’s comparison theorem in the special case for Gm (C) = C×says that the pairing
1
H1 (Gm (C), Z) ⊗Z HdR(Gm (C)/C) → C,∫∼1
which sends [γ] ⊗ [ω] to γ ω, induces a comparison isomorphism HdR(Gm (C)/C) →C⊗Z (H1 (Gm (C), Z))∗ (Here, we write (−)∗ for the Z-dual). Note that H1 (Gm (C), Z) =
[ ]
∫ dT
1
Z [γ0 ], HdR(Gm (C)/C) = C dT
T , and γ0 T = 2πi, where we write γ0 for a counterclockwise loop around the origin, and T for a standard coordinate of Gm .
§ A.2.

p-adic Hodge-theoretic Comparison Theorem.

A p-adic analogue of the above comparison paring (in the special case for Gm overQp ) in the p-adic Hodge theory is the pairing
1
Tp Gm ⊗Zp HdR(Gm /Qp ) → Bcrys ,

[ ]
∫ dT
which sends  ⊗ dTto(“= ”) log [] = t(= t ), where we write Tp for the p-adic
T T
Tate module,  = (n )n is a system of p-power roots of unity (i.e., 0 = 1, 1 6= 1, andpn+1 = n ), Bcrys is Fontaine’s p-adic period ring (cf. also [Fo3]), and t = log [] is anelement in Bcrys defined by  (cf. also [Fo3]). The above pairing induces a comparison

365

a proof of the abc conjecture after Mochizuki∼

1isomorphism Bcrys ⊗Qp HdR(Gm /Qp ) → Bcrys ⊗Zp (Tp Gm )∗ (Here, we write (−)∗ for theZp -dual). Note that  = (n )n is consdered as a kind of analytic path around the origin.
We consider the pairing in the special case for an elliptic curve E over Zp . We
have the universal extension 0 → (LieEQ∨p )∗ → EQ† p → EQp → 0 (cf. [Mess] for theuniversal extension) of EQp := E ⊗Zp Qp (Here, we write (−)∗ for the Qp -dual, andEQ∨p (∼= EQp ) is the dual abelian variety of EQp ). By taking the tangent space atthe origin, we obtain an extension 0 → (LieEQ∨p )∗ → LieEQ† p → LieEQp → 0 whose
Qp -dual is canonically identified with the Hodge filtration of the de Rham cohomol1
ogy 0 → (LieEQp )∗ → HdR(EQp /Qp ) → LieEQ∨p → 0 under a canonical isomorphism†
H 1 (EQ /Qp ) ∼= (LieE )∗ (cf. also [MM] for the relation between the universal exdRQpptension and the first crystalline cohomology; [BO1] and [BO2] for the isomorphism
between the crystalline cohomology and the de Rham cohomology). For an elementdb a/Q such thatωE † of (LieE † )∗ , we have a natural homomorphism log: E† → G
Qp

ωE †

Qp

p

d
the pull-back (logω † ) dT is equal to ωE † , where EQ† p is the formal completion of EQ† p
E
b a/Q is the formal additive group over Qp .
at the origin, and G∗

pNow, the pairing in the p-adic Hodge theory is
Tp E ⊗ (LieEQ† p )∗ → Bcrys ,∫which sends P ⊗ ωE † to (“ P ωE † = ”) logω † [P ], where P = (Pn )n satisfies thatE

Pn ∈ E(Qp ), P0 = 0, and pPn+1 = Pn . The above pairing induces a comparison∼
1
isomorphism Bcrys ⊗Qp HdR(Gm /Qp ) → Bcrys ⊗Zp (Tp Gm )∗ (Here, we write (−)∗ for theZp -dual). Note again that P = (Pn )n is consdered as a kind of analytic path in E. cf.
also [BO1] and [BO2] for the isomorphism between the de Rham cohomology and the
crystalline cohomology; [MM] for the relation between the first crystalline cohomology
and the universal extension; [Mess] for the relation between the universal extension
and the Dieudonné module; [Fo2, Proposition 6.4] and [Fo1, Chapitre V, Proposition
1.5] for the relation between the Dieudonné module and the Tate module (the aboveisomorphism is a combination of these relations).
§ A.3.

Hodge-Arakelov-theoretic Comparison Theorem.

Mochizuki studied a global and “discretised” analogue of the above p-adic Hodge
comparison map (cf. [HASurI], [HASurII]). Let E be an elliptic curve over a number
field F , l > 2 a prime number. Assume that we have a nontrivial 2-torsion pointP ∈ E(F )[2] (we can treat the case where P ∈ E(F ) is order d > 0 and d is prime to l;
however, we treat the case where d = 2 for the simplicity). Write L = O(l [P ]). Thenroughly speaking, the main theorem of Hodge-Arakelov theory says that the evaluation
map on E † [l](= E[l])∼Γ(E † , L|E † )deg<l −→ L|E † [l] (= L|E[l] = ⊕E[l] F )366

Go Yamashita

is an isomorphism of F -vector spaces, and preserves specified integral structures (weomit the details) at non-Archimedean and Archimedean places. Here, we write Γ(E † , L|E † )deg<lfor the part of Γ(E † , L|E † ) whose relative degree is less than l (Note that Zariski locallyE † is isomorphic to E × A1 = Spec OE [T ]). Note that dimF Γ(E † , L|E † )deg<l = l2 ,since dimF Γ(E, L) = l, and that dimF L|E[l] = l2 since #E[l] = l2 . The left-handside is the de Rham side, and the right-hand side is the étale side. The discretasationmeans that we consider l-torsion points E[l], not the Tate module, and in philosophy,
we consider E[l] as a kind of approximation of “underling analytic manifold” of E (like = (n )n and P = (Pn )n were considered as a kind of analytic paths in Gm and E
respectively). We also note that in the étale side we consider the space of functions
on E[l], not E[l] itself, which is a common method of quantisations (like considering
universal enveloping algebra of Lie algebra, not Lie algebra itself, or like consideringgroup algebra, not group itself).
(For the purpose of the reader’s easy getting the feeling of the above map, we also
note that the Gm -case (i.e., degenerated case) of the above map is the evaluation map∼

F [T ]deg<l −→ ⊕ζ∈µl F
sending f (T ) to (f (ζ))ζ∈µl , which is an isomorphism since the Vandermonde determinant is non-vanishing.)
For j ≥ 0, the graded quotient Fil−j /Fil−j+1 (in which the derivations of theta
function live) with respect to the Hodge filtration given by the relative degree on the⊗(−j)
de Rham side (=theta function side) is isomorphic to ωE, where ωE is the pull-backof the cotangent bundle of E to the origin of E. On the other hand, in the étale side2(=theta value side), we have a Gaussian pole q j /8l OF in the specified integral structurenear the infinity (i.e., q = 0) of Mell . This Gaussian pole comes from the values of theta
functions at torsion points. We consider the degrees of the corresponding vector bundles
on the moduli of elliptic curves to the both sides of the Hodge-Arakelov comparison map.
The left-hand side is
−

l−1
∑

j[ωE ] ≈ −

j=0

l2
l2
[ωE ] = − [log q],2
24⊗2
since [ωE
] = [ΩMell ] = 16 [log q], where ΩMell is the cotangent bundle of Mell and 6 isthe degree of the λ-line over the j-line. The right-hand side is

1 ∑ 2
l2
−
j [log q] ≈ − [log q].
8l j=024l−1Note that these can be considered as a discrete analogue of the calculation of Gaussian
integral∫ ∞√2
e−x dx = π−∞

a proof of the abc conjecture after Mochizuki
1from the point of view that − 8l

367

∑l−1

2
2
j=0 j [log q] is a Gaussian distribution (i.e., j 7→ j )∑l−1
2
in the cartesian coordinate, and − j=0 j[ωE ] ≈ − l2 [ωE ] is a calculation in the polar√⊗2 ∼coordinate and [ωE ] is an analogue of π, since we have ωE= ΩMell and the integrationof ΩMell around the infinity (i.e., q = 0) is 2πi. cf. also Remark 1.15.1§ A.4.

Motivation of the Definition of the Θ-Link.

In the situation as in the Hodge-Arakelov setting, we assume that E has everywherestable reduction. In general, E[l] does not have a global multiplicative subspace, i.e., a
submodule M ⊂ E[l] of rank 1 such that it coincides with the multiplicative subspace µl
for each non-Archimedean bad places. However, let us assume such a global multiplicative subspace M ⊂ E[l] exists in suﬃciently general E in the moduli of elliptic curves.
Let M × N ∼= E[l] be an isomorphism as finite flat group schemes over F (not as Galois
modules). Then by applying the Hodge-Arakelov comparison theorem to E 0 := E/Nover K := F (E[l]), we obtain an isomorphism∼

Γ((E 0 )† , L|(E 0 )† )deg<l −→⊕j2(q 2l OK ) ⊗OK K,

l−1>>(− l−12 =)−l ≤j≤l (= 2 )where q = (qv )v:bad is the q-parameters of the non-Archimedean bad places. Then by
the incompatibility of the Hodge filtration on the left-hand side with the direct sum
decomposition in the right-hand side, the projection to the j-th factor is nontrivial for
most j:2
Fil0 = qOK ,→ q j OK ,1

where we write q := q 2l . This morphism of arithmetic line bundles is considered as
an arithmetic analogue of Kodaira-Spencer morphism. In the context of (Diophantine
applications of) inter-universal Teichmüller theory, we take l to be a prime number inthe order of the height of the elliptic curve, thus, l is very large (cf. Section 10). Hencethe degree of the right-hand side in the above inclusion of the arithmetic line bundles
is negative number of a very large absolute value, and the degree of the left-hand side
is almost zero comparatively to the order of l. Therefore, the above inclusion implies
0 . −(large number) (≈ −ht),
which gives us a upper bound of the height ht . 0 in suﬃciently general E in the moduli
of elliptic curves.
However, there never exists such a global multiplicative in suﬃciently general E in
the moduli of elliptic curves (If it existed, then the above argument showed that theheight is bounded from the above, which implies the number of isomorphism class of
E is finite (cf. also Proposition C.1)). If we respect the scheme theory, then we cannot

368

Go Yamashita2

obtain the inclusion qOK ,→ q j OK . Mochizuki’s ingenious idea is: Instead, we respect2

the inclusion qOK ,→ q j OK , and we say a good-bye to the scheme theory. The Θ-linkin inter-universal Teichmüller theory is a kind of identification(Θ-link) :2{q j }1≤j≤l> (= l−1 )
2

7→

q

in the outside of the scheme theory (In inter-universal Teichmüller theory, we also
construct a kind of “global multiplicative subspace” in the outside of the scheme theory).
So, it identifies an arithmetic line bundle of negative degree of a very large absolutevalue with an arithmetic line bundle of almost degree zero (in the outside of the scheme
theory). This does not mean a contradiction, because both sides of the arithmetic line
bundles belong to the diﬀerent scheme theories, and we cannot compare their degrees.
The main theorem of the multiradial algorithm in inter-universal Teichmüller theory
implies that we can compare their degrees after admitting mild indeterminacies by using
mono-anabelian reconstruction algorithms (and other techniques). We can calculatethat the indeterminacies are (roughly) log-diﬀ+log-cond by concrete calculations. Hence
we obtain
0 . −ht + log-diﬀ + log-cond,i.e., ht . log-diﬀ + log-cond. We have the following remark: We need not only to reconstruct (up to some indeterminacies) mathematical objects in the scheme theory of one
side of a Θ-link from the ones in the scheme theory of the other side, but also to reducethe indeterminacies to mild ones. In order to do so, we need to control them, to reducethem by some rigidities, to kill them by some operations like taking p-adic logarithms
for the roots of unity (cf. Proposition 13.7 (2c), Proposition 13.11 (2)), to estimate
them by considering that some images are contained in some containers even though
they are not precisely determinable (cf. Proposition 13.7 (2), Corollary 13.13), and tosynchronise some indeterminacies to others (cf. Lemma 11.9, and Corollary 11.16 (1))
and so on. This is a new kind of geometry – a geometry of controlling indeterminacies
which arise from changing scheme theories i.e., changing unverses. This is Mochizuki’s
inter-universal geometry.
Finally, we give some explanations on “multiradial algorithm” a little bit. In
the classical terminology, we can consider diﬀerent holomorphic structures on R2 , i.e.,C ∼= C, where one C is an analytic (not holomorphic) dilation of another C,= R2 ∼and the underlying analytic structure R2 is shared. We can calculate the amount of
the non-holomorphic dilation C ∼= R2 ∼= C based on the shared underlying analytic
2
structure R (If we consider only holomorphic structres and we do not consider theunderlying analytic structure R2 , then we cannot compare the holomorphic structures
nor calculate the non-holomorphic dilation). This is a prototype of the multiradialalgorithm. In philosophy, scheme theories are “arithmetically holomorphic structures”

369

a proof of the abc conjecture after Mochizukiof a number field, and by going out the scheme theory, we can consider “underlying
analytic structure” of the number field. The Θ-link is a kind of Teichmüller dilationof “arithmetically holomorphic structures” of the number field sharing the “underlying
analytic structure”. The shared “underlying analytic structure” is called core, and each“arithmetically holomorphic structure” is called radial data. The multiradial algorithm
means that we can compare “arithmetically holomorphic structures” (of the both sidesof Θ-link) based on the shared “underlying analytic structure” of the number field after
admitting mild indeterminacies (In some sense, this is a partial (meaningful) realisationof the philosophy of “the field of one element” F1 ). Mochizuki’s ideas of “underlying
analytic structure” and the multiradial algorithm are really amazing discoveries.
Appendix B.

