REPORT ON DISCUSSIONS, HELD DURING THE
PERIOD MARCH 15 – 20, 2018, CONCERNINGINTER-UNIVERSAL TEICHMÜLLER THEORY (IUTCH)

Shinichi Mochizuki
February 2019

§1. The present document is a report on discussions held during the period March
15 – 20, 2018, concerning inter-universal Teichmüller theory (IUTch). These
discussions were held in a seminar room on the ﬁfth ﬂoor of Maskawa Hall, Kyoto
University, according to the following schedule:· March 15 (Thurs.): 2PM ∼ between 5PM and 6PM,
· March 16 (Fri.): 10AM ∼ between 5PM and 6PM,
· March 17 (Sat.): 10AM ∼ between 5PM and 6PM,
· March 19 (Mon.): 10AM ∼ between 5PM and 6PM,
· March 20 (Tues.): 10AM ∼ between 5PM and 6PM.
(On the days when the discussions began at 10AM, there was a lunch break for
one and a half to two hours.) Participation in these discussions was restricted
to the following mathematicians (listed in order of age): Peter Scholze, Yuichiro
Hoshi, Jakob Stix, and Shinichi Mochizuki. All four mathematicians participatedin all of the sessions listed above (except for Hoshi, who was absent on March
16). The existence of these discussions was kept conﬁdential until the conclusionof the ﬁnal session. From an organizational point of view, the discussions took theform of “negotations” between two “teams”: one team (HM), consisting of Hoshi
and Mochizuki, played the role of explaining various aspects of IUTch; the otherteam (SS), consisting of Scholze and Stix, played the role of challenging various
aspects of the explanations of HM. Most of the sessions were conducted in thefollowing format: Mochizuki would stand and explain various aspects of IUTch,
often supplementing oral explanations by writing on whiteboards using markers invarious colors; the other participants remained seated, for the most part, but would,
at times, make questions or comments or brieﬂy stand to write on the whiteboards.
§2. Scholze has, for some time, taken a somewhat negative position concerning
IUTch, and his position, and indeed the position of SS, remained negative evenafter the March discussions. My own conclusion, and indeed the conclusion of HM,
after engaging in the March discussions, is as follows:
The negative position of SS is a consequence of certain fundamental
misunderstandings (to be explained in more detail in the remainder ofTypeset by AMS-TEX

1

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SHINICHI MOCHIZUKI

the present report — cf. §17 for a brief summary) on the part of SS
concerning IUTch, and, in particular, does not imply the existence ofany ﬂaws whatsoever in IUTch.
The essential gist of these misunderstandings — many of which center arounderroneous attempts to “simplify” IUTeich — may be summarized very roughly
as follows:(Smm) Suppose that A and B are positive real numbers, which are deﬁned so as
to satisfy the relation−2B = −A
(which corresponds to the Θ-link). One then proves a theorem−2B ≤ −2A + 1(which corresponds to the multiradial representation of [IUTchIII],
Theorem 3.11). This theorem, together with the above deﬁning relation,
implies a bound on A
−A ≤ −2A + 1,

i.e.,

A ≤ 1

(which corresponds to [IUTchIII], Corollary 3.12). From the point of viewof this (very rough!) summary of IUTch, the misunderstandings of SS
amount to the assertion thatthe theory remains essentially unaﬀected even if one takes A = B,
which implies (in light of the above deﬁning relation) that A = B = 0,
in contradiction to the initial assumption that A and B are positive real
numbers. In fact, however, the essential content (i.e., main results) of
IUTch fail(s) to hold under the assumption “A = B”; moreover, the“contradiction” A = B = 0 is nothing more than a superﬁcial consequenceof the extraneous assumption “A = B” and, in particular, does not imply
the existence of any ﬂaws whatsoever in IUTch. (We refer to (SSIdEx),(ModEll), (HstMod) below for a “slightly less rough” explanation of theessential logical structure of an issue that is closely related to the extraneous assumption “A = B” in terms of· complex structures on real vector spaces
or, alternatively (and essentially equivalently), in terms of the well-knownclassical theory of· moduli of complex elliptic curves.
Additional comparisons with well-known classical topics such as· the invariance of heights of elliptic curves over number ﬁelds
with respect to isogeny,· Grothendieck’s deﬁnition of the notion of a connection, and· the diﬀerential geometry surrounding SL2 (R)may be found in §16.)

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)3

Indeed, in the present context, it is perhaps useful to recall the following well-known
generalities concerning logical reasoning:(GLR1) Given any mathematical argument, it is always easy to derive a contradiction by arbitrarily identifying mathematical objects that must be
regarded as distinct in the situation discussed in the argument. On theother hand, this does not, by any means, imply the existence of any
logical ﬂaws in the original mathematical argument!
(GLR2) Put another way, the correct interpretation of the contradiction
obtained in (GLR1) is — not the conclusion that the original argument,
in which the arbitrary identiﬁcations of (GLR1) were not in force, has logical ﬂaws (!), but rather — the conclusion that the contradiction obtainedin (GLR1) implies that the distinct mathematical objects that were arbitrarily identiﬁed are indeed distinct, i.e., must be treated (in order, forinstance, to arrive at an accurate understanding of the original argument!)
as distinct mathematical objects!
It is most unfortunate indeed that the March discussions were insuﬃcient fromthe point of view of overcoming these misunderstandings. On the other hand, myown experience over the past six years with regard to exposing IUTch to othermathematicians is that this sort of short period (roughly a week) is never suﬃcient,i.e., that
substantial progress in understanding IUTch always requires discussions
over an extended period of time, typically on the order of months.
Indeed, the issue of lack of time became especially conspicuous during the afternoonof the ﬁnal day of discussions. Typically, short periods of interaction center aroundreacting in real time and do not leave participants the time to reﬂect deeply on
various aspects of the mathematics under discussion. This sort of deep reﬂection,
which is absolutely necessary to achieve fundamental progress in understanding,can only occur in situations where the participants are aﬀorded the opportunity
to think at their leisure and forget about any time or deadline factors. (In thiscontext, it is perhaps of interest to note that Scholze contacted me in May 2015 bye-mail concerning a question he had regarding the non-commutativity of the logtheta-lattice in IUTch (i.e., in eﬀect, “(Ind3)”). This contact resulted in a shortseries of e-mail exchanges in May 2015, in which I addressed his (somewhat vaguely
worded) question as best I could, but this did not satisfy him at the time. On theother hand, the March 2018 discussions centered around quite diﬀerent issues, suchas (Ind1, 2), as will be described in detail below.)§3. On the other hand, it seems that the March discussions may in fact be regarded
as constituting substantial progress in the following sense. Prior to the March
discussions, (at least to my knowledge)
negative positions concerning IUTch were always discussed in highly nonmathematical terms, i.e., by focusing on various aspects of the situationthat were quite far removed from any sort of detailed, well-deﬁned,
mathematically substantive content.

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SHINICHI MOCHIZUKI

That is to say, although it is most regrettable that it was not possible to resolve thefundamental misunderstandings of SS during the March discussions, nevertheless,the March discussions were highly meaningful in that, to my knowledge,
they constitute the ﬁrst detailed, well-deﬁned, mathematically substantive discussions concerning negative positions with regard to IUTch.
Put another way, as a result of the absence, during the past six years, of suchdetailed, well-deﬁned, mathematically substantive discussions concerning negative
positions with regard to IUTch, the highly non-mathematical tone that has
appeared, up till now, in statements by mathematicians critical of IUTch has had
the eﬀect of giving the impression thatany existing criticism of IUTch is entirely devoid of any substantive mathematical content, i.e., based on entirely non-mathematicalconsiderations.
On the other hand, as I have emphasized on numerous occasions throughout thepast six years (cf., e.g., [Rpt2014], (7); [QC2016], (4), (5)),
the only way to make meaningful, substantive progress with regard to differences of opinion concerning IUTch is by means of discussions concerning detailed, well-deﬁned, mathematically substantive content.
From this point of view, it seems most desirable that the mathematical content
discussed in the present report be made available for further discussion by all interested mathematicians (i.e., not just the participants in the March discussions!),in the hope that the present report might play the role of serving to stimulate further detailed, well-deﬁned, mathematically substantive dialogue concerning issues
related to IUTch. In this context, it seems also of fundamental importance to keep
the following historical point of view in mind:
It is only by supplementing negative positions concerning IUTch with detailed, mathematically substantive, accessible records of the mathematical content underlying such negative positions that humanity can
avoid creating an unfortunate situation — i.e., of the sort that arose concerning the “proof” asserted by Fermat of “Fermat’s Last Theorem”! —in which an accurate evaluation of the substantive mathematical content
underlying such negative positions will remain impossible indeﬁnitely.
Other historical examples of interest in the present context — i.e., especially fromthe point of view of the explicit documentation of the fascinating phenomenonof transition from social rejection to social acceptance of a scientiﬁc theory— include such well-documented cases as the defense by Galileo of the theory of
heliocentrism (cf., e.g., [Gll]) and the defense by Einstein of the theory of relativity(cf., e.g., [Rtv]). Again, the following (essentially tautological!) fact cannot be
overemphasized:
It is possible for us today to study in detail this fascinating process ofrefutation by Galileo of various denials of the theory of heliocentrism orrefutation by Einstein of various denials — e.g., in the form of various alleged “paradoxes” — of the theory of relativity precisely because of the existence of detailed, explicit, accessible records of the logical structure

