INTER-UNIVERSAL TEICHMÜLLER THEORY
AS AN ANABELIAN GATEWAY TO
DIOPHANTINE GEOMETRY AND ANALYTICNUMBER THEORY (MFO-RIMS23 VERSION)

Shinichi Mochizuki (RIMS, Kyoto University)
September 2023

https://www.kurims.kyoto-u.ac.jp/~motizuki/IUT%20as%20an%20
Anabelian%20Gateway%20(MFO-RIMS23%20Version).pdf§1. Overview via a famous quote of Poincaré§2. Galois groups as abstract groups: the example of the N -th power map§3. Analogy with the projective line/Riemann sphere§4. Brief preview of the Galois-orbit version of IUT

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SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)§1. Overview via a famous quote of Poincaré(cf. [Alien]; [EssLgc], §1.5; [EssLgc], Examples 2.4.7, 2.4.8, 3.3.2;[ClsIUT], §4)· In this talk, we give an overview of various aspects of IUT, manyof which may be regarded as striking examples of the famous
quote of Poincaré to the eﬀect that“mathematics is the art of giving the same name to diﬀerent things”.
— which was apparently originally motivated by various mathematical observations on the part of Poincaré concerning certain
remarkable similarities betw’n transformation group symmetries
of modular functions such as theta functions, on the one hand,and symmetry groups of the hyperbolic geometry of the upper
half-plane, on the other — all of which are closely related to IUT!
· Here, we note that there are three ways in which this quote ofPoincaré is related to IUT:
· the original motivation of Poincaré (mentioned above),· the key IUT notions of coricity/multiradiality (cf. §2, §3),· new applications of the Galois-orbit version of IUT (cf. §4).
· One important theme: it is possible to acquire a survey-level
understanding of IUT using only a knowledge of such elementary
topics as· the elem. notions of rings/ﬁelds/groups/monoids (cf. §2),· the elem. geom. of the proj. line/Riemann sphere (cf. §3).
· A more detailed exposition of IUT may be found in· the survey texts [Alien], [EssLgc], as well as in· the videos/slides available at the following URLs:
https://www.kurims.kyoto-u.ac.jp/~motizuki/ExpHoriz
IUT21/WS3/ExpHorizIUT21-InvitationIUT-notes.html
https://www.kurims.kyoto-u.ac.jp/~motizuki/ExpHoriz
IUT21/WS4/ExpHorizIUT21-IUTSummit-notes.html

IUT AS AN ANABELIAN GATEWAY§2. Galois groups as abstract groups: the example of the N -th
power map
(cf. [EssLgc], Example 2.4.8; [EssLgc], §3.2, §3.8)· Let R be an integral domain (e.g., Z ⊆ Q) equipped with theaction of a group G, (Z ) N ≥ 2. For simplicity, assume thatN = 1+· · ·+1 = 0 ∈ R; R has no nontrivial N -th roots of unity.
Write R ⊆ R for the multiplicative monoid R \ {0}. Then
let us observe that the N -th power map on R determines anisomorphism of multiplicative monoids equipped with actions
by G
G  R

∼→(R )N (⊆ R )  G

that does not arise from a ring homomorphism, i.e., it is clearly
not compatible with addition (cf. our assumption on N !).
· Let † R, ‡ R be two distinct copies of the integral domain R,
equipped with respective actions by two distinct copies † G, ‡ Gof the group G. We shall use similar notation for objects withlabels “†”, “‡” to the previously introduced notation. Then
one may use the isomorphism of multiplicative monoids arisingfrom the N -th power map discussed above to glue together†G  † R ⊇ ( † R  )N

∼

←

‡

R ⊆ ‡ R  ‡ G

the ring † R to the ring ‡ R along the multiplicative monoid∼(† R )N ← ‡ R . This gluing is compatible with the respective∼
actions of † G, ‡ G relative to the isomorphism † G → ‡ G given byforgetting the labels “†”, “‡”, but, since the N -th power map is∼
not compatible with addition (!), this isomorphism † G → ‡ G may
be regarded either as an isomorphism of (“coric”, i.e., invariant
with respect to the N -th power map) abstract groups (cf. thenotion of “inter-universality”, as discussed in [EssLgc], §3.2,§3.8!) or as an isomorphism of groups equipped with actions oncertain multiplicative monoids, but not as an isomorphism of(“Galois” — cf. the inner automorphism indeterminacies of
SGA1!) groups equipped with actions on rings/ﬁelds.