Anabelian Geometry.

For a (pro-)veriety X over a field K, let ΠX (resp. ∆X ) be the arithmetic fundamental group of X (resp. the geometric fundamental group of X) for some basepoint.
(p)
(p)
(p)
Let ∆X be the maximal pro-p quotient of ∆X , and write ΠX := ΠX /ker(∆X → ∆X ).
For (pro-)varieties X, Y over a field K, we write Homdom
K (X, Y ) (resp. IsomK (X, Y ))for the set of dominant K-morphisms (resp. K-isomorphisms) from X to Y . For an
algebraic closure K over K, write GK := Gal(K/K). We write Homopen
GK (ΠX , ΠY ) (resp.
(p)

(p)

(p)

(p)

Out
Out
Homopen
GK (ΠX , ΠY ), resp. IsomGK (∆X , ∆Y ), resp. IsomGK (∆X , ∆Y )) for the set of
(p)

(p)

open continuous homomorphisms from ΠX to ΠY over GK (resp. from ΠX to ΠYover GK , resp. from ∆X to ∆Y compatible with the outer GK -actions up to composi(p)(p)
tion with an inner automorphism arising from ∆Y , resp. from ∆X to ∆Y compatible
with the outer GK -actions up to composition with an inner automorphism arising from
(p)
∆Y ).
Theorem B.1.
(Relative Version of the Grothendieck Conjecture over Sub-padic Fields [pGC, Theorem A]) Let K be a sub-p-adic field (Definition 3.1 (1)). Let
X be a smooth pro-variety over K. Let Y be a hyperbolic pro-curve over K. Then thenatural maps
(p)

(p)

(p)

(p)

(p)

open
open
HomdomK (X, Y ) → HomGK (ΠX , ΠY )/Inn(∆Y ) → HomGK (ΠX , ΠY )/Inn(∆Y )are bijective. In particular, the natural maps
Out
IsomK (X, Y ) → IsomOut
GK (∆X , ∆Y ) → IsomGK (∆X , ∆Y )are also bijective.
Remark B.1.1.
The Isom-part of Theorem B.1 holds for a larger class of field
which is called generalised sub-p-adic field ([TopAnb, Theorem 4.12]). Here, a field K370

Go Yamashita

is called generalised sub-p-adic if there is a finitely generated extension L of thefractional field of W (Fp ) such that we have an injective homomorphism K ,→ L offields. ([TopAnb, Definition 4.11]), where we write W (Fp ) for the ring of Witt vectors
with coeﬃcients in Fp .
Appendix C.
§ C.1.

Miscellany.

On the Height Function.

Proposition C.1.
([GenEll, Proposition 1.4 (iv)]) Let L = (L, || − ||L ) be anarithmetic line bundle such that LQ is ample. Then we have #{x ∈ X(Q)≤d | htL (x) ≤C} < ∞ for any d ∈ Z≥1 and C ∈ R.
N
Proof. By using L⊗n
Q for n >> 0, we have an embedding XQ ,→ PQ for some N .
e → X, this embedding extends to g : X
e ,→ PN
By taking a suitable blowing-up f : X
Z∼
e is normal, Z-proper, Z-flat, and fQ : X
eQ → XQ . Then the propoover Spec Z, where X
e f ∗ L). As is shown in Section 1.1, thesition for (X, L) is reduced to the one for (X,bounded discrepancy class of htf ∗ L depends only on (f ∗ L)Q . Thus, the propositione f ∗ L) is equivalent to the one for (X,e g ∗ OPN (1)), where OPN (1) is the line bunfor (X,ZZ

dle OPN(1) equipped with the standard Fubini-Study metric || − ||FS . Then it suﬃcesZ(1)). For 1 ≤ e ≤ d, we put Q := (PNto show the proposition for (PN
Z ×Spec Z
Z , OPN
Z
N· · · (e-times) · · · ×Spec Z PZ )/(e-th symmetric group), which is normal Z-proper, Z-flat.
The arithmetic line bundle ⊗1≤i≤e pr∗i OPN(1) on PN ×Spec Z · · · (e-times) · · · ×Spec Z PN
Z
descends to LQ = (LQ , ||−||LQ ) on Q with (LQ )Q ample, where pri is the i-th projection.
For any x ∈ PN (F ) where [F : Q] = e, the conjugates of x over Q determine a pointxQ ∈ Q(Q), and, in turn, a point y ∈ Q(Q) determines a point x ∈ PN (F ) up to a finite
number of possibilities. Hence it suﬃces to show that #{y ∈ Q(Q) | htLQ (y) ≤ C} < ∞⊗mfor any C ∈ R. We embed Q ,→ PM
for m >> 0. Then by the
Z for some M by (LQ )Q
M
same argument as above, it suﬃces to show that #{x ∈ P (Q) | htO M (1) (x) ≤ C} < ∞P

for any C ∈ R. For x ∈ PM (Z)(= PM (Q)), we have htO M (1) (x) = degQ x∗ OPM (1) byP∼

definition. We have degQ : APic(Spec Z) → R since any projective Z-module is free (Qhas class number 1), where an arithmetic line bundle LZ,C on Spec Z in the isomorphismclass corresponding to C ∈ R via this isomorphism is (OSpec Z , e−C | − |) (Here | − | is theusual absolute value). The set of global sections Γ(LZ,C ) is {a ∈ Z | |a| ≤ eC } which isa finite set (cf. Section 1.1 for the definition of Γ(L)). We also have LZ,C1 ,→ LZ,C2 forC1 ≤ C2 . Let x0 , . . . , xM ∈ Γ(PM(1)) be the standard generating sections (“theZ , OPMZ
M
coordinate (x0 : . . . : xM ) ∈ PZ ”) with ||xi ||FS ≤ 1 for 0 ≤ i ≤ M i.e., x0 , . . . , xM ∈(1)). Then for x ∈ PM (Z)(= PM (Q)) with htO M (1) (x) ≤ C, we have a map
Γ(OPM
Z
P

x∗ OPM (1) ,→ LZ,C , which sends x0 , . . . , xM ∈ Γ(OPM(1)) to x∗ (x0 ), . . . , x∗ (xM ) ∈Z

a proof of the abc conjecture after Mochizuki

371

Γ(LZ,C ). This map {x ∈ PM (Z) | htO M (1) (x) ≤ C} → Γ(LZ,C )⊕(M +1) , which sends x
Pto (x∗ (x0 ), . . . , x∗ (xM )), is injective since x0 , . . . , xM ∈ Γ(PM(1)) are generating
Z , OPMZ
M
sections. In short, we have {x ∈ P (Q) | htO M (1) (x) ≤ C} ⊂ {(x0 : . . . : xM ) ∈P

PM (Q) | xi ∈ Z, |xi | ≤ eC (0 ≤ i ≤ M )}. Now, the proposition follows from thefiniteness of Γ(LZ,C )⊕(M +1) .
§ C.2.

Non-critical Belyi Maps.

The following theorem, which is a refinement of a classical theorem of Belyi, is usedin Proposition 1.2.
Theorem C.2.
([Belyi, Theorem 2.5], non-critical Belyi map) Let X be a propersmooth connected curve over Q, and S, T ⊂ X(Q) finite sets such that S ∩ T = ∅. Then1
there exists a morphism φ : X → PQ
such that (a) φ is unramified over P1Q \ {0, 1, ∞},(b) φ(S) ⊂ {0, 1, ∞}, and (c) φ(T ) ⊂ P1 (Q) \ {0, 1, ∞}.

Proof. (Step 1): By adjoining points of X(Q) to T , we may assume that #T ≥2gX + 1, where gX is the genus of X. We consider T as a reduced eﬀective divisor on
X by abuse of notation. Let s0 ∈ Γ(X, OX (T )) be such that (s0 )0 = T , where we write(s0 )0 for the zero divisor of s0 . We have H 1 (X, OX (T − x)) = H 0 (X, ωX (x − T ))∗ = 0
for any x ∈ X(Q) since deg(ωX (x − T )) ≤ 2gX − 2 − (2gX + 1) + 1 = −2. Thus,
the homomorphism Γ(X, OX (T )) → OX (T ) ⊗ k(x) induced by the short exact sequnce
0 → OX (T − x) → OX (T ) → OX (T ) ⊗ k(x) → 0 is surjective. Hence there exists
an s1 ∈ Γ(X, OX (T )) such that s1 (t) 6= 0 for all t ∈ T since Q is infinite. Then(s0 : s1 ) has no basepoints, and gives us a finite morphism ψ : X → P1Q such that
ψ ∗ OP1 (1) = OX (T ), and ψ(t) = 0 for all t ∈ T since (s0 )0 = T . Here, ψ is unramified
over 0 ∈ P1Q , since ψ ∗ OP1 (1) = OX (T ) and T is reduced. We also have 0 ∈/ ψ(S)
1since (s0 )0 = T and S ∩ T = ∅. Then by replacing X, T , and S by PQ , 0, andψ(S) ∩ {x ∈ P1Q | ψ ramifies over x} respectively, the theorem is reduced to the casewhere X = P1Q , T = {t} for some t ∈ P1 (Q) \ {∞}(Step 2): Next, we reduce the theorem to the case where X = P1Q , S ⊂ P1 (Q),T = {t} for some t ∈ P1 (Q) \ {∞} as follows: We will construct a non-zero rational1function f (x) ∈ Q(x) which defines a morphism φ : PQ
→ P1Q such that φ(S) ⊂ P1 (Q),φ(t) ∈/ φ(S), and φ is unramified over φ(t). By replacing S by the union of all Gal(Q/Q)conjugates of S, we may assume that S is Gal(Q/Q)-stable (Note that t ∈/ (new S) since1t ∈ P (Q) and t ∈/ (old S)). Write m(S) := maxF ([F : Q] − 1), where F runs through∑the fields of definition of the points in S, and d(S) :=F ([F : Q] − 1), where F
runs thrhough the fields of definition of the points in S with [F : Q] − 1 = m(S).
Thus, S ⊂ P1 (Q) is equivalent to d(S) = 0, which holds if and only if m(S) = 0. We