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)5underlying arguments produced by scholars on either side of the debate,i.e., both scholars who denied (at times with a surprising degree of hostility!) the validity of these theories and scholars involved in the refutationof these denials.
Further remarks concerning the involvement of other mathematicians may be foundin §18 below.
§4. Before proceeding to our exposition of the mathematical content of the March
discussions, we pause to list brieﬂy the various topics that seem to have been themain themes of the March discussions:(T1) the treatment, in IUTch, of histories of various operations performedon mathematical objects;(T2) the treatment, in IUTch, of types of mathematical objects (i.e.,“species”, in the sense of [IUTchIV], §3);
(T3) the “id-version”, i.e., a variant of IUTch obtained by identifying
various copies (such as Frobenius-like and étale-like versions, as well as
copies appearing in diﬀerent Hodge theaters) of familiar objects by meansof the “identity morphism” (cf. the discussion of §2!);(T4) opposition by SS to the use of poly-morphisms in IUTch(T4-1) as a matter of taste,(T4-2) on the grounds that the introduction of indeterminacies such as(Ind1, 2) seemed to SS to be logically unnecessary or “meaningless”;(T5) opposition by SS to the use of labels in IUTch to distinguish distinct
copies of various familiar objects(T5-1) as a matter of taste,(T5-2) on the grounds that the use of such labels seemed to SS to belogically unnecessary or “meaningless”;(T6) refusal, on the part of SS, to consider various key ideas and notions of IUTch such as distinct arithmetic holomorphic structures (i.e., inessence, distinct ring structures);(T7) the issue of simpliﬁcation;(T8) occasional misinterpretation by SS of statements by HM of the form
(T8-1) as deﬁnitive declarations of (T8-2):(T8-1) “you may consider such and such a modiﬁed version of IUTch
if you wish, but it is by no means clear that the essential content
of IUTch is valid for such a modiﬁed version”,(T8-2) “you may consider such and such a modiﬁed version of IUTch
if you wish, and, moreover, I aﬃrm that the essential content of
IUTch is completely valid for such a modiﬁed version”;

6

SHINICHI MOCHIZUKI(T9) detailed exposition of the multiradial representation of [IUTchIII],
Theorem 3.11.
Of these themes, it seems that (T1) and (T2) are, in some sense, the most essential
or fundamental. Then (T3), (T4), (T5) may be understood as being essentially“corollaries” of (T1), (T2). Moreover, (T4), (T5) may be understood as particular
aspects of (T3). By contrast, (T6), (T7), (T8) refer to certain procedural aspectsof the March discussions, which, nonetheless, had a substantial inﬂuence on themathematical content of the discussions. Here, it should be mentioned that theissue of simpliﬁcation mentioned in (T7) refers to the general issue of just what sortof simpliﬁcations in the mathematical content under discussion are mathematicallycorrect, meaningful, and helpful (from the point of view eliminating details that are
unnecessary and irrelevant to the central points at issue). Thus, one aspect of thisgeneral theme (T7) is the erroneous attempts by SS to simplify IUTch (cf. (T3),(T4), (T5), (T6)) to such an extent that it leads to meaningless contradictions (assummarized in (Smm), (GLR1), (GLR2)). On the other hand, (cf. the discussionof §8, §13 below), (T7) also refers to situations where the techniques introduced
in IUTch actually do yield simpliﬁcations, relative to more naive approaches tovarious situations that arise in IUTch. Finally, (T9) proceeded relatively smoothly,in the sense that it consisted essentially of a straightforward exposition of thecontent of the multiradial algorithms of [IUTchIII], Theorem 3.11. That is to say,the position of SS with regard to (T9) was that they did not dispute the validityof these algorithms, but rather the non-triviality, or substantive content, of thesealgorithms, on account of their positions with regard to (T1), (T2), (T3), (T4),(T5), (T6).
§5. In some sense, it seems that the logical starting point of the diﬀerences inpoint of view between SS and HM (and hence of various fundamental misunderstandings of SS) may be understood as a consequence of the diﬀerence between
the following two approaches to considering histories of operations performed onmathematical objects in a given discussion of mathematics (cf. (T1)):(H1) The conventional approach to histories of operations: The conventional approach that is typically taken, with regard to histories of operations performed on various mathematical objects, consists of regarding
all of these operations as being embedded within a single history.
•

→

···

• → • → • →

•




..

→


.

···
..

.

In this approach, all previously executed operations are regarded as
being permanently accessible, regardless of the content of subsequent
operations.
(H2) The approach taken in IUTch to histories of operations: By contrast, the approach to treating such histories of operations that is taken

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)7

throughout IUTch involves the frequent use of re-initialization operations (i.e., “”)
•

→

···

• → • → •



•




..

→ ···


.

..

.

— that is to say, situations where one forgets the previous history ofsome object (such as, for instance, some previously endowed mathematical structure on the object) and regards this previous history as being
inaccessible in subsequent discussions. Such re-initialization operations
then require the use of distinct labels (cf. (T5)) to denote the “versions”
of an object that arise prior to and subsequent to the execution of such
re-initialization operations. Another aspect of central importance in thecontext of such re-initialization operations is the explicit speciﬁcationof the type of mathematical objects (i.e., “species”, in the terminology of [IUTchIV], §3 — cf. (T2)) that one considers, for instance, prior toand subsequent to the execution of such re-initialization operations (e.g.,“groups of automorphisms of some speciﬁed ﬁeld” versus “abstract topological groups”).
§6. The two main examples in IUTch (cf. §15 below for a slightly more detailed— though still quite brief! — review of certain aspects of IUTch) of the sort of
re-initialization operation discussed in (H2) occur in the context of the gluing
operations that arise in the deﬁnition of the log- and Θ-links:(HEx1) The log-link: Here, the gluing operation consists of regarding the“Π’s”(i.e., in the notation of [IUTchI], Fig. I1.2; [IUTchI], Deﬁnition 3.1, (e),(f), “Πv ’s”, for v ∈ V) on either side of the log-link as being known only as
abstract topological groups, i.e., of forgetting the way in which theseabstract topological groups are conventionally related to ring/scheme theory, namely, as groups of ﬁeld automorphisms.
(HEx2) The Θ-link: Here, the gluing operation consists of portions:(HEx2-1) regarding the“G’s”(i.e., in the notation of [IUTchI], Fig. I1.2; [IUTchI], Deﬁnition
3.1, (e), “Gv ’s”, for v ∈ V) on either side of the Θ-link as being
known only as abstract topological groups, i.e., forgetting
the way in which these abstract topological groups are conventionally related to ring/scheme theory, namely, as groups of ﬁeld
automorphisms;

8

SHINICHI MOCHIZUKI(HEx2-2) regarding the“O×μ ’s”(i.e., in the notation of [IUTchI], Fig. I1.2, “OF×μ ’s”, for v ∈v

Vnon ) on either side of the Θ-link as being known only as abstract topological monoids (that are also equipped with certain Gv -actions, as well as certain collections of submodules, as
discussed in [IUTchII], Deﬁnition 4.9, (i), (ii), (iii), (iv), (vi),(vii)), i.e., forgetting the way in which these abstract topological monoids are conventionally related to ring/scheme theory,
namely, as subquotients of multiplicative groups of certain ﬁelds.
In this context, we note that there are several diﬀerences between (HEx1) and(HEx2-1). Indeed, in the case of the Θ-link, although “G” (cf. (HEx2-1)) is notregarded as a group of ﬁeld automorphisms, it is regarded as a group of automorphisms of the abstract topological monoid “O×μ ” (i.e., equipped with a certain collection of submodules — cf. (HEx2-2)). In particular, the Θ-link is constructed by considering the full poly-isomorphism between distinct copies of thedata “G  O×μ ” (where O×μ is always regarded as being equipped with a certain
collection of submodules). This full poly-isomorphism gives rise to the indeterminacies (Ind1, 2), which play a central role in IUTch. By contrast, in the case of the
log-link, the gluing between Frobenius-like data in the codomain and domain (ofthe log-link) involves a speciﬁc bijection between topological sets (arising, respectively, from multiplicative and additive structures in the domain and codomainof the log-link). That is to say, (unlike the case of the Θ-link) one does not performthe gluing in the log-link by considering, say, full poly-isomorphisms between topological sets. A closely related fact is the fact that these topological sets are neverused as vertical cores, i.e., as invariants (up to indeterminate isomorphisms of
topological sets) with respect to the log-link. Nevertheless, these topological sets
(regarded up to indeterminate isomorphisms of topological sets) may, in some sense,be thought of as appearing implicitly, in the sense that the point of view in which
one thinks of the “Π’s” as abstract topological groups is closely related to the pointof view in which one thinks of the “Π’s” as groups of automorphisms of these
topological sets that appear in the gluing of Frobenius-like data that occurs in
the case of the log-link. This state of aﬀairs diﬀers somewhat from the situationin the case of the Θ-link, in which the O×μ ’s (regarded up to the indeterminacy(Ind2)) do indeed play an important role as horizontal cores, i.e., as invariants
with respect to the Θ-link.
§7. Before proceeding, it is of interest to note, in the context of §5, §6, that the sortof re-initialization operations discussed in (H2) — i.e., operations that consist offorgetting mathematical structures in such a way that the forgotten mathematicalstructures cannot (at least in any sort of a priori, or general nonsense, sense) berecovered — in fact occur in various contexts of mathematics other than IUTch. Indeed, the following two “classical” examples are of particular interest in the context
of IUTch:(HC1) Classical complex Teichmüller theory: The fundamental set-up
of classical complex Teichmüller theory consists of considering distinct

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)9holomorphic (i.e., Riemann surface) structures on a given topological (orreal analytic) surface, i.e., of forgetting the way in which the Riemannsurface (i.e., holomorphic structure) gives rise to such a topological (orreal analytic) surface.
(HC2) Anabelian geometry: The fundamental set-up of anabelian geometry
consists of considering various topological (typically proﬁnite) groups thatarise as various types of arithmetic (i.e., étale) fundamental groups ofschemes or Galois groups of ﬁelds as abstract topological groups, i.e., offorgetting the way in which such topological groups arose as arithmetic
fundamental groups or Galois groups.
Indeed, it appears that SS did not dispute the logical feasibility or consistency of
the approach of (H2) in the context of IUTch. Rather, their main objection to
the approach of (H2) in the context of IUTch appears to be to the eﬀect thatthey believed this approach of (H2) in the context of IUTch, i.e., in particular,
in the context of (HEx1), (HEx2-1), (HEx2-2), to be unnecessary/superﬂuous. In
particular, it appears that one of the fundamental assertions of SS is to the eﬀect
that(SSInd) the essential content (e.g., multiradial algorithms) of IUTch is entirely
unaﬀected even if the indeterminacies (Ind1, 2) (cf. the discussion of§6) are eliminated.