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SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)· The problem of describing (certain portions of the) ring structureof † R in terms of the ring structure of ‡ R — in a fashion thatis compatible with the gluing and via a single algorithm that may
be applied to the common (cf. logical AND ∧!) glued data to reconstruct simultaneously (certain portions of) the ring structuresof both † R and ‡ R, up to suitable relatively mild indeterminacies(cf. the theory of crystals!) — seems (at ﬁrst glance/in general)
to be hopelessly intractable (cf. the case of Z)!
One well-known elementary example: when N = p, working
modulo p (cf. indeterminacies, analogy with crystals!), where
there is a common ring structure that is compatible with thep-th power map!
This is precisely what is achieved in IUT (cf. quote of Poincaré!)by means of the multiradial algorithm for the Θ-pilot via· anabelian geometry (cf. the abstract groups † G, ‡ G!);
· the p-adic/complex logarithm, theta functions;· Kummer theory, to relate Frob.-/étale-like versions of objects.
· Main point:
The multiplicative monoid and abstract group structures (but not
the ring structures!) are common (cf. “logical AND ∧!”) to †, ‡.
On the other hand, once one deletes the labels “†”, “‡” to secure
a “common R”, one obtains a meaningless situation, where thecommon glued data may be related via “†” OR “∨” via “‡” tothe common R, but not simultaneously to both!
· When R = Z (or, in fact, more generally, the ring of integers“OF ” of a number ﬁeld F — cf. the multiplicative norm map
NF/Q : F → Q), one may consider the “height”
log(|x|) ∈ R
for 0 = x ∈ Z. Then the N -th power map of (i), (ii) corresponds,
after passing to heights, to multiplying real numbers by N ; themultiradial algorithm corresponds to saying that the height is
unaﬀected (up to a mild error term!) by multiplication by N ,hence that the height is bounded!

IUT AS AN ANABELIAN GATEWAY§3. Analogy with the projective line/Riemann sphere(cf. [EssLgc], Example 2.4.7; [Alien], §3.1; [EssLgc], §1.5, §3.5, §3.8,§3.9, §3.10)· Let k be a ﬁeld (in fact, could be taken to be an arbitrary ring),
R a k-algebra. Denote units of a ring by a superscript “×”. Write
A1 for the aﬃne line Spec(k[T ]) over k,
Gm for the open subscheme Spec(k[T, T −1 ]) of A1 obtained byremoving the origin.
Recall that A1 is equipped with a well-known natural structureof ring scheme over k, while Gm is equipped with a well-known
natural structure of (multiplicative) group scheme over k. Moreover, we observe that the standard coordinate T on A1 and Gm
determines natural bijections:∼

A1 (R) → R,∼

Gm (R) → R×

· Write † A1 , ‡ A1 for the k-ring schemes given by copies of A1
equipped with labels “†”, “‡”. Observe that there exists a∼
unique isomorphism of k-ring schemes † A1 → ‡ A1 ; moreover,there exists a unique isomorphism of k-group schemes∼(−)−1 : † Gm → ‡ Gm
that maps † T → ‡ T −1 . Note that (−)−1 does not extend to an∼
isomorphism † A1 → ‡ A1 and is clearly not compatible with the
k-ring scheme structures of † A1 (⊇ † Gm ), ‡ A1 (⊇ ‡ Gm ).