372

Go Yamashita

use an induction on m(S), and for each fixed m(S), we use an induction on d(S). If
m(S), d(S) 6= 0, take α ∈ S \ P1 (Q) such that d := [Q(α) : Q] is equal to m(S) + 1.
We choose a1 ∈ Q such that 0 < |t − a1 | < (mins∈S\{∞} |s − a1 |)/d(1 + d.d!). Thenby applying an automorphism f1 (x) := (mins∈S\{∞} |s − a1 |)/(x − a1 ) of P1Q (andreplacing t and S by f1 (t) and f1 (S) respectively), we may assume that |s| ≤ 1 for alls ∈ S(= S\{∞}) and |t| > d(1+d.d!) (Note that the property (new t) ∈ P1 (Q)\{∞} stillholds since |(old t)−a1 | > 0 and f1 (x) ∈ Q(x)). Let g(x) = xd +c1 xd−1 +· · ·+cd ∈ Q[x]be the monic minimal polynomial of α over Q. Then |ci | ≤ d! for 1 ≤ i ≤ d since ci is a
( )summation of di (≤ d!) products of i conjugates of α. Thus, |g(s)| ≤ 1+|c1 |+· · ·+|cd | ≤1 + d.d! and |g 0 (s)| ≤ d + d|c1 | + · · · + d|cd | ≤ d(1 + d.d!) for all s ∈ S(= S \ {∞}) since|s| ≤ 1 (Here g 0 (x) is the derivative of g(x)). Hence t ∈/ g(S) ∪ g(Sα ) =: S 0 , whereSα := {β ∈ Q | g 0 (β) = 0}. We also have [Q(α0 ) : Q] < d for any α0 ∈ g(Sα ) sinceg(x), g 0 (x) ∈ Q[x] and deg(g 0 (x)) < d. Therefore, S 0 is Gal(Q/Q)-stable and we havem(S 0 ) < m(S) or (m(S 0 ) = m(S) and d(S 0 ) < d(S)). This completes the induction,
and we get a desired morphism φ by composing the constructed maps as above.
1(Step 3): Now, we reduced the theorem to the case where X = PQ
, S ⊂ P1 (Q),and T = {t} for some t ∈ P1 (Q) \ {∞} with S ∩ T = ∅. We choose a2 ∈ Q suchthat 0 < |t − a2 | < (mins∈S\{∞} |s − a2 |)/4. Then by applying an automorphism
f2 (x) := 1/(x − a2 ) of P1Q (and replacing t and S by f2 (t) and f2 (S) respectively),
we may assume that |t| ≥ 4|s| for all s ∈ S(= S \ {∞}). (Note that the property(new t) ∈ P1 (Q) \ {∞} still holds since |(old t) − a2 | > 0 and f2 (x) ∈ Q(x)). New
t is not equal to 0 since old t is not equal to ∞. By applying the automorphismx 7→ −x of P1Q , we may assume that t > 0 (still t ∈ P1 (Q) \ {0, ∞}). By applying an
automorphism f3 (x) := x + a3 of P1Q , where a3 := maxS\{∞}3s0 <0 |s0 | (a3 := 0 when{s0 ∈ S \ {∞} | s0 < 0} = ∅) and replacing t and S by f3 (t) and f3 (S) respectively, we
may assume that s ≥ 0 for all s ∈ S(= S \ {∞}) and t ≥ 2s for all s ∈ S(= S \ {∞}),
since (t + a3 )/(s + a3 ) ≥ t/(s + a3 ) ≥ t/2a3 ≥ 2 where t, s are old ones (still (new t) ∈P1 (Q)\{0, ∞}). By adjoing {0, ∞} (if necessary for 0), we may assume that S ⊃ {0, ∞}since t ∈/ {0, ∞}.
1
, {0, ∞} ⊂(Step 4): Thus, now we reduced the theorem to the case where X = PQ
11S ⊂ P (Q), T = {t} for some t ∈ P (Q) \ {∞} with S ∩ T = ∅, and s > 1 , t ≥ 2s
for every s ∈ S \ {0, ∞}. We show the theorem in this case (hence the theorem in
the general case) by the induction on #S. If #S ≤ 3 then we are done. We assume
that #S > 3. Let a4 ∈ Q be the second smallest s ∈ S \ {0, ∞}. By applying
an automorphism f4 (x) := x/a4 of P1Q (and replacing t and S by f4 (t) and f4 (S)respectively), we may assume moreover that 0 < r < 1 for some r ∈ S and s > 1 for everys ∈ S \ {0, r, 1, ∞} Write r = m/(m + n) where m, n ∈ Z>0 . We consider the function
h(x) := xm (x − 1)n and the morphisms ψ, ψ 0 : P1Q → P1Q defined by h(x) and h(x) + a5

a proof of the abc conjecture after Mochizuki

373

respectively, where a5 := − mins∈S\{∞} h(s). We have h({0, 1, r, ∞}) ⊂ {0, h(r), ∞}.
Thus #ψ(S) < #S and hence #ψ 0 (S) < #S. Any root of the derivative h0 (x) =xm−1 (x − 1)n−1 ((m + n)x − m) = 0 is in {0, r, 1, ∞} ⊂ S. Thus ψ is unramified outside
ψ(S), and hence ψ 0 is unramified outside ψ 0 (S). Now h(x) is monotone increasing forx > 1 since h0 (x) > 0 for x > 1. Thus we have h(t) > h(s) for s ∈ S \ {∞} with s > 1
since t ≥ 2s > s. We also have h(t) > h(2) > 1 since t ≥ 2 (which comes from t ≥ 2s
for s = 1 ∈ S). Thus, ψ(t) ∈/ ψ(S) since |h(x)| ≤ 1 for 0 ≤ x ≤ 1. Hence we also00have ψ (t) ∈/ ψ (S). Now we claim that (h(t) + a5 )/(h(s) + a5 ) ≥ 2 for all s ∈ S \ {∞}such that h(s) + a5 6= 0. If this claim is proved, then by replacing S, t by ψ 0 (S), ψ 0 (t)respectively, we are in the situation with smaller #S where we can use the induction
hypothesis, and we are done. We show the claim. First we observe that we haveh(t)/h(s) = (t/s)m ((t−1)/(s−1))n ≥ (t/s)m+n ≥ (t/s)2 (*) for s ∈ S \{∞}, since t ≥ simplies (t − 1)/(s − 1) ≥ t/s. In the case where n is even, we have a5 = 0 since h(s) ≥ 0
for all s ∈ S \ {∞} and h(0) = 0. Thus, we have (h(t) + a5 )/(h(s) + a5 ) = h(t)/h(s) ≥(t/s)2 ≥ t/s ≥ 2 for 1 < s ∈ S \ {∞} by (*). On the other hand, h(s) + a5 = h(s) = 0
for s = 0, 1 and (h(t) + a5 )/(h(r) + a5 ) = h(t)/h(r) ≥ h(t) = tm (t − 1)n ≥ t ≥ 2
by 0 < h(r) < 1 and t ≥ 2. Hence the claim holds for even n. In the case where nnm)m ( m+n)n , since h(x) ≤ 0 for 0 ≤ x ≤ 1 and,is odd, we have a5 = |h(r)| = ( m+nmnx = r ⇔ h0 (x) = 0 for 0 < x < 1. We also have 0 < a5 = ( m+n)m ( m+n
)n ≤
m
n
mn
mn
1
m+n m+n = (m−n)2 +4mn ≤ 4mn = 4 . Then for 1 < s ∈ S \ {∞} with h(s) ≥ a5 , wehave (h(t) + a5 )/(h(s) + a5 ) ≥ h(t)/2h(s) ≥ (t/s)2 /2 ≥ 2 by (*). For 1 < s ∈ S \ {∞}with h(s) ≤ a5 , we have (h(t)+a5 )/(h(s)+a5 ) ≥ h(t)/2a5 ≥ 2h(t) = 2tm (t−1)n ≥ t ≥ 2
by 0 < a5 ≤ 1/4 and t ≥ 2. For s = r ∈ S, we have h(r) + a5 = −a5 + a5 = 0. For
s = 0, 1 ∈ S, we have (h(t)+a5 )/(h(s)+a5 ) = (h(t)+a5 )/a5 ≥ h(t) = tm (t−1)n ≥ t ≥ 2
by 0 < a5 ≤ 1/4 and t ≥ 2. Thus, we show the claim, and hence the theorem.
§ C.3.
Lemma C.3.
of characteristic 0.

k-Cores.

([CanLift, Proposition 2.7]) Let k be an algebraically closed field(1) If a semi-elliptic (cf. Section 3.1) orbicurve X has a nontrivial automorphism, thenit does not admit k-core.
(2) There exist precisely 4 isomorphism classes of semi-elliptic orbicurves over k which
do not admit k-core.
Proof. (Sketch) For algebraically closed fields k ⊂ k 0 , the natural functor fromthe category Ét(X) of finite étale coverings over X to the category Ét(X ×k k 0 ) offinite étale coverings over X ×k k 0 is an equivalence of categories, and the natural map

374

Go Yamashita

Isomk (Y1 , Y2 ) → Isomk0 (Y1 ×k k 0 , Y2 ×k k 0 ) is a bijection for Y1 , Y2 ∈ Ob(Ét(X)) by
the standard arguments of algebraic geometry, i.e., For some k-variety V such that thefunction field k(V ) of V is a sub-field of k 0 , the diagrams of finite log-étale morphisms
over (X ×k k 0 , D ×k k 0 ) (Here, X is a compactification and D is the complement) underconsideration is the base-change of the diagrams of finite étale morphisms over V withrespect to Spec k 0 → Spec k(V ) → V , we specialise them to a closed point v of V , wedeform them to a formal completion Vbv at v, and we algebrise them (cf. also [CanLift,
Proposition 2.3], [SGA1, Exposé X, Corollaire 1.8]), and the above bijection is also∨(−D)|Y ) = 0 for any finite morphismshown in a similar way by noting H 0 (Y , ωX/k
Y → X in the argments of deforming the diagrams under consideration to Vbv . Thus,
the natural functor Lock (X) → Lock0 (X ×k k 0 ) is an equivalence categories. Hence thelemma is reduced to the case where k = C.
We assume that k = C. Note also that the following four statements are equivalent:(i) X does not admit k-core,(ii) π1 (X) is of inifinite index in the commensurator CPSL2 (R)0 (π1 (X)) in PSL2 (R)0 (∼=0Aut(H)) (Here, we write PSL2 (R) for the connected component of the identity of
PSL2 (R), and we write H for the upper half plane),(iii) X is Margulis-arithmetic (cf. [Corr, Definition 2.2]), and(iv) X is Shimura-arithmetic (cf. [Corr, Definition 2.3]).
The equivalence of (i) and (ii) comes from that if X admits k-core, then the morphism
to k-core X  Xcore is isomorphic to H/π1 (X)  H/CPSL2 (R)0 (π1 (X)), and that if
π1 (X) is of finite index in CPSL2 (R)0 (π1 (X)), then H/π1 (X)  H/CPSL2 (R)0 (π1 (X)) is
k-core (cf. also [CanLift, Remark 2.1.2, Remark 2.5.1]). The equivalence of (ii) and(iii) is due to Margulis ([Marg, Theorem 27 in p.337, Lemma 3.1.1 (v) in p.60], [Corr,
Theorem 2.5]). The equivalence of (iii) and (iv) is [Corr, Proposition 2.4].
(1): We assume that X admits a k-core Xcore . Let Y → X be the unique double
covering such that Y is a once-punctured elliptic curve. We write Y , Xcore for thesmooth compactifications of Y, Xcore respectively. Here, we have Y \ Y = {y}, and apoint of Y is equal to y if and only if its image is in Xcore \ Xcore . Thus, we have
Xcore \ Xcore = {x}. The coarsification (or “coarse moduli space”) of Xcore is the
projective line P1k over k. By taking the coarification of a unique morphism Y  Xcore ,we obtain a finite ramified covering Y  P1k . Since this finite ramified covering Y  P1kcomes from a finite étale covering Y → Xcore , the ramification index of Y  P1k is thesame as all points of Y lying over a given point of P1k . Thus, by the Riemann-Hurwitz∑
formula, we obtain −2d + i edi (ei − 1), where ei ’s are the ramification indices over
the ramification points of P1k , and d is the degree of the morphism Y  P1k . Hence by375

a proof of the abc conjecture after Mochizuki∑

1
i ei (ei − 1) = 2, the possibility of ei ’s are (2, 2, 2, 2), (2, 3, 6), (2, 4, 4), and (3, 3, 3).

Since y is the unique point over x, the largest ei is equal to d. In the case of (2, 2, 2, 2),
we have X = Xcore , and X has no nontrivial automorphism. In other three cases, Y is a
finte étale covering of the orbicurve determind by a triangle group (cf. [Take1]) of type(2, 3, ∞), (2, 4, ∞), and (3, 3, ∞). By [Take1, Theorem 3 (ii)], this implies that Y is
Shimura-arithmetic, hence X is Shimura-arithmetic as well. This is a contradiction (cf.
also [CanLift, Remark 2.1.2, Remark 2.5.1]) by the above equivalence of (i) and (iv).
(2): If X does not admit k-core, then X is Shimura-arithmetic by the above equivalence of (i) and (iv). Then by [Take2, Theorem 4.1 (i)], this implies that, in thenotation of [Take2], the arithmetic Fuchsian group π1 (X) has signature (1; ∞) such
√ √
√ √ √
√ √
that (tr(α), tr(β), tr(αβ)) is equal to ( 5, 2 5, 5), ( 6, 2 3, 3 2), (2 2, 2 2, 4), and(3, 3, 3). This gives us precisely 4 isomorphism classes.
§ C.4.

On the Prime Number Theorem.

For x > 0, write π(x) := #{p | p : prime ≤ x} and ϑ(x) :=(Chebychev’s ϑ-function). The prime number theorem says that
π(x) ∼∑

prime: p≤x log px(x → ∞),
log x

where, ∼ means that the ratio of the both side goes to 1. In this subsection, we show
the following proposition, which is used in Proposition 1.15.
Lemma C.4.

π(x) ∼ logx x (x → ∞) if and only if ϑ(x) ∼ x (x → ∞).