§8. The central reason for the introduction, in IUTch, of various types of indeterminacy lies in the goal (cf. [IUTchIII], Theorem 3.11) of obtaining multiradialalgorithms for representing the Θ-pilot object, i.e., algorithms for representing the Θ-pilot object that are(SW) compatible with — that is to say, invariant, up to suitable indeterminacies, with respect to — the operation of switching/interchanging
corresponding collections of data (e.g., Θ-pilot objects) in the domain andcodomain of the Θ-link, in a fashion that ﬁxes the gluing of data that
constitutes the Θ-link.
This switching property (SW), i.e., multiradiality, may also be thought of as a sort ofsymmetry between certain data in the domain and codomain of the Θ-link. This
sort of symmetry is achieved precisely by introducing various indeterminacies —i.e., via the operation of “re-initialization”, or “forgetting certain mathematicalstructures”, as discussed in (H2). That is to say,(Sym) unlike the very rigid history diagrams discussed in (H1), i.e., where
one does not allow oneself to perform re-initialization operations, history
diagrams such as those in (H2), i.e., diagrams that include re-initialization
operations “” (which typically give rise to certain indeterminacies!),are much more ﬂexible/less rigid, hence have far fewer obstructions
to admitting symmetries.
Put another way, the re-initialization operations “” in history diagrams such as
those in (H2) may be visualized as ﬂexible/rotary joints — such as those in thehuman skeleton or in robot arms — whose ﬂexibility renders possible various types

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SHINICHI MOCHIZUKI

of symmetry. Perhaps the most fundamental example of this sort of phenomenon
is the following very classical/elementary example:(SWC1) Consider the ordered set E = {0, 1}, equipped with the ordering “0 < 1”.
Then the operation
E → S
given by forgetting the ordering, i.e., passing from ordered sets to underlying sets, gives rise to an object “S” (i.e., a set of cardinality 2) that
admits symmetries (i.e., the symmetric group on 2 letters!) that may
only be considered if one completely forgets the ordering on “E”!
Another observation of fundamental importance in the present context is the following: Although at ﬁrst glance, the introduction of re-initialization operations as in(H2), together with the resulting indeterminacies, may appear to give rise to mathematical structures that are more complicated (cf. the “issue of simpliﬁcation”, i.e.,(T7)!) than the mathematical structures that arise in rigid history diagrams of thesort discussed in (H1), in fact,
re-initialization operations — i.e., operations of forgetting mathematical structures that obstruct desired symmetries — typically yield
mathematical structures that are much simpler/more tractable/more
likely to admit symmetries, in that they allow one to concentrate on
structures of interest while carrying around much less “unnecessary
baggage”!
This sort of phenomenon — i.e., achieving simplicity by forgetting! — may beseen in numerous classical/elementary examples, such as the following one:
(SWC2) Consider the geometry of topological manifolds, i.e., which involves various continuous maps between topological manifolds. At ﬁrst glance, considering topological manifolds equipped with atlases, i.e., equipped withsystems of local coordinates that yield local embeddings into some Euclidean space, may appear to be “simpler” mathematical structures (i.e.,than topological manifolds that are not equipped with atlases) in thatthey contain speciﬁc data that relates such a topological manifold to Eucidean space, whose geometry is more explicit and easier to grasp than
the geometry of an arbitrary topological manifold. In fact, however, theoperation of forgetting atlases, i.e., of regarding topological manifolds as
not necessarily being equipped with explicit atlases, yields mathematical structures that are much simpler/more tractable/more likely toadmit symmetries, especially, for instance, when one considers various
continuous maps between topological manifolds (which are not necessarily
compatible with given atlases in the domain and codomain!), than thegeometries/mathematical structures that arise if one insists (e.g., in thename of “simplicity” — cf. (T7)!) on considering topological manifolds
equipped with atlases.
In the context of IUTch, the two main examples of this general mathematical phenomenon are the following (cf. also the discussion of §15 below for more details):

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)11(SWE1) (Frobenius-like) ring structures surrounding the log-link (cf.
(HEx1)): The usual “Galois-theoretic” relationship (i.e., via the interpretation as a group of ﬁeld automorphisms) with the ring/ﬁeld structuresin the domains and codomains of the log-links in a single vertical line of
the log-theta-lattice of the“Π’s”
that appear in such a vertical line is a mathematical structure that is far
from being symmetric with respect to switching operations between thetwo vertical lines (in the log-theta-lattice) that appear on either side of aΘ-link (cf. (LbMn) below). On the other hand, such a switching symmetry may be achieved precisely by forgetting about these (Frobeniuslike) ring/ﬁeld structures, i.e., by thinking of the various “Π’s” as abstract topological groups — cf. (HEx1), as well as (EtMn) below.
(SWE2)(Frobenius-like) ring structures surrounding the Θ-link (cf.
(HEx2)): In a similar vein, the usual “Galois-theoretic” relationship(i.e., via the interpretation as a group of ﬁeld automorphisms) with thering/ﬁeld structures in the domain and codomain of a Θ-link of the“G’s”
that appear in the domain and codomain of a Θ-link, as well as the usual
relationship — i.e., relative to the local and global value group portionsof the gluing data that appears in a Θ-link — with the ring/ﬁeld structuresin the domain and codomain of a Θ-link of the“O×μ ’s”
that appear in the domain and codomain of a Θ-link, are mathematicalstructures that are far from being symmetric with respect to switching
operations between the domain and codomain of the Θ-link (cf. (LbΘ) below). On the other hand, such a switching symmetry may be achieved
precisely by forgetting about these (Frobenius-like) ring/ﬁeld structures,i.e., by thinking of the various “G’s” as abstract topological groups
and of the various “O×μ ’s” as abstract topological monoids (equippedwith certain collections of submodules), that is to say, by introducing theindeterminacies (Ind1, 2) — cf. (HEx2), as well as (VUC), (EtΘ) below.

§9. The gluing (poly-)isomorphism constituted by the Θ-link may be thoughtof, in essence, as a(n) (poly-)isomorphism between collections of data as follows:
⎛

G

⎞

⎜
⎜
⎝



⎟
⎟
⎠

2

{q j }N · O×μΘ-hol. str.

full
∼
→
polyisom.

⎛

G

⎞

⎜
⎜
⎝



⎟
⎟
⎠

q N · O×μq-hol. str.

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SHINICHI MOCHIZUKI
def— where the “j” in “{−}” varies from 1 to l = 12 (l − 1) (cf. the notation of, e.g.,[Alien], §3.3, (vii), as well as the notation of the above discussion). One fundamental
aspect of IUTch is the following observation:(ΘNR) The gluing isomorphism constituted by the Θ-link is not compatible
with the ring structures — i.e., does not arise from a ring homomorphism
relative to the ring structures — on the algebraic closures of local ﬁelds“k’s”(i.e., in the notation of the discussion surrounding [IUTchI], Fig. I1.2,“F v ’s”, for v ∈ V) that lie on either side of the Θ-link. We shall referto these ring structures on either side of the Θ-link by the terms “Θholomorphic structure” and “q-holomorphic structure” (and abbreviate the lengthy term “arithmetic holomorphic structure” by “hol.
str.”). Here, from the point of view of the data in large parentheses “(−)”
in the ﬁrst display of the present §9, the “ring structures” under consideration may be thought of as consisting of the following three structures:(ΘNR1) the additive structure on (each) k,(ΘNR2) the multiplicative structure on (each) k,(ΘNR3) the “Galois-theoretic interpretation” of each “G” as a group ofﬁeld automorphisms of the corresponding “k”.
On the other hand, in the present context, it is important to recall that
(GIUT) the central goal of IUTch is precisely to compute the Θ-hol. str.
in terms of the q-hol. str. (where we regard these two hol. strs. —i.e., ring structures — as being related via the gluing (poly-)isomorphism
constituted by the Θ-link).
From the point of view of implementing (GIUT) via the multiradial representation algorithms developed in IUTch (i.e., [IUTchIII], Theorem 3.11), the following
property is of central importance:(ΘCR) The Θ-link is deﬁned in such a way as to be compatible with aslarge a portion of the ring structures on either side of the Θ-link (i.e.,the Θ-, q-hol. strs.) as is possible. These ring structures are necessary
in order to deﬁne the respective log-links associated to the Θ-, q-hol.
strs.; the use of these log-links is an essential portion of the multiradial
representation algorithms developed in IUTch. Here, the “portion”
of these ring structures that is compatible with the Θ-link consists of thefollowing two structures:(ΘCR1) the subquotients (equipped with certain collections of sub×
modules) of the multiplicative monoid k (i.e., of nonzero
2elements of the ring k) given by the monoids “{q j }N · O×μ ”,“q N · O×μ ”;(ΘCR2) the “interpretation” of each “G” as a group of automorphisms ofthe monoids considered in (ΘCR1) — an “interpretation” which,in this case, just happens to be equivalent to simply thinking ofeach “G” as an abstract topological group.