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SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)· The standard construction of the projective line P1 may be understood as the result of gluing † A1 to ‡ A1 along the isomorphism
†

1

A

⊇

†

(−)−1 ‡Gm −→G m ⊆ ‡ A1

— i.e., at the level of R-rational points
†

R ⊇

†

R

× (−)

−1

−→ ‡ R× ⊆ ‡ R— where  R =  A1 (R),  R× =  Gm (R), for  ∈ {†, ‡} (cf.
the gluing situation discussed in §2, where “(−)−1 ” correspondsto “(−)N ”!). Thus, relative to this gluing, we observe that there
exists a single rational function on the copy of “Gm ” that appears
in the gluing that is simultaneously equal to the rational function†
T on † A1 AND [cf. “∧”!] to the rational function ‡ T −1 on ‡ A1 .
· Summary:
The standard construction of the projective line may be regarded
as consisting of a gluing of two ring schemes along an isomorphismof multiplicative group schemes that is not compatible with thering scheme structures on either side of the gluing.
Finally, we observe that if, in the gluing under discussion, one
arbitrarily deletes the distinct labels “†”, “‡” (e.g., on the grounds
that both ring schemes represent “THE” structure sheaf “OX ”
of a k-scheme X!), then the resulting “gluing without labels”
amounts to a gluing of a single copy of A1 to itself that maps thestandard coordinate T on A1 (regarded, say, as a rational function on A1 ) to T −1 . That is to say, such a deletion of labels (evenwhen restricted to the (abstractly isomorphic) multiplicative
monoids † T Z , ‡ T Z !) immediately results in a contradiction (i.e.,since T = T −1 !), unless one passes to some sort of quotient of
A1 . On the other hand, passing to such a quotient amounts,
from a foundational/logical point of view, to the introduction ofsome sort of indeterminacy, i.e., to the consideration of some
sort of collection of possibilities [cf. “∨”!].

IUT AS AN ANABELIAN GATEWAY· When k = C (i.e., the complex number ﬁeld), one may think of
the projective line P1 as the Riemann sphere S2 equipped with
the Fubini-Study metric and of the gluing under discussion as
the gluing, along the equator E, of the northern hemisphere H+
to the southern hemisphere H− . Then the discussion above of
the standard coordinates “† T ”, “‡ T ” translates into the following(at ﬁrst glance, self-contradictory!) phenomenon:
an oriented ﬂow along the equator — which may be thoughtof physically as a sort of east-to-west wind current — appears
simultaneously to be ﬂowing in the clockwise direction, fromthe point of view of H+ ⊆ S2 , AND in the counterclockwise
direction, from the point of view of H− ⊆ S2 .
In particular, if one arbitrarily deletes the labels “+”, “−” andidentiﬁes H− with H+ , then one does indeed literally obtain acontradiction. On the other hand, one may relate H− to H+(not by such an arbitrary deletion of labels (!), but rather) byapplying
the metric/geodesic geometry of S2 — i.e., by considering
the geodesic ﬂow along great circles/lines of longitude — torepresent, up to a relatively mild distortion, the entirety of S2 ,i.e., including H− ⊆ S2 , as a sort of deformation/displacementof H+ (cf. the point of view of cartography!).
It is precisely this metric/geodesic approach that corresponds tothe anabelian geom.-based multiradial algorithm for the Θ-pilot
in IUT (cf. the analogy discussed in [Alien], §3.1, (iv), (v), as
well as in [EssLgc], §3.5, §3.10, between multiradiality andconnections/parallel transport/crystals!).