This is well-known for analytic number theorists. However, we include a proof here
for the convenience for arithmetic geometers.
∫x
Proof. We show the “only if” part: Note that ϑ(x) = 1 log t·d(π(t)) = π(x) log x−∫x∫ x π(t)π(1) log 1 − 1 π(t)
dt
=
π(x)
log
x
−
2). Thenit suftt dt (since π(t) = 0 for t < (
2
)
∫
π(t)
1 x π(t)1fices to show that limx→∞ x 2 t dt = 0. By assumption t = O log t , we have
( ∫
)
√
√
∫
∫ x dt∫ √x dt ∫ x dtxx− √x
1 x π(t)1 x dt
√
dt
=
O
.
By
=
+
≤
+
, we obtainx 2tx 2 log t
log t
log 2
2 log t2
x log t
log x
∫
∫
x
x
1
limx→∞ x1 2 π(t)t dt = 0. We show the “if” part: Note that π(x) = 3/2 log t d(ϑ(t)) =∫x∫ x π(t)ϑ(x)ϑ(3/2)π(t)ϑ(x)
log x − log(3/2) + 3/2 t(log t)2 dt = log x + 2 t(log t)2 dt (since ϑ(t) = 0 for t < 2). Then it∫ x ϑ(t)suﬃces to show that limx→∞ logx x 2 t(log
2 dt = 0. By assumption ϑ(t) = O(t), we have() t) ∫∫∫∫ √x dt∫xx
log x xlog x x ϑ(t)
dt
dt
1
√
dt
=
O
.
By
dt
=
≤
x
x
(log t)2 +2x (log t)22 t(log t)22 (log t)22 (log t)2
√
√
∫
x
x−√ x
log x x ϑ(t)dt = 0.
(log 2)2 + (log x)2 , we obtain limx→∞ x
2 t(log t)2§ C.5.

On the Residual Finiteness of Free Groups.

376

Go Yamashita

Proposition C.5.
(Residual Finiteness of Free Groups) Let F be a free group.
Then the natural homomorphism F → Fb to its profinite completion Fb is injective.
Proof. Let a ∈ F \ {1}. It suﬃces to show that there exists a normal subgroup
H ⊂ F of finite index such that a 6∈ H. Let Gen (⊂ F ) be a set of free generators of F .
Write Gen−1 := {a−1 | a ∈ Gen} ⊂ F . Thus, any element of F may be written as a finite
product of elements of Gen ∪ Gen−1 . Let a = aN aN −1 · · · a1 , where ai ∈ Gen ∪ Gen−1 ,be such a representation of a (i.e., as a finite product of elements of Gen ∪ Gen−1 ) of
minimal length. Let φ : Gen → SN +1 be a map such that, for x ∈ Gen, φ(x) ∈ SN +1
sends i 7→ i + 1 if x = ai and j 7→ j − 1 if x = a−1
j−1 . (To see that such a φ exists,
it suﬃces to observe that since the representation a = aN aN −1 · · · a1 is of minimal
length, the equations ai = x, ai−1 = x−1 cannot hold simultaneously.) Since Gen isa set of free generators of F , the map φ : Gen → SN +1 extends to a homomorphism
φF : F → SN +1 such that, for i = 1, . . . , N , the permutation φF (ai ) sends i 7→ i + 1.
Write H for the kernel of φF . Since φF induces an injection of F/H into the finitegroup SN +1 , it follows that H is a normal subgroup of finite index in F . Then φF (a)sends 1 7→ N + 1, hence in particular, is nontrivial, i.e., a 6∈ H, as desired.
§ C.6.

Some Lists on Inter-universal Teichmüller Theory.

Model Objects
Local:
Vbad (Example 8.8)

Vgood ∩ Vnon (Example 8.7)

Varc (Example 8.11)

Dv

B temp (X v )0 (Πv )0
B(X→− v ) (Πv )

X
−
→
v

Dv`

B(Kv )0 (Gv )B(Kv )0 (Gv )(O (Cv` ), spl`v )Cv(F v )base-field (Πv y (OF )pf )Πv y (OF )pf

Fv

temp. Fr’d F v ( Θ-fct.)equal to Cv(Cv , Dv , κv )Cv`

Gv y OF× · q N

Gv y OF× · pNv

equal to Cv

Fv`(Cv` , spl`v )(Cv` , spl`v )(Cv` , Dv` , spl`v )

v

v

v

v

v

Arch. Fr’d Cv (ang. region)

We use Cv (not F v ) with v ∈ Vnon and F v with v ∈ Varc for F-prime-strips (cf.
Definition 10.9 (3)), and F v ’s with v ∈ V for Θ-Hodge theatres.

377

a proof of the abc conjecture after Mochizuki

Global :D} := B(C K )0 ,D}± := B(X K )0 ,∼Fmod := (Cmod , Prime(Cmod ) → V, {Fv` }v∈V , {ρv : ΦCmod ,v

gl. to loc.
∼

−→

ΦR
C ` }v∈V )v( ρv : log`mod (pv ) 7→ [Kv :(F1mod )v ] logΦ (pv ) ).
Some Model Bridges, and Bridges

poly

}
NF
}
• (model D-NF-bridge, Def. 10.16) φNF
v := Aut (D ) ◦ φ•,v ◦ Aut(Dv ) : Dv −→ D ,
poly

poly

NF
}
NF
φNF
:= (action of j) ◦ φNF: Dj −→ D} ,1 := {φv }v∈V : D1 −→ D , φj
1
poly

NF
}
φNF
> := {φj }j∈F> : D> := {Dj }j∈F> −→ D .
ll• (model D-Θ-bridge, Def. 10.17)
eval. section
−→
B temp (Πv )0B temp (Πv )0labelledbyjφΘ
full polyv j := Aut(D>,v ) ◦∼B temp (Π )0 →B temp (Πv )0vpoly: Dvj −→ D>,v ,

poly

Θ
φΘ
j := {φv j }v∈V : Dj −→ D> ,(v ∈ Vbad ) (v ∈ Vgood)◦ Aut(Dvj )

poly

Θ
φΘ
> := {φj }j∈F> : D> −→ D> .
l

poly

}±
• (model Θell -bridge, Def. 10.31) φΘ
)◦φΘ
v 0 := Autcusp (D•,v ◦Aut+ (Dv 0 ) : Dv 0 −→
D}± ,
ell
ell
ell
ell
poly
poly
}±
:= (action of t) ◦ φΘ
: Dt −→ D}± ,φΘ
:= {φΘ
, φΘ
v 0 }v∈V : D0 −→ D
t
0
0
ell

ell

φΘ
±

ell

poly

:= {φΘ
}t∈Fl : D± −→ D}± .
tell• (model Θ±±±-bridge, Def. 10.30) φΘ
vt

: Dvt

±

+-full poly∼−→D,v ,

±
φΘ
t

‡

(‡ FJ

: Dvt

+-full poly
∼

−→

poly

Θ
D,v , φΘ± := {φt }t∈Fl : D± := {Dt }t∈Fl −→ D .

• (NF-,Θ-bridge, Def. 10.24) (‡ FJ‡
HT Θ ).
Θ• (Θell -,Θ± -bridge, Def. 10.36) † ψ±

‡

NF
ψ>

−→

ell

‡

F}
poly

99K

: † FT −→ † D}± ,F ~ ),
†

±

‡

Θ
ψ>

−→ ‡ F>

99K

poly

Θ
ψ±
: † FT −→ ‡ F .

Theatres• (Θ-Hodge theatre, Def. 10.7) † HT Θ = ({† F v }v∈V , † Fmod ).
†

†

φNF
>

φΘ
>

• (D--Hodge theatre, Def. 10.18 (3)) † HT D- = († D} ←− † DJ −→ † D> ).
‡

NF
ψ>

‡

Θ
ψ>

• (-Hodge theatre, Def. 10.24 (3)) ‡ HT  = (‡ F ~ L99 ‡ F } ←− ‡ FJ −→ ‡ F> 99K‡
HT Θ ).

378

Go Yamashita
†

±

†

φΘ
±

ell

φΘ
±

• (D--Hodge theatre, Def. 10.32 (3)) † HT D- = († D ←− † DT −→ † D}± ).
†• (-Hodge theatre, Def. 10.24 (3)) HT

†

†

±

Θ
ψ±

†

†

ell

Θ
ψ±

= ( F ←− FT −→ † D}± ).
gluing• (D-  -Hodge theatre, Def. 10.40 (1)) † HT D- = († HT D- 99K † HT D- ).
gluing• (-Hodge theatre, Def. 10.40 (2)) † HT  = († HT  99K † HT  ).
Properties(Proposition 10.20, Lemma 10.25, Proposition 10.34, Lemma 10.37)>‡ NF• Isom(† φNF> , φ> ) : an Fl -torsor.
‡ NF• #Isom(† φNF> , φ> ) = 1.

• #Isom(† HT D- , ‡ HT D- ) = 1.
†

NF †• Isomcapsule-full poly († DJ , † DJ 0 ) φ> , φ> form a D--Hodge theatre : an F>l -torsor.
• † φNF>†ΘHT D- , up to F>
l -indeterminacy.
∼

2 NF
NF 2 NF, ψ> ) → Isom(1 φNF• Isom(1 ψ>> , φ> ).
∼

Θ 2 Θ
2 Θ
• Isom(1 ψ>, ψ> ) → Isom(1 φΘ
> , φ> ).
∼• Isom(1 HT Θ , 2 HT Θ ) → Isom(1 D> , 2 D> ).
∼• Isom(1 HT  , 2 HT  ) → Isom(1 HT D- , 2 HT D- ).
‡NF ‡• Isomcapsule-full poly (‡ FJ , ‡ FJ 0 ) ψ> , ψ> form a -Hodge theatre : an F>l -torsor.
±

Θ

±

‡ Θ
V
• Isom(† φΘ± , φ± ) : a {±1} × {±1} -torsor.
o±‡ NF• Isom(† φNF> , φ> ) : an Fl -torsor. we have a natural isomorphism• Isom(† HT D- , ‡ HT D- ) : an {±1}-torsor.
†

Θ± †

Θell

• Isomcapsule-+-full poly († DT , † DT 0 ) φ± , φ±torsor.
• † φΘ
±ell†form a D--Hodge theatre

HT D- , up to Fo±l -indeterminacy.

±

±

∼

±

ell

ell

∼

ell

±

Θ
Θ
Θ
• Isom(1 ψ±, 2 ψ±) → Isom(1 φ±, 2 φΘ
± ).
ell

Θ
Θ
2 Θ
• Isom(1 ψ±, 2 ψ±) → Isom(1 φΘ
± , φ± ).
∼• Isom(1 HT  , 2 HT  ) → Isom(1 HT D- , 2 HT D- ).

V: an Fo±l ×{±1} -379

a proof of the abc conjecture after Mochizuki
‡

Θ± ‡

Θell

D

‡

HT D-

D

‡

HT

D-

D-

D

−→

HT

†

• Isomcapsule-+-full poly (‡ FT , ‡ FT 0 ) ψ± , ψ±torsor.

: an Fo±× {±1}V lform a -Hodge theatreLinks• (D--link, Def. 10.21) † HT D- −→†• (D--link, Def. 10.35) HT

D-

−→

†

• (D-  -link, Cor. 11.24 (4)) HT†• (Θ-link, Def. 10.8) HT

Θ

Θ

‡

−→

• (Θ×µ -link, Cor. 11.24 (3)) † HT †• (Θ×µlgp -link, Def. 13.9 (2)) HT• (log-link, Def. 12.3) † HT  −→
log

∼

‡

−→
Θ×µ
gau

Θ×µ
LGP

‡

HT

∗

−→

full poly
∼
‡

−→

D-

full poly∼‡−→D`> ).
D`> ).
full poly
∼

†

( D`∆

−→

full poly
∼
‡

HT 

I×µ
(† Fgau−→D`∆ ).

F∆I×µ ).

full poly
∼
‡

−→

I×µ
HT  ((‡→)† FLGP

HT 

‡

Fmod ).

I×µ(† Fenv

∗

−→

−→

( D`>

full poly
∼
‡

HT 

‡

−→

Θ×µ
lgp

†

( Ftheta

Θ×µ
†• (Θ×µgau -link, Cor. 11.24 (3)) HT†• (Θ×µLGP -link, Def. 13.9 (2)) HTΘ‡(† D`>

I×µ((‡→)† Flgp

F∆I×µ ).

full poly∼∗−→full poly∼∗−→F∆I×µ ).

F∆I×µ ).