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)13

Note, moreover, that (ΘCR1), (ΘCR2) are consistent with the approachto switching symmetrization discussed in (SWE2), hence give rise to theindeterminacies (Ind1, 2).
Here, it is important to note that, although the splittings∼

2

2{q j }N · O×μ → {q j }N × O×μ ;∼q N · O×μ → q N × O×μ2

might, at ﬁrst glance, give the impression that the “value group portions” {q j }N ,q N and “unit group portions” O×μ are treated (in the gluing (poly-)isomorphismthat constitutes the Θ-link) as independent, unrelated objects, in fact this is simply
not the case. That is to say:
(VUSQ) These splittings exist merely as a consequence of the (simple!) monoid
structure of the particular monoids involved. The observation of central
importance, from the point of view of (ΘCR), is that the entire monoid2“{q j }N · O×μ ” (respectively, “q N · O×μ ”) — i.e., which contains both the“value group portion” and “unit group portion” discussed above×— is a subquotient of the single multiplicative monoid k (i.e., ofnonzero elements) that arises from the ring k.
At a somewhat more concrete level, the above discussion may be summarized asfollows:(VUC) In the following, we use left-hand superscripts “0” and “1” to denote,
respectively, objects in the domain and codomain of various modiﬁed versions of the Θ-link:(VUC1) Consider the modiﬁed version of the Θ-link
2{0 q j }N · 0 O×μ2

in which {0 q j } →

1

∼

→

1 N

q · 1 O×μq, and one takes the (poly-)isomorphism∼

O×μ → 1 O×μ to be the “identity isomorphism”, i.e., theisomorphism arising from some ﬁxed choice of (“rigidifying”) isomorphisms of both sides with some ﬁxed “model” copy of O×μarising from a ﬁxed “model” copy of k. This “identity iso∼morphism” 0 O×μ → 1 O×μ is then equivariant with respect tosome (uniquely determined) isomorphism of topological groups∼0
G → 1 G. The rigidifying isomorphisms just mentioned deterimine, by applying the elementary construction reviewed in(RLFU) below to the invariants of i O×μ with respect to various
open subgroups of i G, for i = 0, 1, an “identity isomorphism”∼∼
0
k → 1 k. This “identity isomorphism” 0 k → 1 k is incompat2
ible with the assignment {0 q j }N → 1 q in the sense that, for0∼

j = 2, . . . , l , this “identity isomorphism” 0 k →2
map 0 k 0 q j → 1 q ∈ 1 k.

1

k does not(VUC2) The incompatibility discussed in (VUC1) may be eliminated by2
regarding 0 q j (for j = 1, . . . , l ) as belonging to yet another

copy “0 k” of k that is regarded as not being related to 0 k via a

14

SHINICHI MOCHIZUKI

ﬁeld isomorphism (and, for the sake of “symmetry”, regarding1
q as belonging to yet another copy “1 k” of k that is regarded
as not being related to 1 k via a ﬁeld isomorphism) — cf. the discussion above of the issue of treating the “value group portions”
2{q j }N , q N and “unit group portions” O×μ as independent, unrelated objects. This approach to eliminating the incompatibility
discussed in (VUC1) is not the approach adopted in IUTch.
(VUC3) On the other hand, one may also eliminate the incompatibility discussed in (VUC1) by abandoning the “identity isomorphism” and considering instead the (“full”) poly-isomorphism
given by the collection of all isomorphisms of topological monoids(equipped with certain collections of submodules)
2{0 q j }N · 0 O×μ

∼

→

1 N

q · 1 O×μ

that are equivariant with respect to some (uniquely determined)∼
isomorphism of topological groups 0 G → 1 G. The signiﬁcanceof this approach (cf. (VUSQ)!) lies in the fact that it involves
2
isomorphisms between a single subquotient {0 q j }N · 0 O×μ of
×(copies indexed by labels j of) the topological monoid 0 k and×a single subquotient 1 q N · 1 O×μ of the topological monoid 1 k .
This is the approach that is actually adopted in IUTch.
Finally, we review a certain elementary construction that was applied in the above
discussion:(RLFU) Reconstruction of p-adic local ﬁelds from groups of units: Let K
be a ﬁnite extension of the ﬁeld of p-adic numbers Qp . Write OK ⊆ K for×μ×⊆ OK for the group of units of OK , OKforthe ring of integers of K, OK×μ×
the quotient of OK
by its torsion subgroup, WK = OK
⊗ Q (where thetensor product is taken with respect to the multiplicative module structure
def
def
×μ
×
on OK
), MK = p2 · OK ⊆ OK , UK = 1 + MK ⊆ OK
. Thus, one veriﬁes
×μ×
immediately that the natural composite map UK ⊆ OK
 OK
→ WK
determines an injection UK → WK , whose image we denote by VK (so∼we obtain a natural isomorphism γK : UK → VK ). Next, let us observethat by considering the natural surjection UK × UK  MK that maps
(u, v) → u − v ∈ MK , one may think of the set MK as a
UK × UK−1)
quotient set of the product set UK × UK , hence also (by applying γKof the product set VK × VK . In particular, one may think of the additive
structure on the module MK as a binary operation on some quotient of the
product set VK × VK . Moreover, by observing that (u1 − v1 ) · (u2 − v2 ) =(u1 ·u2 −u1 ·v2 )+(v1 ·v2 −v1 ·u2 ), where u1 , v1 , u2 , v2 ∈ UK , one concludes
that this additive structure on MK (regarded as a quotient of VK × VK !),together with the multiplicative structure on VK , allows one to reconstruct
the multiplicative structure on MK (i.e., the multiplicative structure thatarises by restricting the multiplicative structure on OK to MK ⊆ OK ).
Thus, in summary, by thinking of K as MK ⊗ Q, we conclude that:def

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)15

Suppose that we are given an arbitrary topological group † W , to∼gether with an isomorphism of topological groups α : † W → WK .
Then we obtain a construction(† W, α)→†Kof a topological ﬁeld † K that is functorial in the pair († W, α)(i.e., consisting of a topological group and an isomorphism of
topological groups).
Indeed, α determinesdef· a subgroup † V = α−1 (VK ) ⊆ † W ;· a quotient set † M of the product set † V × † V , together with a∼natural bijection † M → MK that is compatible with the bijection∼†
V × † V → VK × VK determined by α;· an additive structure on † M such that the natural bijection∼†
M → MK is compatible with the respective additive structures;· a multiplicative structure on † M such that the natural bijec∼
tion † M → MK is compatible with the respective multiplicative
structures;def· a topological ﬁeld structure on † K = † M ⊗ Q such that the∼∼natural bijection † M → MK induces an isomorphism † K → K
of topological ﬁelds.
The functoriality of this construction († V, α) → † K is a tautological consequence of the construction.
§10. During the afternoon session of the ﬁnal day of the March discussions, SS madethe following assertion, which may be thought of as a sort of concrete realization of
their assertion (SSInd):(SSId) The multiradial algorithms of IUTch may be applied to relate the Θ-,q-hol. strs. that appear in the following “id-version” (cf. (T3)) of theΘ-link2{q j }N−−−−−−−−−G  O×μqN−−−−−−−−−G  O×μ“identity”∼→
isom. on
G, O×μΘ-hol. str.

q-hol. str.
∼

— where the “−−” on either side of the “ → ” are intended as a notational
device to document the understanding that the ring structure that gives
2
rise to the (“value group”) monoid “{q j }N ” (respectively, “q N ”) is tobe regarded as distinct and unrelated to the ring str. — i.e., the Θ(respectively, q-) hol. str. — that gives rise to the data “G  O×μ ” on∼the same side of the “ → ” (cf. (VUC2)!).
Here, we observe that one may introduce terminology “† Θ-hol. str.” (respectively,“† q-hol. str.”) to denote the ring structure that gives rise to the (“value group”)

16

SHINICHI MOCHIZUKI
2

monoid “{q j }N ” (respectively, “q N ”). (SS deeply objected to discussing and, indeed,refused to consider (cf. (T6)) these ring structures, on the grounds that they wishedto keep the discussion as “simple” (cf. (T7)) as possible — cf. also (T8), thediscussion of (T7) in §8.) From the point of view of this terminology, the “idversion” of the Θ-link discussed in (SSId) may be described via the diagram
⎛

⎞
G
⎝
⎠

2
{q j }N · O×μ†Θ-hol. str.
−−−−−−−−−G  O×μ

⎛

⎞
G
⎝
⎠

q N · O×μ
“identity”∼→
isom. on
G, O×μΘ-hol. str.

†

q-hol. str.
−−−−−−−−−G  O×μq-hol. str.