northern hemisphere H+——— equator E ———southern hemisphere H−

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SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)· In this context, it is important to remember that, just like SGA,
IUT is formulated entirely in the framework of“ZFCG”(i.e., ZFC + Grothendieck’s axiom on the existence of universes),
especially when considering various set-theoretic/foundational
subtleties (?) of “gluing” operations in IUT (cf. [EssLgc], §1.5,§3.8, §3.9, as well as [EssLgc], §3.10, especially the discussionof “log-shift adjustment” in (Stp 7)):· gluing is performed at the abstract level of diagrams (cf. graphof groups/anabelioids), is not equipped with an embedding intosome familiar ambient space (like a sphere);· output of reconstruction algorithms only well-deﬁned at thelevel of objects up to isomorphism (+ suitable indeterminacies),
i.e., “types/packages of data” (such as groups, rings, monoids,
diagrams, etc.) called “species”,=⇒

hence the (subtle?) importance of closed loops
in order to obtain set-theoretic comparisonsthat are not possible at intermediate steps... note importance of working with “types/packages of data”(cf., e.g., the diagrams referred to above!) — as opposedto certain particular underlying sets of interest! — cf. theclassical functoriality of resolutions in cohomology, as well
as algebraic closures of ﬁelds up to conjugacy indeterminacies(which become unnecessary, e.g., if one considers norms!)... note importance of working with “closed loops” —cf. norms in Galois theory, as well as the classical theory ofanalytic continuation/Riemann surfaces (which isreminiscent of the classical Riemann-Weierstrass dispute!),the geodesic completeness/closed geodesics of the sphere.

IUT AS AN ANABELIAN GATEWAY§4. Brief preview of the Galois-orbit version of IUT(cf. “Expanding Horizons” videos/slides cited in §1; [GSCsp];[AnPf]; [Alien], §3.11, (iii))· Brief preview of various new enhanced versions of IUT, which is
closely related to recent progress (joint work in progress!) on the
Section Conjecture (“SC”):· [GSCsp]: reduces, using RNS (cf. [RNSPM]), together witha result of Stoll, geometricity of an arbitrary Galois
section of a hyperbolic curve over a number ﬁeld to· local geometricity at each nonarchimedean prime, plus
· 3 global conditions, which correspond, respectively, to
3 new enhanced versions of IUT!
· [GSCsp]+[AnPf]: substantial progress on the p-adic SC that isclosely related to the use of Raynaud-Tamagawa“new-ordinariness” in the theory of RNS(cf. [RNSPM]), which functions as a sort oflocal analogue of IUT — via the analogy“N · (−) ≈ (−)” ←→ “Norm(−) = (−)”!
· One such new enhanced version of IUT is the
Galois-orbit version of IUT (GalOrbIUT), which implies:· one of the 3 global conditions mentioned above in the discussionof the Section Conjecture (“intersection-ﬁniteness”);· nonexistence of Siegel zeroes of Dirichlet L-functions associatedto imaginary quadratic number ﬁelds (i.e., by applying the workof Colmez/Granville-Stark/Táfula);· numerically stronger version of abc/Szpiro inequalities.
· That is to say, we obtain three a priori diﬀerent applications to· anabelian geometry (“local-global” Section Conjecture),· analytic number theory (nonexistence of Siegel zeroes),· diophantine geometry (abc/Szpiro inequalities)— a striking example of Poincaré’s quote, i.e., all three are essentially the same mathematical phenomenon of bounding heights,
i.e., bounding “local denominators”!

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SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)· Here, the local-global Section Conjecture application is also
noteworthy in that· it exhibits IUT as “anabelian geometry applied to obtain
more anabelian geometry!” (less psychologically/intuitively
surprising than the other two applications!);· it is technically the most diﬃcult/essential (!) of the three, i.e.,to the extent that the other two applications may be thoughtof, to a substantial extent, as being “inessential by-products”;· the historical point of view (cf., e.g., of Grothendieck’s famous“letter to Faltings”) suggests (without any proof !) that theSection Conjecture might imply results in diophantine geometry (such as the Mordell Conjecture).
· In this context, it is interesting to recall (cf. [Alien], §3.11, (iii))
that the essential content of anabelian geometry may be understood as a sort of “conceptual translation” of the abc inequality:· anabelian geometry:
addition reconstructed from multiplication[i.e., addition “dominated by” multiplication!]· abc inequality:
height (“additive size”)

conductor (“multiplicative size”)[i.e., addition “dominated by” multiplication!]. . . cf. conceptual Weil Conjectures versus numerical inequalities
for the number of rational points of a variety over a ﬁnite ﬁeld!