HT († HT D- → ‡ HT D- , † F> → ‡ F> , † F → ‡ F , {† Fj → ‡ Fj }j∈J , {† Ft →‡Ft }t∈T ).
log

log

log

log

Index of Terminologies- F `×µ -prime-strip, 319
bounded discrepancy class, 16
bridge
D-NF- -, 228
D-Θ- -, 228
D-Θ- - is glued to D-Θ± - -, 252
D-Θell - -, 245
D-Θ± - -, 245
NF- -, 236
Θ- -, 237
Θ- - is glued to Θ± - -, 252Θell - -, 249
Θ± - -, 249
base-NF- -, 228
base-Θ- -, 228
base-Θell - -, 245
base-Θ± - -, 245
isomorphism of D-NF- -s, 228
isomorphism of D-Θ- -s, 228
isomorphism of D-Θell - -s, 245
isomorphism of D-Θ± - -s, 245
isomorphism of NF- -s, 237
isomorphism of Θ- -s, 237
isomorphism of Θell - -s, 249
isomorphism of Θ± - -s, 249
model D-NF- -, 226
model D-Θ- -, 228
model D-Θell - -, 245
model D-Θ± - -, 244
model NF- -, 234
model Θ- -, 233
model base-NF- -, 226
model base-Θ- -, 228
model base-Θell - -, 245
model base-Θ± - -, 244

abc Conjecture, 5
uniform -, 47
abstractly equivalent, 11
algorithm
multiradial, 196, 243, 368
uniradial, 196
α-signed automorphism- of † D, 241anabelioid
connected -, 112
morphism of -s, 113
angular region, 177
arithmetic
Margulis-, 374
Shimura-, 374
arithmetically holomorphic, 87
arithmetic divisor, 15, 179
R-, 15eﬀective -, 15, 179
principal -, 15
arithmetic line bundle, 15
Aut-holomorphic disc, 91
Aut-holomorphic orbispace, 92
Aut-holomorphic space, 91
elliptically admissible -, 91
hyperbolic - of finite type, 91
local morphism of -s, 91(U, V)- -, 91
co-holomorpic (U, V)- -, 92
finite étale (U, V)- -, 91
morphism to † D} , 219
morphism to † D}± , 219
Aut-holomorphic structure, 91
U-local pre- -, 91bi-anabelian, 64
bi-coric

CAF, 12381

382

capsule, 11#J- -, 11- -full poly-isomorphism, 12- -full poly-morphism, 12morphism of -, 11Cauchy sequence, 102
equivalent -, 102closed point
algebraic -, 62
co-holomorphicisation, 92
pre- -, 92
commensurably terminal, 13commensurator, 13
compactly bounded subset, 18
support of -, 18
condition-(Cusp)X , 63-(Delta)X , 63-(Delta)’X , 63-(GC), 63-(slim), 63
co-orientation, 92
pre- -, 92
co-oriented, 92
coric, 253∞ κ- -, 184∞ κ- - structure, 189, 191, 193∞ κ×- -, 184∞ κ×- - structure, 189, 191, 193
κ- -, 184
- category, 252- data, 252-ally defined, 253
bi- -, 101
horizontally -, 101, 214, 243vertically -, 101
critical point, 184
strictly -, 184

Go Yamashita

cusp±-label class of -s of Πv , 268±-label class of -s of Π±v , 268† }±-label class of -s of D , 241±-label class of -s of † Dv , 240
b v , 268±-label class of -s of Πb±±-label class of -s of Πv , 268† }label class of -s of D , 224label class of -s of † Dv , 224
non-zero -, 146
set of -s of † D} , 224
set of -s of † D}± , 241
set of -s of † Dv , 224, 240
set of -s of † D±v , 240†set of -s of Dv , 224zero -, 146
cuspidalisation
Belyi -, 70
elliptic -, 67
cuspidal quotient, 66
cyclotome, 14- of Gk , 84
- of M , 86
- of P , 183- of ΠX as orientation, 75- of K, 14- of Π[µN ], 153
external - of † M, 162
internal - of † M, 162
cyclotomic envelope, 153
cyclotomic rigidity- for inertia subgroup, 76- in tempered Frobenioid, 181- of mono-theta environment, 162
b × = {1}, 190, 192- via Q>0 ∩ Z- via LCFT, 85, 86- via positive rational structure and
LCFT, 124

a proof of the abc conjecture after Mochizuki

classical -, 85, 86
decent data from Z to F1 , 361
domaincompact -, 17edge-like subgroup, 117element
negative - of Fo±l , 239o±positive - of Fl , 239
étale-like object, 99étale theta class, 134- of standard type, 143, 152
standard set of values of -, 143étale theta function, 137étale-transport, 101, 329, 354
indeterminacies, 354
evaluation isomorphism, 283, 284, 286,
292, 295, 299
formal -, 290
global formal -, 296
of F -prime-strips, 305
evaluation points- of X v , 227- of X v , 227
Faltings height, 38
F±l -group, 239
F±
l -torsor, 239
positive automorphism of, 239
frame, 96-d, 96orthogonal -, 96
Frobenioid, 170µN -split pre- -, 169
×-Kummer pre- -, 172
×µ-Kummer pre- -, 172
p-adic -, 174
Archimedean -, 178base category of elementary -, 169383

base category of pre- -, 169
base-field-theoretic hull of tempered-, 176birationalisation of model -, 171
divisor monoid of model -, 170
divisor monoid of pre- -, 169
elementary -, 169
global non-realified -, 187
global realified -, 179
isomorphism of pre- -s, 170
model -, 170
pre- -, 169
pre- - structure, 169
rational function monoid of model -,170
realification of model -, 171
split pre- -, 169
split-×-Kummer pre- -, 172
split-×µ-Kummer pre- -, 172
tempered -, 176vertically coric étale-like pre- -, 349
Frobeniusabsolute -, 360
Frobenius-like object, 99
fundamental groupadmissible - , 115
Galois evaluation
principle of -, 272, 362
graphdual - , 113
dual semi- - , 113
semi- - of anabelioids, 114
graph of Ξ, 253
height function, 16
Hodge theatre
D-ΘNF- -, 228
D-Θ±ell - -, 246

384

Go Yamashita

D-Θ±ell NF- -, 252
D-- -, 252
D-- -, 246
D-- -, 228
Θ- -, 213
ΘNF- -, 238
Θ±ell - -, 249
Θ±ell NF- -, 252
- -, 249
- -, 238
- -, 252
base-ΘNF- -, 228
base-Θ±ell - -, 246
base-Θ±ell NF- -, 252
isomorphism of - -, 249
isomorphism of - -, 238
isomorphism of - -s, 252
isomorphism of D-- -s, 246
isomorphism of D-- -s, 229
isomorphism of D-  - -s, 252
holomorphic hull, 26
indeterminacy
horizontal -, 30, 31
permutation -, 30, 31
vertical -, 30, 31
initial Θ-data, 41, 205
integral element, 12inter-universal Melline transformation,47
inversion automorphism, 138, 146, 262
pointed -, 263, 264
isometry of O×µ (G), 172
isomorph, 11
isomorphismof categories, 206
k-core, 62
admit -, 62

Kummer-detachment, 101, 329
indeterminacy, 355
Kummer-faithful, 60
Kummer isomorphism- by Kummer structure, 193, 194- for M , 86
- for F-prime-strips, 303- for Ik , 105, 108×- for k (ΠX ), 105- for algebraic closure of number
fields, 189, 302- for constant monoids, 279, 291, 297- for labelled Frobenioids, 303- for labelled constant monoids, 291,
298
- for labelled number fields, 303- for labelled pseudo-monoids, 303- for local LGP-monoids, 340- for monoids, 192- for number fields, 302- for pseudo-monoids, 189, 192, 302- for theta monoids, 279- of F ⊥ -prime-strip, 351
Kummer structure
×- -, 172×µ- -, 172- of an Aut-holomorphic space, 98
model - of an Aut-holomorphic
space, 97
morphism of elliptically admissible
Aut-holomorphic orbispaces
with -s, 98
l-cyclotomically full, 59, 60
line bundle
- -, 198
- -, 198
elementary morphism of - -s, 198
elementary morphism of - -s, 198

a proof of the abc conjecture after Mochizuki

morphism of - -s, 199
morphism of - -s, 198
tensor product of - -, 198
tensor product of - -, 198
line segment, 95- tangent to S · p, 96
endpoint of -, 95
parallel -s, 95
link
D-ΘNF- -, 231
D-Θ±ell - -, 248
D-- -, 248
D-- -, 231
D-  - -, 306
Θ- -, 214Θ×µ - -, 306
Θ×µLGP - -, 345
Θ×µgau - -, 306
Θ×µlgp - -, 345
log- -, 104, 108
log- - from † Fv to ‡ Fv , 308, 310
log- - from † F to ‡ F, 310
log- - from † HT  to ‡ HT  , 316
base-ΘNF- -, 231
base-Θ±ell - -, 248
full log- - from † Fv to ‡ Fv , 308, 310
full log- - from † F to ‡ F, 310
full log- - from † HT  to ‡ HT  ,316
generalised Θ×µ
LGP - -, 362
tautological log- - associated to † Fv ,308, 309
tautological log- - associated to † F,
310
local additive structure, 95
local field, 12
local linear holomorphic structure, 97
system of -s, 97385

local structure, 91
log-conductor function, 17
log-diﬀerent function, 17
log-divisor
eﬀective Cartier, 175
log-Kummer correspondence
global -, 351
local -, 342
log-meromorphic function, 175
log-orbicurveof type (1, (Z/lZ)Θ ), 149of type (1, Z/lZ), 149of type (1, Z/lZ)± , 149of type (1, l-tors−−→), 167of type (1, l-tors), 146of type (1, l-tors)± , 146of type (1, l-torsΘ ), 147
log-shell, 26, 108étale-like holomorphic -, 105, 108étale-like mono-analytic -, 106, 110étale-like mono-analytic - associatedto † Dv` , 313
Frobenius-like holomorphic -, 105
Frobenius-like holomorphic associated to † Fv , 309, 310
Frobenius-like holomorphic associated to † F, 311
Frobenius-like mono-analytic associated to † Fv`×µ , 311
Frobenius-like mono-analytic associated to † F`×µ , 311
vertically coric étale-like
holomorphic - associated to ∗ D,
314
log-theta-lattice, 317
LGP-Gaussian -, 346
lgp-Gaussian -, 346
Gaussian -, 317

386

Go Yamashita

non-Gaussian -, 317
log-volume function, 23, 106
global -, 348
radial -, 23, 109
maximal cuspidally central quotient, 75
miracle identity, 46
MLF, 12mono-analytic, 87
mono-anabelian, 49, 64, 97
mono-anabelian transport, 194monoid- on D, 168
Frobenioid-theoretic constant -, 279
Frobenioid-theoretic Gaussian -, 284
Frobenioid-theoretic theta -, 278
Frobenius-like global realified
-logarithmic Gaussian
procession -, 344
Frobenius-like global realified
-logarithmic Gaussian
procession -, 343
Frobenius-like global realified LGP-, 343
Frobenius-like global realified lgp- -,344
Frobenius-like local LGP- -, 339
Frobenius-like local logarithmic
Gaussian procession -, 339
group-like - on D, 168
mono-theta-theoretic constant -, 278
mono-theta-theoretic Gaussian -, 282
mono-theta-theoretic theta -, 277
morphism of split -s, 98
primary element of -, 179
prime of -, 179
split -, 98
vertically coric étale-like -, 349

vertically coric étale-like local LGP-, 340vertically coric étale-like local
logarithmic Gaussian procession-, 340multiradial, 253- environment, 253-ly defined, 253µN -conjugacy class, 153
mutations, 64, 252
negative automorphism- of † Dv , 241
NF, 12-constant, 80-curve, 80-point, 80-rational function, 80
normalisation
packet- -, 347
procession- -, 347
normally terminal, 13
number field, 12vertically coric étale-like -, 349
orbicurve- of strictly Belyi type, 62
elliptically admissible -, 62
hyperbolic -, 14semi-elliptic -, 62
orientation, 96
outer semi-direct product, 14
parallelogram, 95
pre-∂- , 95
side of -, 96
parallel transport, 253
pictureétale- - of D--Hodge theatres, 249
étale- - of D--Hodge theatres, 231

a proof of the abc conjecture after Mochizuki

étale- - of Θ-Hodge theatres, 215étale- - of -Hodge theatres, 307étale- - of multiradial theta monoids,325étale- - of the étale theta functions,265
Frobenius- - of -Hodge theatres,
306
Frobenius- - of log-links for-Hodge theatres, 316
Frobenius- - of log-links for
F-prime-strips, 315
Frobenius- - of Θ-Hodge theatres,214
pilot object
Θ- -, 345
q- -, 345
poly+-full - -isomorphism, 241
α-signed +-full - -automorphism, 241- -action, 11- -automorphism, 11- -isomorphism, 11- -morphism, 11
capsule-+-full - -morphism, 241
full - -isomorphism, 11
negative +-full - -automorphism, 241
positive +-full - -automorphism, 241
positive automorphism- of F±l -torsor, 239†- of Dv , 241- of † D, 241
positive rational structure, 124, 126
prime number theorem, 35, 42, 375
prime-strip
D- -, 215
D` - -, 215
F- -, 216