∼

— where the “ → ” only relates the value group portions of the † Θ-, † q-hol. strs.,i.e., it relates (via the “identity” isomorphism) the “G  O×μ ” portions of theΘ-, q-hol. strs., but does not relate the “G  O×μ ” portions of the † Θ-, † q-hol.
strs. It appears that SS introduced the (mathematical content represented by the)“−−’s”(SSId1) precisely in order to ensure that the “id-version” of the Θ-link discussed
in (SSId) satisﬁes the switching property (SW) (cf. also (VUC2)).
Moreover, it appears that SS felt justiﬁed in introducing the “−−’s” precisely as aconsequence of the misunderstanding, on the part of SS, that
2

(SSId2) the “value group portions” {q j }N , q N and “unit group portions” O×μ are
treated (in the gluing (poly-)isomorphism that constitutes the Θ-link) as
independent, unrelated objects (cf. the discussion immediately preceding(VUSQ); the discussion of (VUC)).
On the other hand, one fundamental property of this “id-version” of the Θ-link
is the following:(† ΘCR) The multiradial algorithms of IUTch only may be applied to relate thevalue group portion of the † Θ-hol. str. to the value group portion ofsome other hol. str. that is linked to the † Θ-hol. str. via data that
contains the “G  O×μ ” portion of the † Θ-hol. str. (cf. (ΘCR),(VUSQ), (VUC)). That is to say, the multiradial algorithms of IUTchcannot be applied (cf. (ΘCR), (VUSQ), (VUC)) in such a way that the“G  O×μ ” that appears in these algorithms may be taken to be therigidiﬁed “G  O×μ ” portion of the Θ-, q-hol. strs. of (SSId) (i.e.,
which is not subject to the indeterminacies (Ind1, 2)!).
In particular,(SSIdFs) the assertion (SSId) is false, i.e., in summary, the use of “−−” to ensurethat (SW) is satisﬁed (cf. (SSId1)) tautologically gives rise to fundamental obstacles to satisfying (ΘCR) (cf. also (VUSQ), (VUC), (SSId2)),
hence also with respect to implementing (GIUT).

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)17

In this context, we remark that one way to understand the essential logical structure of the problem with the “id-version” of the Θ-link discussed in (SSId) is toconsider the following elementary example concerning complex structures on real
vector spaces:
def(SSIdEx) Write V = R2 (i.e., a copy of two-dimensional Euclidean space), e1 =
def

def

def

(1, 0) ∈ V , e2 = (0, 1) ∈ V , V1 = R · e1 , V2 = R · e2 . Let t < 1 be apositive real number. Consider the following two complex structures (i.e.,
structures as a vector space over the ﬁeld of complex numbers C) on V :· the “U -structure”: i · e1 = e2 , i · e2 = −e1 ;· the “W -structure”: i · e1 = t · e2 , i · e2 = −t−1 · e1 .
Thus, the U -, W -structures determine, respectively, complex vector spaces∼
U , W whose underlying real vector space is V . Write φ : U → W for theisomorphism of real vector spaces arising from the fact that the underlying
real vector space of U , W is V . For i = 1, 2, write Ui ⊆ U , Wi ⊆ W for the∼real subspaces determined by Vi ⊆ V and φi : Ui → Wi for the restrictionof φ to Ui . In the following, we shall use a subscript “C” to denote thetensor product over R with C. Write∼ιU : U2 → i · U1 ⊆ (U1 )C
for the R-linear morphism that maps e2 → i · e1 ,∼

ιW : W2 → i · W1 ⊆ (W1 )C
for the R-linear morphism that maps t · e2 → i · e1 . Thus, one may thinkof ιU as the restriction to U2 of the unique C-linear isomorphism∼

ζU : U → (U1 )C
that restricts to the identity on U1 ; one may think of ιW as the restriction
to W2 of the unique C-linear isomorphism∼

ζW : W → (W1 )C
that restricts to the identity on W1 . Next, let us consider the gluing
isomorphism of collections of datadef

Φ = (φ,φ1 , φ2 , e1 → e1 , e2 → e2 ) :∼(U, U1 , U2 , e1 ∈ U1 , e2 ∈ U2 ) → (W, W1 , W2 , e1 ∈ W1 , e2 ∈ W2 )— where each collection of data consists of the underlying real vector spaceassociated to some complex vector space, together with two real subspacesof the real vector space and distinguished elements of each of the subspaces.
Now suppose that one attempts toapproach the issue of understanding the relationship between
the complex structures of U , W by identifying U , W with
(U1 )C , (W1 )C via ζU , ζW .

18

SHINICHI MOCHIZUKIThis approach then encounters various fundamental diﬃculties, as follows:First of all,(∼=) the collections of data (U, U1 , U2 , e1 ∈ U1 , e2 ∈ U2 , ζU ) and(W, W1 , W2 , e1 ∈ W1 , e2 ∈ W2 , ζW ) — or, essentially equivalently, the collections of data (U, U1 , U2 , e1 ∈ U1 , e2 ∈ U2 , ιU )and (W, W1 , W2 , e1 ∈ W1 , e2 ∈ W2 , ιW ) — (where each collectionof data consists of the underlying real vector space associated tosome complex vector space, together with two real subspaces of
the real vector space, distinguished elements of each of the subspace, and a linear morphism between real vector spaces) are notisomorphic.
(Indeed, (φ1 )C ◦ ιU = ιW ◦ φ2 , where we note that the R-linear morphisms
φ1 and φ2 are determined by the respective images, via these morphisms, ofthe distinguished elements. Also, we recall that ιU = ζU |U2 , ιW = ζW |W2 .)Of course, one can modify the collections of data that appear in (∼=) byregarding, for i = 1, 2, Ui , Wi as independent real vector spaces — which,to avoid confusion, we shall denote by [Ui ], [Wi ], respectively — that are
not equipped with their respective inclusions into U , W . It then follows
tautologically from the deﬁnition of “[−]” that we obtain an isomorphismof (ordered) pairs[Φ] = ([φ1 ],[φ2 ], e1 → e1 , e2 → e2 ) :∼([U1 ], [U2 ], e1 ∈ [U1 ], e2 ∈ [U2 ]) → ([W1 ], [W2 ], e1 ∈ [W1 ], e2 ∈ [W2 ])— where each pair consists of two real vector spaces, each of which isequipped with a distinguished element. On the other hand,(Dilat) working with [Φ] and thinking of U , W as being identiﬁed
with (U1 )C , (W1 )C by means of ζU , ζW does not allow one tocompute the nonzero dilatation of the “quasiconformal map”∼

φ:U →Wby replacing φ by the C-linear (i.e., complex holomorphic!) map∼(φ1 )C : (U1 )C → (W1 )C
by means of ζU , ζW , i.e., whose dilatation is zero.
Relative to the analogy with IUTch:· V1 corresponds to the unit group portion of the gluing data thatappears in the Θ-link;· V2 corresponds to the value group portion of the gluing datathat appears in the Θ-link;· the “U -structure” corresponds to the q-hol. str.;· the “W -structure” corresponds to the Θ-hol. str.;· the property (∼=) corresponds to (VUC1);· the gluing isomorphism [Φ] corresponds to the gluing isomorphism that appears in the “id-version” of the Θ-link discussed inDISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)19(SSId) (cf. (VUC2));· the gluing isomorphism Φ corresponds to the gluing isomorphism that appears in the Θ-link (cf. (VUC3));· the property (Dilat) corresponds to (SSIdFs).

In the context of IUTch, it is also perhaps of interest to observe that the essential mathematical content of this (very elementary!) example (SSIdEx) may, by
introducing slightly more sophisticated (but still very classical!) terminology, beinterpreted in terms of the well-known classical theory of moduli of complex elliptic
curves:(ModEll) (Holomorphic moduli of elliptic curves vs. Teichmüller moduli
of elliptic curves): There are two classical approaches to understanding
the moduli of complex elliptic curves:(HolMod) One may start with a ﬁxed copy “C” of the ﬁeld of complex
numbers (or, essentially equivalently, with a one-dimensional Cvector space) and think of an elliptic curve over C as a quotientof C by a lattice
C/(Z + Z · τ )— where the “period” τ ∈ C is a complex number whose imaginary part Im(τ ) > 0. This approach to elliptic curves over C
allows one to think of elliptic curves over C in terms of holomorphic moduli. Then:· From the point of view of the natural interpretation, in termsof inﬁnitesimal moduli, of the ﬁrst cohomology module of the(trivial!) tangent bundle of an elliptic curve over C, this approachcorresponds to computing this ﬁrst cohomology module via theholomorphic de Rham complex.
· From the point of view of the discussion of (SSIdEx), this approach — which centers around ﬁxing a copy “C” of the ﬁeldof complex numbers! — corresponds to the gluing isomorphism [Φ], i.e., to thinking in terms of the identiﬁcations via the
C-linear (hence, in particular, holomorphic!) isomorphisms∼∼(U1 )C → (V1 )C → (W1 )C , together with an additional, auxiliarydefdatum “t−1 · i · e1 ” (which corresponds to taking τ = t−1 · i in
the present discussion).
· In particular, this approach corresponds to the the gluing isomorphism that appears in the “id-version” of the Θ-link discussed in (SSId) (cf. (VUC2)).
(TchMod) One may start with a ﬁxed latticedefΛ = Z⊕Zdef(⊆ ΛR = R ⊕ R)

and think of an elliptic curve over C as a holomorphic structure on this ﬁxed lattice Λ (i.e., on the ﬁxed real vector space

20

SHINICHI MOCHIZUKI

ΛR ). Thus, for instance, the holomorphic structure given byi · (1, 0) = (0, t) corresponds to taking the “τ ” in (HolMod) to bet−1 · i. This (non-holomorphic, quasi-conformal/real analytic!)
approach to elliptic curves over C is the approach taken in classical complex Teichmüller theory. Then:· From the point of view of the natural interpretation, in termsof inﬁnitesimal moduli, of the ﬁrst cohomology module of the(trivial!) tangent bundle of an elliptic curve over C, this approachcorresponds to computing this ﬁrst cohomology module via the
Dolbeault complex.
· From the point of view of the discussion of (SSIdEx), this approach — which centers around ﬁxing the lattice “Λ ⊆ ΛR ”!
— corresponds to the gluing isomorphism Φ, i.e., to thinking
in terms of the identiﬁcations via the R-linear (i.e., quasicon∼∼
formal, non-holomorphic!) isomorphisms U → V → W between the underlying real vector spaces of various complex vector
spaces.
· In particular, this approach corresponds to the gluing isomorphism that appears in the Θ-link (cf. (VUC3)).