IUT AS AN ANABELIAN GATEWAY

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References[IUAni1] E. Farcot, I. Fesenko, S. Mochizuki, The Multiradial Representation ofInter-universal Teichmüller Theory, animation available at the following URL:
https://www.kurims.kyoto-u.ac.jp/~motizuki/IUTanimation-Thm-A-black.wmv[IUAni2] E. Farcot, I. Fesenko, S. Mochizuki, Computation of the log-volume ofthe q-pilot via the multiradial representation, animation available at thefollowing URL:
https://www.kurims.kyoto-u.ac.jp/~motizuki/202001%20Computation%20of%20q-pilot%20(animation).mp4[IUTchI] S. Mochizuki, Inter-universal Teichmüller Theory I: Construction of
Hodge Theaters, Publ. Res. Inst. Math. Sci. 57 (2021), pp. 3-207.
[IUTchII] S. Mochizuki, Inter-universal Teichmüller Theory II: Hodge-Arakelovtheoretic Evaluation, Publ. Res. Inst. Math. Sci. 57 (2021), pp. 209-401.
[IUTchIII] S. Mochizuki, Inter-universal Teichmüller Theory III: Canonical Splittings of the Log-theta-lattice, Publ. Res. Inst. Math. Sci. 57 (2021),
pp. 403-626.
[IUTchIV] S. Mochizuki, Inter-universal Teichmüller Theory IV: Log-volume Computations and Set-theoretic Foundations, Publ. Res. Inst. Math. Sci. 57(2021), pp. 627-723.
[ExpEst] S. Mochizuki, I. Fesenko, Y. Hoshi, A. Minamide, W. Porowski, Explicit
Estimates in Inter-universal Teichmüller Theory, Kodai Math. J. 45(2022), pp. 175-236.
[Pano] S. Mochizuki, A Panoramic Overview of Inter-universal Teichmüller
Theory, Algebraic number theory and related topics 2012, RIMS Kōkyūroku Bessatsu B51, Res. Inst. Math. Sci. (RIMS), Kyoto (2014), pp.
301-345.

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SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)[Alien] S. Mochizuki, The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmüller Theory, Inter-universal Teichmuller Theory Summit 2016, RIMS Kōkyūroku Bessatsu B84, Res.
Inst. Math. Sci. (RIMS), Kyoto (2021), pp. 23-192; available at thefollowing URL:
https://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20
Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf[EssLgc] S. Mochizuki, On the essential logical structure of inter-universal Teichmüller theory in terms of logical AND “∧”/logical OR “∨” relations:
Report on the occasion of the publication of the four main papers on
inter-universal Teichmüller theory, preprint available at the following
URL:
https://www.kurims.kyoto-u.ac.jp/~motizuki/Essential%20Logical%20Structure%20of%20Inter-universal%20Teichmuller%20Theory.pdf[ClsIUT] S. Mochizuki, Classical roots of inter-universal Teichmüller theory, lecture notes for Berkeley Colloquium talk given in November 2020, available at the following URL:
https://www.kurims.kyoto-u.ac.jp/~motizuki/2020-11%20Classical%20roots%20of%20IUT.pdf[RNSPM] S. Mochizuki, S. Tsujimura, Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility for p-adic Hyperbolic Curves, RIMS
Preprint 1974 (June 2023).
[GSCsp] S. Mochizuki, Y. Hoshi, Arithmetic cuspidalization of Galois sectionsof hyperbolic curves, manuscript in preparation.
[AnPf] S. Mochizuki, Y. Hoshi, S. Tsujimura, G. Yamashita, Anabelian geometry over complete discrete valuation rings with perfect residue ﬁelds,
manuscript in preparation.