387

F - -, 216
F ` - -, 216
F I×µ - -, 222
F I - -, 223
F ⊥ - -, 223
F `I×µ - -, 221
F `I - -, 223
F `⊥ - -, 223
F `×µ - -, 221
F `× - -, 221
arrow of F- -s lying over φ, 219
global realified mono-analytic
Frobenioid- -, 216
holomorphic base- -, 215
holomorphic Frobenioid- -, 216
isomorphism of F- -s, 216
isomorphism of F - -s, 216
isomorphism of F ` - -s, 216
isomorphism of F I×µ - -s, 222
isomorphism of F I - -s, 223
isomorphism of F ⊥ - -s, 223
isomorphism of F `I×µ - -s, 221
isomorphism of F `I - -s, 223
isomorphism of F `⊥ - -s, 223
isomorphism of F `×µ - -s, 221
isomorphism of F `× - -s, 221mono-analytic base- -, 215
mono-analytic Frobenioid- -, 216
morphism of D- -s, 215
morphism of D` - -s, 215
poly-morphism from D- - to † D} ,220
poly-morphism from D- - to † D}± ,220
poly-morphism from F- - to † F } ,232poly-morphism from a capsule of F- to † F } , 233

388

Go Yamashita

poly-morphism from a capsule of F- to an F- -, 233
poly-morphism from a cupsule of D- to † D} , 220
poly-morphism from a cupsule of D- to † D}± , 220
poly-morphism from a cupsule of D- to a D- -, 220vertically coric étale-like F - -, 349
primitive automorphisms, 26
- -, 26
processionn- -, 331
morphism of -s, 331
normalised average, 33
pseudo-monoid, 183
cyclotomic -, 183
divisible -, 183
topological -, 183
radial- algorithm, 252- category, 252- data, 252- environment, 252
RC-holomorphic morphism, 91
Riemann hypothesis, 47
section- associated with a tangential
basepoint, 135
mod N algebraic -, 153, 154
mod N theta -, 155slim, 53
relatively -, 53
species, 64, 252
structure of Dyµ2l - , 135
compatible with -, 136{±1}- , 135
compatible with -, 136
canonical discrete- , 135
canonical integral- , 135
compatible with -, 135
canonical tame integral- , 135
sub-p-adic, 60
generalised -, 370
symmetrising isomorphismF>l - -, 301
Fo±l - -, 282, 289, 294
synchronisation
global {±1}- -, 292, 294, 316
Teichmüller dilation, 25, 31, 88
temp-slim, 112
relatively -, 112
tempered covering, 111
tempered filter, 175
tempered group, 112
temperoid, 115
temperoid
connected -, 112
morphism of -s, 113
tensor packet
global j- - associated to S>
j and a-Hodge theatre, 336
local holomorphic 1- - associated toan F-prime-strip, 332
local holomorphic j- - associated to acollection of F-prime-strips, 333
local mono-analytic 1- - associatedto an D` -prime-strip, 334
local mono-analytic 1- - associatedto an F `×µ -prime-strip, 335
local mono-analytic j- - associatedto a collection ofD` -prime-strips, 334
local mono-analytic j- - associateda proof of the abc conjecture after Mochizuki

to a collection ofF `×µ -prime-strips, 335
Θ-approach, 185
theta environment
homomorphism of mono- - , 156
mod N bi- - , 157
mod N model mono- - , 156
mod N mono- - , 156
isomorphism of - , 156
theta quotient, 127
theta trivialisation, 133
tripod, 14
uniradial, 253- environment, 253-ly defined, 253
upper semi-compatibility, 105, 108
valuation
δ- -, 232
value-profile, 282
verticial subgroup, 117
Vojta’s Conjecture- for curves, 5389

Index of Symbols−| log(Θ)|, 36−| log(Θ)|vQ , 33−| log(Θ)|vQ ,j , 32−| log(Θ)|{v0 ,...,vj } , 30−| log(q)|, 36#(−), 10
≈, 16
&, 16
., 16
`, 100ADivR (F ), 15
APic(X), 15
Aut+ (D}± ), 242
Aut+ († Dv ), 241
Aut+ († D), 241
Aut+ (T ), 239
Aut− († Dv ), 241
Autα († D), 241
Autcusp (D}± ), 242
AutD (−), 216
AutD` (−), 216
Aut (C K ), 225
Aut (D} ), 225
AutF (−), 216
AutF (−), 216
AutF I×µ (−), 222
AutF ` (−), 216
AutF `I×µ (−), 222
AutF `× (−), 222
AutF `×µ (−), 222
Aut(U top ), 90
Authol (U ), 90
Aut± (D}± ), 242
Aut± (T ), 239
AutΘ
Dv (S), 181α̈δ , 157αδ , 157±αΘ , 244

outo , 13−1Fl , 244(−)0 , 11(−)ab , 13(−)gp , 13(−)gp , 105, 108(∞) (−), 279(−)ι , 264(−)pf , 13(−)× , 13(−)> , 11(−)⊥ , 11(−)tors , 13|| · ||Falt
Ev , 37
av , 29
A∞ , 172, 278
AΘ∞ , 278‡
A∞ , 220
Abirat , 180
Ap , 96
Ap , 97
AX , 97
AX , 90
AX , 91
AX , 97
ADiv(F ), 15

BI , 24, 31
Base(φ), 169
B(Π), 112
B temp (Π), 112
B temp (X), 112
B(X), 112
B(G), 114
B cov (G), 115
391

392

Go Yamashita

B temp (G), 115
B, 176
B0 , 175
Bconst
, 175
0
const
B
, 176
R
B , 171
C, 142, 146
CF , 205
CG (H), 13
CK , 205†C∆ , 304‡ log †

Clgp (( −→) HT  ), 344‡ log †

CLGP (( −→) HT  ), 343
C, 146C F , 205
C F , 205
C, 252
C birat , 171
Cenv († HT Θ ), 299
Cgau († HT Θ ), 299†
Clgp , 344(‡→)†
CLGP , 343†
Cenv , 305†
Cgau , 305†
C , 216† `Cmod , 213†
Cv , 213
Cmod , 179
C ` , 257
Cv`R , 208
Cv` , 174, 177, 178
C R , 171
Ctheta , 211
CvΘ , 210
Cv , 174, 176, 177
C(† D} , w), 219◦,◦
C, 325†C, 324n,◦C, 325
χcyc , 14χcyc,l , 14†
χ, 229d∗mod , 29d∗ , 41
degF , 15
degFr (φ), 169
Div(φ), 169
Div+ (Z∞ ), 175
Dµ− , 263Dµδ − , 270
Dx , 145Dtδ , 270
δ
Dt,µ
, 270
−
D0 , 174
D0ell , 175
D>,v , 227
D} , 208D}± , 208
† ~D , 186
Denv († D`> ), 321
Denv († D` ), 295
DF , 182
Dgau († D` ), 296
D (‡ D` ), 294
D († D` )j , 296
Dvj , 226
† `Dv , 213, 215† Θ
Dv , 213† ±Dv , 239† ±Dv , 240†
Dv , 223†
Dv , 190†
Dv , 213, 215
DMΘ
, 256
M
Dmod , 212
Dv` , 173, 174, 178

a proof of the abc conjecture after Mochizuki

DvΘ , 209, 210†
Dv , 192† rat
Dv , 193
Dv , 173, 174, 178
DY , 156
D> , 227
Dj , 226†
D, 215†
D> , 251†
DhJi , 234† `D , 215†
DT > , 251
DN , 132n,m `D∆ , 318
D± , 244D , 243
Dt , 243poly
ED −→ † D} , 220
polyE
D −→ † D}± , 220
polyE
D −→ † D, 220
poly†
D −→ † D} , 220
poly†
D −→ † D}± , 220dL , 29
dx , 17∆vÏ , 269∆vI , 269∆v¨•t , 269∆v•t , 269∆CF , 205∆}± , 273
∆cor
v , 268, 287
(MΘ
∆temp
∗ ), 257C
temp∆C (MΘ
N ), 257∆ , 166∆− , 166∆+
 , 166∆G , 115(Σ)∆G , 115(Σ),temp

∆G
, 115
temp
∆G , 115∆[µN ], 153∆C , 146ell∆C , 146∆C , 146∆Θ , 145∆X , 145ell∆X , 145∆±vI , 269∆±v•t , 269∆±v , 268, 287
∆MΘ
, 257
N
∆Θ , 127, 145∆cusp-cent, 75
U∆v , 117∆v , 268, 287∆ell
X , 127, 145∆XF , 205
∆X,H , 120∆temp
X,H , 120∆ΣX , 116(Σ),temp
∆X
, 116
temp
∆X , 127(∆temp)ell , 127X(∆temp)Θ , 127X∆Θ
X , 127, 145
∆temp
(MΘ
∗ ), 257X
temp∆X (MΘ
N ), 257∆X , 205∆temp(MΘ∗ ), 257
X
∆temp
(MΘ
N ), 257Xell(∆tempYN ) , 128Θ(∆tempYN ) , 128
∆temp
, 128
Y
temp ell(∆Y ) , 128(∆temp)Θ , 128
Y
(MΘ
∆temp
∗ ), 257
Y

393

394

∆temp
(MΘ
N ), 257Yell(∆temp
ZN ) , 130Θ(∆temp
ZN ) , 130ev , 12emod , 29e∗mod , 29
expk , 108
expk(X) , 108
EF , 205
E, 166E v , 175
E env , 265
E LCFT , 261
E Θ , 258
EN , 181
EΠ
N , 182
ExcK,d, , 41
ιX , 146
, 166
0 , 166
00 , 166
0 , 166
, 206[† ], 230
v , 206η̈ Θ , 134η̈ Θ , 152η̈ Θ,lZ×µ2 , 152η̈ Θ,Z×µ2 , 143† 0η v , 240† ±
η v , 240†
η v , 224

ηZb , 126
ηZb0 , 125
ηZp , 125
fv , 12F , 205

Go Yamashita

Fmod , 205
F , 205
Ftpd , 27
Fv , 12F>
l , 11Fo±l , 11FΦ , 169
F ~ († D} ), 187
~
FMOD
, 198
‡ `I×µ
Fv
, 221
‡ `×µ
Fv , 221‡ `×Fv , 221‡ `Fv , 221† }F , 187† }F , δ232† ~FhJi , 235† ~FJ , 235† ~Fj , 235† ~R
FhJi , 235

FJ~R , 235† ~R
Fj , 235† ~Fmod , 187† ~( Fmod )hF> i , 303†l~R( Fmod)hF> i , 303l† `Fv , 213, 216† ΘFv , 213†F v , 213†
Fv , 191, 193, 216~
Fmod(† D} )hF> i , 301l~Fmod(† D} )j , 301~
FMOD(n,◦ HT D- )α , 349~
Fmod(n,◦ HT D- )α , 349~R † }
Fmod( D ), 300~R † }
Fmod ( D )hF> i , 301l~R † }
Fmod( D )j , 301
~
Fmod
, 199
~
Fmod (n,◦ HT D- )α , 349
Fv` , 174, 178†

a proof of the abc conjecture after Mochizuki

FvΘ , 210
F base-field, 176
v
F v , 174, 176, 178~)α , 343((‡→)† FMOD(‡→)† ~(
Fmod )α , 343~R†
FhJi → † FR
hJi , 236

FJ~R → † FR
J , 236† ~R† R
Fj → Fj , 236† ~Fj → † FJ , 236† `F∆,v , 345† ~( FMOD )α , 342~R(† FMOD)α , 342† ~( Fmod )α , 343~R)α , 343(† Fmod†log†

Fv → ‡ Fv , 308, 310

†

Fv

†

full log ‡→Fv , 308, 310

log

Fv → log(† Fv ), 308, 309∼~† ~Fmod → † FhJi
→ † FvhJi , 235
†

∼

~
~
Fmod
→ † FhJi→ † FhJi , 236
∼

~
Fmod
→ † Fj~ → † Fvj , 236∼† ~Fmod → † Fj~ → † Fj , 236
F`> (n,m D> ), 318
FD , 213†
FD , 213
F`∆ (n,m D ), 318
n,mF`×D ), 318
∆ (`×µ n,mF∆ ( D ), 318
F(n,◦ D )t , 339
F(∗ D), 314
Fenv († D> ), 321
F`env († D> ), 321~
Fmod(n,◦ HT D- )α , 349~
Fmod(n,◦ HT D- )α , 349~
Fmod(n,◦ HT D- )α , 349‡
F∆I×µ , 305‡
F , 297‡ `I
F , 222‡ `I×µ
F
, 221
†