Before proceeding, it is perhaps worthwhile to recall that, historically,(HstMod) of these two approaches discussed in (ModEll), the “holomorphic approach” (HolMod) has been the “preferred approach” in (scheme-theoretic)arithmetic geometry, presumably because holomorphic structures may
be related very directly to algebraic/scheme-theoretic structures,
whereas the central role played in the “quasiconformal approach” (TchMod) by consideration of the two underlying dimensions of a holomorphic structure did not, prior to the appearance of IUTch (cf. also
the Introduction to [AbsTopIII]), have any evident analogue in arithmetic
geometry.
Finally, in this context, it should be pointed out that SS only arrived at (SSId),
as formulated above, during the afternoon session of the ﬁnal day of the March
discussions precisely as a consequence of the fact that the details of the formulationof their “id-version” evolved substantially over the course of the March discussions.
For instance, the introduction of “−−” in (SSId) arose precisely from the discussionof (SW), (Sym) (cf. (SSId1)), which in turn arose as a consequence of the discussions
on previous days summarized in §5, §6, §7. This state of aﬀairs suggests that if the
March discussions had continued further (perhaps after a period of rest, to allow
the participants to digest the course of the previous discussions), it is quite possiblethat further discussion of (ΘCR) could have resulted in a common understanding(i.e., as summarized in (SSIdFs)) concerning (SSId) (cf. the discussion following(GLR2)).
§11. In the context of the discussion of §10, it is important to remember that:

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)21(ΘFrd) The data that appears in the Θ-link contains global realiﬁed Frobenioids, i.e., certain globalized versions of the local value group portions2“{q j }N ”, “q N ” whose purpose is to category-theoretically encode the notion of the global arithmetic degree of an arithmetic line bundle (cf.,e.g., [Alien], §3.3, (vii)). In particular, the multiradial representation
algorithms developed in IUTch (i.e., [IUTchIII], Theorem 3.11), as wellas the inequalities derived by applying these algorithms, are phenomenathat are essentially global in nature — i.e., not just a sort of result of
summing up essentially local phenomena (cf. the discussion of [IUTchIII],
Remarks 3.6.2; 3.10.1; 3.12.2, (v)).
In particular, the compatibility of the multiradial representation algorithms developed in IUTch (i.e., [IUTchIII], Theorem 3.11) with the correspondence in theΘ-link2∼{q j }N → q N
cannot be ascribed solely to purely local indeterminacies such as (Ind1, 2, 3).
For instance:(ΘInd) This compatibility cannot be understood as being a consequence of the existence of automorphisms appearing in (Ind1, 2) that permute the elementsof “O×μ ” determined by local units of the form
2

{exp(q j )},exp(q)(where j = 1, . . . , l ).
(Indeed, in general, such automorphisms do not exist!) This essentially global natureof the logical structure of IUTch has apparently been a source of misunderstanding
for some mathematicians. In this context, it is perhaps of interest to recall the following well-known examples of non-isomorphic (but comparable!) mathematicalstructures that have isomorphic underlying structures, but which cannot be transformed into one another by means of isomorphisms of the underlying structures:(UnEx1) Classical complex Teichmüller theory: (cf. (HC1)) It is well-known
and easy to see that there exist non-isomorphic compact Riemann surfaces whose associated underlying topological (or real analytic) surfaces are
isomorphic, but whose respective holomorphic structures cannot be transformed into one another by conjugating by some isomorphism between theassociated underlying topological (or real analytic) surfaces.
(UnEx2) Group structures on a set: It is easy to see (for instance, in the caseof groups of order four!) that there exist non-isomorphic groups whose
associated underlying sets are isomorphic (i.e., of the same cardinality),
but whose respective group structures cannot be transformed into one another by conjugating by some isomorphism (i.e., bijective map) between
the associated underlying sets.
(UnEx3) Absolute Galois groups of p-adic local ﬁelds: Let K be a ﬁnite
extension of the ﬁeld of p-adic numbers Qp , K an algebraic closure of K,
def

GK = Gal(K/K). Write K × for multiplicative group of nonzero elements of K, Gab
K for the abelianization of the proﬁnite group GK . Then22

SHINICHI MOCHIZUKI

we recall that the reciprocity map of local class ﬁeld theory yields a naturalembedding K × → Gab
K , whose image may be reconstructed via a purelygroup-theoretic algorithm from GK (cf., e.g., the discussion at the beginning of [Alien], §2.12). Thus, by considering the additive structure (i.e.,
arising from the additive structure of the ﬁeld K) on the union with {0} of
the image of this embedding K × → Gab
K , one may think of the ﬁeld K as
an “auxiliary structure [K]” on the proﬁnite group GK . In particular, one
may think of the proﬁnite group GK as an underlying structure associatedto the pair (GK , [K]) (consisting of a topological group and an additive
structure, as just described) . Now let L be a ﬁnite extension of the ﬁeld
defof p-adic numbers Qp , L an algebraic closure of L, GL = Gal(L/L). We
use similar notation for objects related to L to the notation introduced
above for objects related to K. Then recall (cf., [Ymgt], §2, Theorem)that there exist cases of K and L such that
K is not isomorphic (as a ﬁeld) to L, but GK is isomorphic (asa proﬁnite group) to GL .
Thus, it follows formally that the pairs (GK , [K]) and (GL , [L]) are notisomorphic, but their “underlying topological groups” GK and GL are isomorphic. In particular, the pairs (GK , [K]) and (GL , [L]) cannot be transformed into one another by conjugating by some isomorphism between theunderlying proﬁnite groups GK and GL .
§12. During the afternoon session of the ﬁnal day of the March discussions, SS
proceeded to derive from the “id-version” of (SSId) a “contradiction” concerning
arithmetic degrees (cf. (Smm)). This “contradiction” essentially amounts to thefollowing assertion:(AD) The correspondence that appears in the Θ-link
2{q j }→q(where j = 1, . . . , l ) is, when considered within/relative to a single
hol. str., i.e., a single ring/scheme theory, manifestly incompatible/inconsistent with computing arithmetic degrees, unless one considers arithmetic degrees up to the indeterminacy given by possible multiplication by factors “j 2 ” (where j = 1, . . . , l ) — an indeterminacy whichthen renders the computation meaningless.
Before proceeding further, it should be emphasized that:(IUAD) There is absolutely no dispute whatsoever (between SS and HM)
with respect to the validity of the assertion (AD). Indeed, the point isthat the situation in (AD) occurs whenever one works within a single
hol. str./ring theory. Moreover, this observation (AD) is by no meansa new observation, but rather constitutes the starting point of IUTch.
It is precisely for this reason (cf. (GLR2)!) that, in IUTch, one must treat
the hol. strs. in the domain and codomain of the Θ-link as distinct hol.
strs. that are related in a nontrivial way that may be only be elucidatedby means of a nontrivial computation (cf. (GIUT)).

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)23

Moreover, in the present context, it is of fundamental importance to recall that:(ADItw) Computations of arithmetic degrees of global arithmetic line bundles thatare related to one another via the Θ-link depend, in an essential way, on
the speciﬁcation of a particular relationship, or “intertwining”, between
2
the “value group portions” {q j }N , q N and “unit group portions” O×μ of
the data that appears in the gluing (poly-)isomorphism that constitutes
the Θ-link. Such “intertwinings” arise from the hol. strs. (i.e., ring structures) in the domain and codomain of the Θ-link, but (since the Θ-link is
not a ring homomorphism!) the Θ-link fails, a priori, to be compatible
with the distinct ring structures in its domain and codomain, hence alsowith arithmetic degree computations in its domain and codomain.
When discussing (AD), SS, at times, asserted that:(SSAD) The “contradiction” of (AD) occurs even if one does not invoke (SSId),i.e., even if one applies the multiradial algorithms of IUTch as stated in[IUTchIII] (that is to say, without necessarily invoking the assertion thatthese algorithms may be applied to some sort of “id-version” as in (SSId)).
On the other hand, it should be stated unequivocally that:(SSADFs) The assertion of (SSAD) is false. That is to say, the “contradiction” of(AD) only occurs when one identiﬁes the hol. strs./ring theories inthe domain and codomain of the Θ-link (cf. (ADItw)).
Unfortunately, whenever I tried to explain (SSADFs) during the afternoon sessionof the ﬁnal day of the March discussions, for instance, by using distinct labels todenote the distinct hol. strs./ring theories in the domain and codomain of theΘ-link, SS stubbornly refused to allow the use of such distinct labels in thediscussion (cf. (T5); (T6); (T8); the discussion of (T6) in §10). On the other hand,
it should be stated clearly that:(SSDLFs) Deleting the labels used in IUTch (i.e., as formulated in [IUTchI],[IUTchII], [IUTchIII]) to denote the distinct hol. strs./ring theories in the domain and codomain of the Θ-link amounts to identifying/confusing these distinct hol. strs./ring theories — an operationthat is already stronger than (i.e., “implies, in particular, the contentof”) the operations discussed in (SSInd) (cf. also the discussion of §5, §6,§7) and (SSId), i.e., operations which already lead to situations in which
the multiradial algorithms are no longer applicable (cf. the discussionof §10, especially, (SSIdFs)).
Here, it is perhaps helpful to recall that the labels used in IUTch to denote distinct
hol. strs./ring theories may be thought of, from the point of view of taking logvolumes of regions in tensor packets of log-shells (i.e., the situation that occurs,
for instance, in the ﬁnal portion of [IUTchIII], Corollary 3.12), as(LbLV) markers that denote whether (a certain suitable “generator” called the“pilot object” of) the value group portion of the gluing data in the Θlink is identiﬁed with the log-volume of “q” (i.e., which corresponds to2

the case of the q-hol. str.) or with the log-volume of “{q j }” (i.e., whichcorresponds to the case of the Θ-hol. str.).