‡ `⊥

F , 222F , 221
‡ `×µ
F
, 221
(‡→)†
Flgp , 344
I×µ
(‡→)†
Flgp
, 345
(‡→)† `
Flgp , 344(‡→)†
FLGP , 344
I×µ
(‡→)†
FLGP
, 345
(‡→)† `FLGP , 344†
F, 216†
F> , 231, 251†
FhJi , 234†
F∆ , 304† `F∆ , 304† `×µF∆ , 306†
Fenv , 305
†
I×µ
Fenv
, 305
† `
Fenv , 305† `×µFenv , 306†
Fgau , 305
†
I×µ
Fgau
, 305
† `
Fgau , 305† `×µFgau , 306†
F , 216†
FJ , 233†
Fj , 233†
Fmod , 213† `F , 216†
FT > , 251†
Ftheta , 213
Fmod , 209∗
F∆I×µ , 345∗
F I , 223∗
F I×µ , 222∗
F ⊥ , 223
Ftheta , 212
‡ `×

poly

F −→ † F } , 233
E poly †F −→ F, 233
E

395

396

Go Yamashita

‡

poly

†

log

F −→ † F } , 232
log†
F −→ log(† F), 310
F → ‡ F, 311† full log ‡F → F, 311

htFalt (E)non , 38
htnon
ωP1 ({0,1,∞}) , 38
HomdomK (X, Y ), 369
open
HomGK (ΠX , ΠY ), 369(p)Graph(Ξ), 253
GF , 205
Gv (MΘ∗ ), 257, 278), 274Gv (MΘ∗ÏΘ †Gv (M∗Ï ( F v ))hF> i , 286l†
Gv (MΘ
∗ ( F v ))hF> i , 286l

Gv (MΘ∗ )t , 282
Gv (MΘ
N ), 257
Gv (ΠvÏ ), 271
Gv (ΠvÏ )hF> i , 286lGv (Πcorv ), 287
Gv (Π±v ), 287
Gv (Πv ), 271, 287
GX , 99
Gi , 115
Gi∞ , 118
GX , 114
GX , 113
γ± , 245
ΓI
X , 267Γ•t
X , 268Γ•X , 267
Γ(L), 15
ΓX , 124
ΓX , 267
ΓX , 267
ΓY , 267
ΓŸ , 267
ΓY , 267
ΓŸ , 267
htL , 16
htFalt (E), 38
htFalt (E)arc , 38(p)

Homopen
GK (ΠX , ΠY ), 369
1∞ H (−), 189, 191, 263, 265, 271, 275,
276, 287, 288
H(L̈lN , Z̈lN )180
H(OZ̈lN ), 180
H(Z̈lN ), 180‡
HT  , 238†
HT  , 249†
HT D- , 246†
HT D- , 228, 231†
HT D- , 252†
HT Θ , 213, 231, 279, 280, 283, 286,
290

n,◦
HT
, 339, 349, 352
log

†
HT
−→ ‡ HT  , 316
full log†
HT  −→ ‡ HT  , 316†

Θ×µHT  −→ ‡ HT  , 306
×µ Θgau ‡−→ HT  , 306

†

HT

†

LGP ∗
HT  −→HT  , 345

Θ×µ



Θ×µ
lgp

HT
−→ ∗ HT  , 345
D†
HT D- −→ ‡ HT D- , 248
D†
HT D- −→ ‡ HT D- , 306
D†
HT D- −→ ‡ HT D- , 231
Θ†HT Θ −→ ‡ HT Θ , 214†I(α Dv`Q ), 335, 336
I† Dv` , 313±

I(Sj+1 ,α Dv` ), 335, 336±

I(Sj Dv`Q ), 335, 336
I† D` , 314
I∗ D , 314
I(α Fv`×µ), 336Q
αI( FvQ ), 336

a proof of the abc conjecture after Mochizuki

I† Fv`×µ , 311
I† Fv , 309, 310±

I(Sj+1 ,α Fv`×µ ), 336±

I(Sj+1 ,α Fv ), 336±

I(Sj FvQ ), 336
I† F , 311, 314
I† F`×µ , 311, 314
I(G), 106, 109
Ik , 26, 105, 108
In,m F∆ , 320
I(ΠX ), 105
I(X), 108I Q (α Dv`Q ), 336±

I Q (Sj+1 ,α Dv` ), 336±

I Q (Sj Dv`Q ), 336), 336I Q (α Fv`×µQ
Q α
I ( FvQ ), 336±
Q Sj+1 ,α

I (

I Q(

S±
j+1 ,α±Fv`×µ ), 336
Fv ), 336), 336I Q (Sj Fv`×µ
Q±

I Q (Sj FvQ ), 336±

I Q (Sj+1 ;n,◦ DV`Q ), 353
S±
j+1

I Q(
FVQ ), 348∗
IC ∼ , 109
Ik∗ , 105, 108
Iv1 ,··· ,vn , 26
IvQ1 ,··· ,vn , 26
(Indet →), 30, 31(Indet xy), 30, 31(Indet ↑), 30, 31
Inn(G), 13
IsomK (X, Y ), 369
IsomOut
GK (∆X , ∆Y ), 369
IsomD (−, −), 216
IsomD` (−, −), 216
IsomF (−, −), 216
IsomF (−, −), 216

IsomF I×µ (−, −), 222
IsomF ` (−, −), 216
IsomF `I×µ (−, −), 222
IsomF `× (−, −), 222
IsomF `×µ (−, −), 222(p)(p)
IsomOut
GK (∆X , ∆Y ), 369
Isomet, 172
Isomet(G), 172{(ι, D)}(Π), 264
ιfL ,w , 35
ιX , 262
ιX , 146, 166
ιvQ , 29
ιX , 146
ιY , 138
ιŸ , 262
ιŸ , 262
Iord , 328
JN , 129J¨N , 132
JX , 166
k NF , 80
kI , 24k(ΠX ), 105k ∼ , 104, 107k(X), 108K, 27, 205
K̈, 132
KN , 128
K̈N , 132
KV , 17‡ `×µκv , 221‡ `×
κv , 220† ∼κv , 309†
κv , 213κv , 178l, 205397

398

Go Yamashital> , 11l± , 11(l∆Θ )(MΘ∗ ), 257, 264(l∆Θ )(MΘ), 274∗ÏΘ(l∆Θ )(MN ), 257(l∆Θ )(Π), 258(l∆Θ )(ΠvÏ ), 271(l∆Θ )S , 181
log-condD , 17
log-diﬀ X , 17`
logDv (pv ), 288
log(dL ), 29‡ `log D (pv ), 294†`

log D (Θ), 295
log(† Fv ), 308, 309
log(† F), 310
logGv (pv ), 288
logk , 107
logk , 104log`mod (pv ), 179
logΦ (pv ), 208
logΦ (q ), 177v

log(pv ) log(Θ), 210log(q∀ ), 40log(qL ), 29
log(q-2 ), 40log(sL ), 29log(s≤ ), 29
log(† Dv` ), 313±

log(Sj+1 ,α Dv` ), 335±

log(Sj+1 ,α Fv`×µ ), 335±

log(Sj Dv`Q ), 334log(α Dv`Q ), 334
log(† D` ), 314±), 335log(Sj Fv`×µQ
α `×µlog( FvQ ), 335±

log(Sj FvQ ), 333log(α FvQ ), 332

log(† Fv`×µ ), 311
log(† Fv ), 310±

log(Sj+1 ,α Fv ), 333±

log(Sj FVQ ), 337
log(† F), 310, 314
log(† F`×µ ), 311, 314
log(n,m F∆ ), 320
LabCusp(† D} ), 224
LabCusp± († D}± ), 241
LabCusp(† Dv ), 224
LabCusp(† D), 225
LabCusp± († Dv ), 240
LabCusp± († D ), 247
LabCusp± († Dt ), 246
LabCusp± (Π±v ), 268±
LabCusp (Πv ), 268b±LabCusp± (Πv ), 268± b
LabCusp (Πv ), 268
LocG (Π), 67
Lock (X), 62
LN , 129
L̈N , 132
M , 124
Mero(Z∞ ), 175
Mell , 14
Mell , 14
Mon, 168
mk , 12
mv , 12M, 156~
M († D} ), 186† }M~
κ ( D ), 186
M~(† D} ), 186∞κM~(† D} ), 186∞ κ×M~ († D} ), 186† ~M , 189† ~M∞ κ , 189† ~M∞ κ× , 189

a proof of the abc conjecture after Mochizuki

M(F v ), 182

M(I Q (α Dv`Q )), 347±

M(I Q (Sj+1 ,α Dv` )), 347±

M(I Q (Sj+1 Dv`Q )), 347
M(I Q (α Fv`×µ)), 348
Q
Q α
M(I ( FvQ )), 347±

M(I Q (Sj+1 ,α Fv`×µ )), 348±

M(I Q (Sj+1 ,α Fv )), 347±

M(I Q (Sj+1 Fv`×µ)), 348
Q±

M(I Q (Sj+1 FVQ )), 348±

M(I Q (Sj+1 FvQ )), 347
M∞ κ×v († Dv ), 191, 193
M∞ κv († Dv ), 191, 193† }M~
κ ( D ), 190
Mκv († Dv ), 191, 193M(k ∼ (G)), 110M(k ∼ (G)), 110M(k ∼ (G)G ), 106† ~Mmod , 190(‡→)† ~(
MMOD )α , 343(‡→)† ~
Mmod )α , 343(† ~Mκ , 190† ~( Mmod )hF> i , 303~l(† MMOD )α , 342~(† Mmod )hF> i , 303l~( Mmod )S> , 336
j† ~( Mmod )α , 343
† }
M~
, 301
mod ( D )hF>
l i
~
† }
Mmod ( D )j , 300n,◦M~
HT D- )α , 349
MOD (
n,◦
M~
HT D- )α , 349
mod (
† }
M~
mod ( D ), 190n,◦M~
HT D- )α , 349
mod (~Mmod († D} )hF> i , 301l~† }Mmod ( D )j , 300~
MMOD (n,◦ HT D- )α , 349†~

Mmod (n,◦ HT D- )α , 349~
Mmod (n,◦ HT D- )α , 349
MΘ(† F v ), 286
∗Ï
γ
(MΘ
∗ ) , 274MΘ
∗ (Π), 258
Mv († Dv ), 190, 193
Mκv († Dv ), 192†
Mv , 191, 193†
M∞ κv , 191, 193†
M∞ κ×v , 191, 193†
Mκv , 193†
Mκv , 192µ− , 226(µ− )X , 262(µ− )Ÿ , 262µlog (G), 106µlog
kI , 24µlog
α,vQ , 347, 348µlog
k , 23µlog (G), 110µ̆log (G), 110µlog
k , 23µlog, 347, 348
S± ,α,v
j+1

µlog
, 348
S± ,V
j+1

Q

, 347, 348µlog
S±
j+1 ,vQµN (A), 169µN (A∞ ), 172µN (MΘ
N ), 257µN (Π[µN ]), 153µQ/Z (Gk ), 84µQ/Z (M ), 86µQ/Z (O (ΠX )), 85µQ/Z (K), 14µZb (Gk ), 84µZb (M ), 86µZb (MΘ), 274∗Ï
µZb (O (ΠX )), 84

399

400

Go YamashitaµZb (K), 14µZb (P ), 183µZb (ΠX ), 75µZl (K), 14µZb (MΘ∗ (Π)), 258µZ/nZ (K), 14† }µΘb ( Π ), 185
Z
NG (H), 13
ordv , 12
Orn(Z), 92
Orn(Z, p), 91
Oµ (A∞ ), 172
O×µ (A∞ ), 172
O×µN (A∞ ), 172
OI (−), 222
OI (‡ A∞ ), 220
OI×µ (‡ A∞ ), 220
Out(G), 13
OC×Θ , 210v

OCΘ , 209v

OCΘ , 210v
Oα FvQ , 347
OS± ,α , 347
j+1

OS ±

Fv

j+1 F
vQ

, 347

Oµ (G), 171
O× (G), 171
O×µ (G), 171
O (G), 171
Ok , 12×OK̈· η̈ Θ , 134
Ok (ΠX ), 105
Ok (ΠX ), 86
Ok (ΠX ), 105
Ok×∼ (G), 110
Ok× , 12
Ok× (ΠX ), 105
Ok× (X), 108

Ok , 12
Ok (X), 108† 
Op , 188O⊥ (−), 222
O⊥ (‡ A∞ ), 220† e ~×
O , 188† e
Obp0 , 188
O× (A), 169
O×µ (ΠγvÏ ), 272
O× (ΠγvÏ ), 272
O× ∞ θenv (MΘ∗ (Π)), 264γι×O ∞ θ (ΠvÏ ), 272
O× ∞ θ(Π), 264
O× θι (ΠγvÏ ), 272
O (A), 169
O (Π), 260
Ogcp (G), 260
Ogcp (Π), 260
OA , 171
OŸ , 179
pv , 12prang , 110
prrad , 110
pro-CI , 11Prime(Cmod ), 179
Prime(M ), 179
Proc(† DJ ), 332
Proc(† D`J ), 332
Proc(† D`T ), 332
Proc(† DT ), 332
Proc(† FJ ), 332
Proc(† FT ), 332
π1 (C), 112
π1 (C 0 ), 112π, 12π(x), 375
π1rat († D~ ), 300{π1rat († D~ ) y M~(† D} )}hF> i , 301∞κla proof of the abc conjecture after Mochizuki{π1rat († D~ ) y M~(† D} )}j , 301∞κ{π1rat († D~ ) y † M~
} > , 303∞ κ hFl i
ΠvÏ , 269
ΠvÏ (MΘ), 274∗Ï
ΠvI , 267
Πv¨•t , 269
Πv•t , 269
Πv• , 267
ΠCF , 205
Π}± , 273
Πcorv , 268, 287† ~Π , 186(† Π~ )rat , 186
Πtemp
(MΘ
∗ ), 257Ctemp
ΠC (MΘ
N ), 257†
Πp0 , 188†
Πv , 190
ΠMΘ
, 274
∗
ΠMΘ
, 274
∗I
ΠMΘ , 274
∗Ï
ΠMΘ
, 256
M
Π[µN ], 153
ΠµN ,K , 153
ΠC , 146
ΠC , 146
ΠX , 145ellΠX , 145
Π±vI , 268
Πv•t , 269
Π±v• , 268
Π±v , 268, 287
ΠMΘ
, 257
N
cusp-cent
, 75
ΠU
†
rat
( Πv ) , 191
Πv , 179, 206, 287
ΠXF , 205
ΠX,H , 120
Πtemp
X,H , 120

Πtemp
, 127
X
temp ell(ΠX ) , 127(Πtemp)Θ , 127XΠΘ
X , 145(MΘΠtemp∗ ), 257X
temp
ΠX (MΘ
N ), 257temp
ΠX (MΘ∗ ), 257
Πtemp(MΘN ), 257Xell(ΠtempYN ) , 128Θ(ΠtempYN ) , 128(Πtemp)ell , 128Y(Πtemp)Θ , 128YΠtemp(Π), 262
Ÿ

Πtemp
(MΘ
∗ ), 257
Y
Πtemp
(MΘ
N ), 257Y(Π), 262
Πtemp
Y
ell
(Πtemp
ZN ) , 130Θ(Πtemp
ZN ) , 130† LC
φj , 229† NF
φ> , 228† Θ
φ> , 251† Θ
φ> , 228, 245† Θellφ> , 245
φNF
1 , 226
φNF•,v , 226
φNF
> , 226
φNF
j , 226
φNF
v , 226φΘ
> , 228ell
φ0Θ , 244
Θell
φ•,v
, 244
ell

Θ
φ±
, 245
Θell
φt , 244ellφΘ
v 0 , 244φΘ
j , 227Θ±
φ± , 244±φΘt , 243

401

402
±

φΘ
v t , 243φΘv j , 227
Φ, 176, 252
Φ0 , 175Φconst, 175
0R
Φ0 , 176Φbirat , 170
ΦCv` , 174
ΦCv , 174
ΦCenv († HT Θ ),v , 299
ΦCgau († HT Θ ),v , 299
ΦRCv` , 208
ΦCv` , 177, 210
ΦCtheta , 211
ΦCvΘ , 210
ΦCv , 177
Φconst , 176
Φ~ († D~ ), 187
ΦDenv († D` ),v , 295
ΦDgau († D` ),v , 296
ΦD (‡ D` ),v , 294
ΦLCFT , 261
ΦR , 171, 176ΦΘ , 258‡ NF
ψ> , 236‡ Θψ> , 237† NF
ψ> , 234† NF
ψj , 234† Θψ> , 251† Θ
ψ> , 233† Θellψ± , 249† Θψj , 233
ΨΞenv , 265(Ψ† Cv )hF> i , 285lΨCv , 279
Ψcns (Dv` ), 287
Ψcns (‡ D), 293Ψcns (‡ D)v , 293

Go YamashitaΨcns († D )t , 294
Ψcns (‡ F), 296
Ψcns (‡ F)v , 296
gpΨcns (‡ F), 314
Ψcns (Gv ), 287
Ψcns (MΘ∗ ), 278
Ψcns (MΘ, 283∗ )hF>l i†Ψcns ( Πv )t , 292
Ψcns (P iv ), 288
Ψcns (Πv ), 293Ψcns (Πv )hF> i , 289
l
`
Ψss
(D
),
288
cns
v
‡ `
(
D
), 293Ψss
cns
ss ‡ `
Ψcns ( D )v , 293‡ `Ψsscns ( F ), 297‡ `Ψsscns ( F )v , 297
Ψsscns (Gv ), 288
Ψsscns (Πv ), 288
‡ ` ×
Ψss
cns ( D )v , 294
Ψsscns (Uv ), 288
Ψcns (Πv )× , 288
Ψcns (Uv )× , 288
Ψcns († Uv )t , 292
Ψcns (Uv ), 288
Ψcns (Uv )hF> i , 289lΨcns (MΘ)
t∗ , 282R`Ψcns (Dv ), 288
ΨR
cns (Gv ), 288
Ψenv († D ), 295
Ψenv († D )v , 295
Ψιenv (MΘ∗ ), 277
Ψenv (MΘ∗ ), 277
Ψenv (Uv ), 289
ΨFenv († HT Θ ), 298
ΨFenv († HT Θ )v , 298
ΨFgau († F v ), 292
ΨFξ († F v ), 284

ΨFgau († HT Θ ), 298

a proof of the abc conjecture after Mochizuki

ΨFgau († HT Θ )v , 298
gpΨ† Fv , 308
ΨR† F ` , 291v
Ψss† F ` , 291Θ†∞ ΨFgau ( HT ), 298Θ†∞ ΨFgau ( HT )v , 298
log∞ ΨFLGP ((‡−→)†HT  )v , 339
log⊥(n,m−1−→)n,m
HT  )v , 341∞ ΨFLGP (

v
µN
†F
v

Ψ , 309Ψ∼† F , 308, 309v
Ψ† FvΘ , 292
Ψss† F , 291v

(Ψ† F )t , 292v(Ψ† F )hF> i , 292lvΨ† FjR ,v , 304‡ log †

ΨFlgp (( −→) HT  ), 344
log(‡−→)†HT  )v , 345
Ψ⊥Flgp (
log

ΨFLGP ((‡−→)† HT  )v , 339
log(n,m−1−→)n,m
Ψ⊥HT  )v , 341
FLGP (

ΨFvΘ ,α , 278
ΨFvΘ ,id , 278
ΨFvΘ , 278
ΨFξ († F v ), 284
ΨFξ († F v )×µ , 286
Ψgau († D ), 295
Ψgau († D )v , 295
Ψgau (MΘ∗ ), 282
Ψgau (Πv ), 289
Ψgau (Uv ), 289†∞ Ψenv ( D ), 295†∞ Ψenv ( D )v , 295†∞ Ψgau ( D ), 295†∞ Ψgau ( D )v , 295
Θι∞ Ψenv (M∗ ), 277
Θ∞ Ψenv (M∗ ), 277⊥ †∞ Ψenv ( D> )v , 322
Θ†∞ ΨFenv ( HT ), 298
Θ†∞ ΨFenv ( HT )v , 298
Θ⊥†∞ ΨFenv ( HT )v , 322†∞ ΨFξ ( F v ), 284∞ ΨFvΘ ,α , 278∞ ΨFvΘ ,id , 278∞ ΨFvΘ , 278†∞ ΨFξ ( F v ), 284Θ∞ Ψgau (M∗ ), 282n,◦HT D- )v , 340∞ ΨLGP (⊥n,◦
HT D- )v , 341∞ ΨLGP (Θ∞ Ψξ (M∗ ), 282(Ψ† Cv )×µ , 281

Ψ† Fv` , 290
Ψ† F , 290v(Ψ† FvΘ )×µ , 280

ΨLGP (n,◦ HT D- )v , 340n,◦Ψ⊥
HT D- )v , 341
LGP (Ψlog(† Fv ) , 308, 309,αΨlog(S±
j+1
Fv ), 338
Ψlog(α Fv ) , 333
R S± ,αΨlog(
j+1
Fv ), 338
× S± ,αΨ log( j+1 Fv ), 338
Ψξ (MΘ∗ ), 282
qL , 29
q , 176v

qv , 176
Q, 145R, 252
R ×C R, 253
RLCFT , 260
RΘ , 258†
R, 323† bad
R , 324n,◦R, 325n,◦ LGP
R
, 353

403

404

Rarc (G), 110
R≥0 (Dv` ), 288
R≥0 (‡ D` )v , 294
R≥0 (Gv ), 288(R`≥0 )v , 212, 213
R≥0 (Πv ), 288
R≥0 (Uv ), 288
Rnon (G), 106ρDenv ,v , 321
ρDv , 213‡
ρD ,v , 294(‡→)†
ρlgp,v , 344(‡→)†
ρLGP,v , 344†
ρ∆,v , 304†
ρenv,v , 305†
ρgau,v , 305†
ρv , 213, 216ρΘv , 211ρv , 209salg
Π , 153
, 154
salg
Ÿ
sΘ
F , 182sι , 147
sL , 29s≤ , 29sN , 130su
N , 180
su-gp
, 181
N
st
N , 180
st-gp
, 181
N
t-Π
sN , 182
striv
N , 180
sΘ
, 155
Ÿ
sΘ
, 256
MΘ
M
spl`v , 174, 177, 178
splΘ
v , 210Sect(Dy  GL ), 135

Go Yamashita

Seg(G), 109
Supp(a), 15
S±
j , 332
S>j , 332
τ , 143τN , 133θιbs (Πγ∗Ï ), 276
θιbs (Πγv ), 276
θbs (Πγv ), 276
θtbs (ΠγvÏ ), 276>

l , 282
θFenv
ι
θenv ((MΘ)γ ), 275
∗Ïγθιenv ((MΘ∗ ) ), 275
θ(MΘ∗ ), 264t)γ ), 275
θenv ((MΘ∗Ïιγ∞ θ bs (Πv ), 276γ∞ θ bs (Πv ), 276ιΘ γ∞ θ env ((M∗ ) ), 275γι∞ θ bs (Π∗Ï ), 276γt∞ θ bs (Πv Ï ), 276ιΘ γ∞ θ env ((M∗Ï ) ), 275Θ∞ θ env (M∗ ), 264tΘ γ∞ θ env ((M∗Ï ) ), 275γι∞ θ (Πv Ï ), 271∞ θ(Π), 264γt∞ θ (Πv Ï ), 271
θι (ΠγvÏ ), 271
θ(Π), 264
θt (ΠγvÏ ), 271
ϑ(x), 42, 375
Θα
, 278
v
Θv , 176

UP1 , 14Ucor
v , 287†
Uv , 290U±v , 287
Uv , 287

a proof of the abc conjecture after Mochizuki

U → † D} , 219
U → † D}± , 219
U, 134Ü, 135v, 27, 206
V , 124
V comb , 124
V ét , 124
V mult , 124
V new , 124V(† D} ), 187
V(† D}± ), 219
V(† D} ), 187, 219
V(† D}± ), 219
V(F ), 12
V(F )arc , 12
V(F )bad , 27, 205
V(F )good , 27, 205
V(F )non , 12
V(L)dist , 29
V(L)v , 12
Vgood
mod , 27
Vmod , 27
Vbad
mod , 27, 205
Vdist
mod , 29
Vgood
mod , 205†
Vmod , 187
Vdist
Q , 29V, 205
VhJi , 234
Varc , 205
Vbad , 205
Vgood , 205
VJ , 234
Vj , 234
Vnon , 205
V± , 242
V, 27

X arc , 15Ẍ, 143
XF , 205
X(Q)≤d , 15
X top , 91
X K , 205
X, 147X, 146Ẍ, 143
κ
X x M , 98
Xtop , 91
X(† D}± , w), 219† Θellξvt ,wt , 246† Θ±ξ,vt ,wt , 247† Θ±ξ,v,w , 247

Ξ, 315
Ξenv , 265
ΞLCFT , 261
ΞΘ , 259Ÿ , 132
YN , 128
ŸN , 132
Y , 149Ÿ , 149(ŸN )j , 132
Y, 128Ÿ, 132
YN , 129ŸN , 132
ZG (H), 13
Z̈N , 132
ZN , 130Z̈N , 132b 0 , 125
Z
Z, 127b 127
Z,†
ζ> , 230

405

406
†

ζ± , 247ζ , 294† Θellζt , 246† Θellζvt , 246†† Θ±
ζt , 247† Θ±ζvt , 247

Go Yamashitaa proof of the abc conjecture after Mochizuki

407

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