24

SHINICHI MOCHIZUKI(For more details, we refer to, for instance, the discussion of “toy models” in[IUTchIII], Remark 3.12.2, (ii).) The multiradial algorithms of [IUTchIII], Theorem 3.11, then provide an alternative way to compute the log-volume of theΘ-pilot object in terms of the q-hol. str., up to certain indeterminacies. In thiscontext, it is important to remember that
(MlLV) unlike the linear correspondence induced by the Θ-link — i.e., in eﬀect,
a linear relationship between copies of the topological group of real numbers “R” — between the log-volume of the Θ-pilot object relative to theΘ-hol. str. and the log-volume of the q-pilot object relative to the q-hol.
str., this computation, via the multiradial algorithms of IUTch, of thelog-volume of the Θ-pilot object in terms of the q-hol. str. involves certain indeterminacies, which give rise to a highly non-linear relationshipbetween the log-volume of the Θ-pilot object in the Θ-hol. str. and thelog-volume of the multiradial representation of the Θ-pilot object.
This general qualitative phenomenon of indeterminacies giving rise to a highly
non-linear relationship between the log-volume of a (“rigid”) region not subject toindeterminacies and the log-volume of an approximation (say, from above) of the(“rigid”) region that is subject to indeterminacies may be understood by considering
the following elementary example:(LVEx) Write V = R2 (i.e., a copy of two-dimensional Euclidean space), σ :∼V → V for the automorphism of V given by V (x, y) → (−x, y). Denoteby W the “stack-theoretic quotient” of V by the group G of automorphismsof V of order 2 generated by σ. Thus, we have a natural ﬁnite étalemorphism of degree 2
φ:V → Wof “orbispaces”. Next, let us consider the following region of V and its“symmetrization” with respect to the action of G:
defRa,b = {(x, y) ∈ V | |x| ≤ 1, 0 ≤ |y| ≤ a}
∪ {(x, y) ∈ V | 0 ≤ x ≤ 1, a ≤ |y| ≤ a + b};
def

Sa,b = Ra,b ∪ σ(Ra,b )— where a, b > 0 are positive real numbers. Then one computes easily the
log-volume “μlog (−)” (i.e., the natural logarithm of the volume relative tothe usual Euclidean measure) of these regions as follows:
μlog (Sa,b ) = log(4a + 4b)>

μlog (Ra,b ) = log(4a + 2b).

In particular, we observe thatthe ratio [μlog (Sa,b ) : μlog (Ra,b )] (which may be thought of asa point of the projective line over R) varies continuously —i.e., is far from being constant, as would be the case if therelationship between the two log-volumes were linear — as a, bvary.
Here, since σ(Sa,b ) = Sa,b , one may think of Sa,b as being “deﬁned over W ”
and hence, at a purely formal level, equipped with a “label” W (which we

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)

shall denote by a subscript “W ”). On the other hand, since σ(Ra,b ) = Ra,b ,Ra,b is only “deﬁned over V ”, hence, at a purely formal level, may bethought of as being equipped with a “label” V (which we shall denote bya subscript “V ”). Here, as the inclusion Ra,b ⊆ Sa,b suggests,
one may think of the label “V ” as denoting a sort of “incomplete
portion” of the label “W ”.
Next, let us consider a real number λ ∈ R such that a, b, λ satisfy theinequalities
μlog (Sa,b ) > 0 > λ > μlog (Ra,b )— which imply, in particular, that μlog (Sa,b ) = λ, and that the positivedefreal number 1 < ρ = μlog (Ra,b ) · λ−1 ∈ R is well-deﬁned. We shall thinkof λ as the log-volume of some region “Rλ ” that is “deﬁned over W ”, i.e.,
at a more concrete level, a region of V stable under the action of G. Then,
relative to the analogy with IUTch — i.e., especially the key argument in
Step (xi) of the proof of [IUTchIII], Corollary 3.12 — we have the following
correspondences:· Ra,b corresponds to the Θ-pilot relative to the Θ-hol. str.;· Sa,b corresponds to the holomorphic hull, relative to the qhol. str., of the multiradial representation of the Θ-pilot,
with “indeterminacies” given by the action of the group G;· Rλ corresponds to the q-pilot;· the assignment μlog (Ra,b )V → λW = μlog (Rλ )W determines an∼
R-linear “gluing isomorphism” RV → RW , i.e., given by “dividing
by the factor ρ”; this gluing isomorphism corresponds to the Θlink;· the chain of relationships of log-volumes
μlog (Sa,b )W > μlog (Ra,b )V → λW
— where the “→” is the gluing determined by the assignment
μlog (Ra,b )V → λW = μlog (Rλ )W just discussed — corresponds
to the chain of relationships between log-volumes in the argumentof Step (xi) of the proof of [IUTchIII], Corollary 3.12.
This last chain of relationships of log-volumes is remarkable in that it may
be interpreted in the following way:
the inequality μlog (Sa,b )W > μlog (Ra,b )V arises (not from thegluing given by dividing/multiplying, i.e., depending on the direction in which one is going, by a factor of ρ, but rather) fromthe geometry of the regions Ra,b ⊆ Sa,b and the deﬁnition of
W as a “stack-theoretic quotient” of V by G, but is, nevertheless, entirely logically consistent — i.e., in the sense that thecomposite inequality μlog (Sa,b )W > λW does indeed hold in the
copy “RW ” of R labeled by W — with the gluing given by dividing/multiplying by a factor of ρ.

25

26

SHINICHI MOCHIZUKI

Here, we observe that this logical consistency of the gluing (given by dividing/multiplying by a factor of ρ) with the inequality μlog (Sa,b )W >μlog (Ra,b )V depends, in an essential way, on the geometry of the regions
Ra,b and Sa,b , as well as the inequalities satisﬁed by a, b, and λ. In particular, it is
no means the case that a similar consistency property
is satisﬁed by arbitrary regions of W , V and an arbitrary real
number λ.
Finally, we observe that the labels “V ” and “W ” in the chain of relationships of log-volumes μlog (Sa,b )W > μlog (Ra,b )V → λW may be interpreted
in the following way:
· the V -labeled real number μlog (Ra,b )V is the log-volume of an“incomplete portion” of some “complete object” whose
log-volume is the W -labeled real number μlog (Rλ )W = λW ;· on the other hand, the relationship of “completion” between
the V -labeled “incomplete portion” and the W -labeled “complete
object” may also be estimated from above by considering theinclusion within the symmetrization (Sa,b )W ⊇ (Ra,b )V , which,upon passing to log-volumes, yields an inequality μlog (Sa,b )W >μlog (Ra,b )V .
The situation just described in detail in the present (LVEx) is precisely
the sort of situation that arises in the argument of Step (xi) of the proofof [IUTchIII], Corollary 3.12.
Another interesting and relatively elementary example of the phenomenon of computing the “size” of a region subject to indeterminacies (cf. the multiradial representation of the Θ-pilot object in IUTch) that approximates (say, from above) someregion of interest that is “rigid” and not subject to indeterminacies (cf. the Θ-pilot
relative to the Θ-hol. str. in IUTch) may be seen in the displacement estimatesthat appear in Bogomolov’s proof of the geometric version of the Szpiro Conjecture (cf. the discussion of (DsInd) below). Finally, in this context, we remarkthat the importance of working with distinct hol. strs./ring theories is discussed indetail — and in a fashion directed toward an audience of non-mathematicians —in, roughly, the ﬁnal 30 minutes of the video [FKvid].
§13. One topic to which a substantial amount of time and energy was devoted,
especially during the ﬁrst few days of the March discussions, was the topic of polymorphisms in IUTch (cf. (T4)), i.e., sets (that are not necessarily of cardinality
1!) of morphisms between two objects. SS (especially, Scholze) were substantially
opposed to the use of poly-morphisms in IUTch. This opposition appeared to be
based, to a substantial extent, on “taste/aesthetics” (cf. (T4-1)). In this context,
however, it should be remembered that, although the term “poly-morphism” isapparently new, in fact poly-morphisms — i.e., in eﬀect,(PMInd) situations in which one wishes to regard morphisms/objects of interest
up to/modulo some sort of indeterminacy, the elimination of which is
not a matter of interest

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)27— appear quite frequently in mathematics. Well-known examples include(PMEx1) homotopy, or stable homotopy, classes of continuous maps between
topological spaces;(PMEx2) morphisms in derived categories between complexes, i.e., morphisms that
may be related to morphisms between complexes in the naive sense only
by considering the classes of such naive morphisms obtained by inverting
the quasi-isomorphisms.
Here, before proceeding, it is perhaps of interest to observe the existence of “trivial
examples” of poly-morphisms:(PMTrv) For i = 1, 2, suppose that Gi is a group, and that Hi ⊆ Gi is a normal
subgroup. Then a group homomorphism φ : G1 /H1 → G2 /H2 between
the quotient groups G1 /H1 , G2 /H2 may be regarded as a trivial example of a poly-morphism, i.e., by thinking of φ as the collection of grouphomomorphisms G1 → G2 that lift φ.
A poly-morphism of the sort discussed in (PMTrv) is “trivial” in the sense that it
may be expressed as a (single!) morphism (i.e., map!) between quotient sets (i.e.,the sets “Gi /Hi ” for i = 1, 2). By contrast:(PMQut) The examples of poly-morphisms that occur in IUTch — such as thosethat arise from the indeterminacies (Ind1, 2) — make use of the notion ofa poly-morphism in a nontrivial way precisely because they cannot be expressed as single maps between quotient sets. (Here, we note that theexamples (PMEx1) and (PMEx2) are also nontrivial in this sense. For instance, there is no meaningful sense in which one can deﬁne the “quotientset” of a topological space by its group of homotopy self-equivalences.)
It is precisely this sort of situation that requires the use of formal techniques — such as in the case of algebraic stacks (as opposed to schemes oralgebraic spaces) or orbifolds (as opposed to topological manifolds) — inorder to work with quotients in some sort of abstract or formal sense.
This is precisely what is obtained in IUTch by the introduction of the“formal technique” constituted by the notion of a poly-morphism.
SS at times justiﬁed their opposition to the “apparently substantively meaningless” (cf. (T4-2)) use of poly-morphisms by invoking the issue of simplicity (cf.
(T7)), but, as one may see in the well-known examples (PMEx1), (PMEx2), working with speciﬁc morphisms in situations where it is much more natural to workwith classes of morphisms, tends to have the eﬀect of rendering arguments much
more complicated, or less tractable (cf. the discussion of §8). Put another way, as
was discussed in §5, §6, §7, §8, §9, §10 (cf., especially, the discussion of (SSInd),(SW), (Sym), (SWE1), (SWE2), (VUC), (SSId), (SSIdFs)):(PMEss) The use of poly-morphisms in IUTch — e.g., the introduction of indeterminacies such as (Ind1, 2) — plays the quite substantive role ofyielding structures that admit certain symmetries (cf. the discussion of§8), which form the basis of multiradiality, in a fashion that is compatible with suitable portions of the ring structures in the domain andcodomain of the Θ-link (cf. the discussion of §9, §10).
Further remarks concerning the use of poly-morphisms in IUTch may be found in[Alien], §4.1, (iv).

28

SHINICHI MOCHIZUKI§14. Another topic to which a substantial amount of time and energy was devoted,
especially during the ﬁrst few days of the March discussions, was the topic of labels
in IUTch (cf. (T5)) to distinguish distinct copies of various familiar objects thatplay substantively diﬀerent roles in the various apparatuses treated in IUTch.
SS (especially, Scholze) were substantially opposed to the use of labels in IUTch.
This opposition appeared to be based, to a substantial extent, on “taste/aesthetics”(cf. (T5-1)). In this context, however, it should be remembered that in fact “labels”— i.e., in eﬀect,(LbRl) situations in which one wishes to distinguish distinct copies of various familiar objects that play substantively diﬀerent roles within acomplicated apparatus— appear quite frequently in mathematics. Well-known examples include the following:(LbEx1) The substantive signiﬁcance of the use of labels may be seen in thefollowing fundamental example: One considers the (“line segment”) graphΓseg•—•given by two vertices joined via a single edge. Write Γloop for the (“loop”)
graph obtained from Γseg by identifying the two distinct vertices of Γseg —i.e., which may be thought of as distinct labels — to a single vertex. Thus,Γseg is structurally diﬀerent from Γloop in the sense that Γseg is simply
connected, while Γloop is not. This diﬀerence gives rise to very substantive consequences in many situations, e.g., situations where some sort of“parallel transport” or “analytic continuation” along the loop of
Γloop gives rise to some sort of nontrivial monodromy operator. This
sort of nontrivial monodromy may be resolved precisely by distinguishing the situations that arise prior to and subsequent to the application of
the monodromy operator, i.e., by working over Γseg as opposed to Γloop .
When, moreover, one wishes to distinguish situations that arise prior toand subsequent to multiple applications of the monodromy operator, it isnatural to consider the universal covering Γuni... • — • — • — • ...
of Γloop , which may be thought of as the result of concatenating distinct
copies of Γseg labeled by elements ∈ Z. One fundamental example of thissort of situation, i.e., of nontrivial monodromy around a loop, is the (angular portion of the) logarithm function in one-variable complex analysis.
This example is, moreover, reminiscent of the situation surrounding thevertical columns of log-links that appear in the log-theta-lattice of IUTch(cf. (LbLp) below). Alternatively, one may understandthe incompatibility of the Θ-link (i.e., a horizontal arrow inthe log-theta-lattice) with arithmetic degrees (cf. (AD)) as asort of nontrivial monodromy around the loop (i.e., a special
case of “Γloop ”!) that arises as soon as one identiﬁes the domain
and codomain of the Θ-link — an incompatibility, i.e., “contradiction”, that is in fact avoided in IUTch (cf. (IUAD))DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)29

precisely by distinguishing the (ring structures in the) domain
and codomain of the Θ-link, i.e., by working with “Γseg ”, andthen applying the multiradial algorithms of IUTch to obtainan alternative way to compute this nontrivial monodromy(cf. (GIUT)).
(LbEx2) Suppose that A is a commutative domain with unity, f ∈ A a nonzero,non-unit. Then the localization Af of A with respect to f may beconstructed as the inductive limit of the inductive system...

f·−→ An−1

f·

−→ An

f·

−→ An+1

f·

−→

...

given by copies “An ” of A labeled by elements n ∈ Z and transition morphisms given by multiplication f . Here, we observe that forgetting the
choice of a “speciﬁc basepoint label 0 ∈ Z” does not have any substantive
eﬀect on the construction if one is only interested in the A-module structure of Af . If, on the other hand, one is interested in constructing thering structure on Af , then it is of crucial importance to specify a “speciﬁc basepoint label 0 ∈ Z”, i.e., a speciﬁc copy for which the unity
element 1 ∈ A of A serves as the unity element for the newly constructed
Af . Thus, in summary:
Omitting the labels leads to confusion concerning which copyof the unity element 1 ∈ A is to be regarded as the unity elementfor “Af ”. Such confusion may, of course, be misinterpreted as
an “internal contradiction” in the theory of localizations ofcommutative rings with unity. In fact, however, there is no “internal contradiction” in the theory of such localizations; theapparent “internal contradiction” is nothing more than a superﬁcial consequence of the erroneous operation of omitting thelabels.
def(LbEx3) Consider the polynomial ring A = Fp [t] in one variable over the ﬁeldof cardinality p, for some prime number p. Write φ : A → A for the
Frobenius endomorphism, i.e., the ring endomorphism given by sendingnt → tp . Then the perfection A∞ of A — i.e., Fp [{t1/p }n≥1 ] — may beconstructed as the inductive limit of the inductive system...

φ−→ An−1

φ

−→ An

φ

−→ An+1

φ

−→

...

given by copies “An ” of A labeled by elements n ∈ Z and transition
morphisms given by (copies of) φ. Here, we observe that, if one forgets
the labels n ∈ Z, i.e., and just considers the inductive system given bya single copy of A and the transition morphism φ : A → A, then one
computes easily that the resulting inductive limit is isomorphic to a directsum ⊕ Fp of p copies of Fp (i.e., and hence not isomorphic to A∞ !). Thus,in summary:
Omitting the labels leads to confusion between mutually
non-isomorphic inductive limits (i.e., A∞ , ⊕ Fp ). Such confu-

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SHINICHI MOCHIZUKI

sion may, of course, be misinterpreted as an “internal contradiction” in the theory of perfections in positive characteristic.
In fact, however, there is no “internal contradiction” in thetheory of perfections in positive characteristic; the apparent “internal contradiction” is nothing more than a superﬁcial consequence of the erroneous operation of omitting the labels.
For more detailed discussions of phenomena arising from the Frobenius
morphism in positive characteristic that are more directly related to IUTch,
we refer to (EtFrEx) below, as well as [Alien], Example 2.6.1; the discussion of [Alien], §2.7.
(LbEx4) Write λ ∈ C for the unique positive real root ∈ C of the equation X 3 −def

2 = 0, ω = exp( 23 πi) ∈ C, Q ⊆ C for the ﬁeld of rational numbers,
def

def

def

F1 = Q(λ) ⊆ C, F2 = Q(ω · λ) ⊆ C, K = Q(ω, λ) ⊆ C. Thus, wehave a diagram of ﬁeld extensions
K



F2

F1



Q

such that Q = F1 ∩ F2 , K = F1 · F2 . In particular, one may interpret the
portion of the above diagram involving F1 , F2 , Q as a situation in which
one has two distinct abstract ﬁelds F1 , F2 that are glued together(in the category of sets) along the subﬁelds Q ⊆ F1 , Q ⊆ F2 (viathe unique isomorphism between these subﬁelds) — cf. the descriptionof the Θ-link at the beginning of §9; (ΘNR). Observe that there exists a∼
unique isomorphism of ﬁelds ψ : F1 → F2 , which, of course, is notcompatible with the respective embeddings F1 → K, F2 → K. Then:
If one identiﬁes F1 and F2 by means of ψ and then uses thisidentiﬁcation to omit the labels “1”, “2” that are used to distinguish the ﬁelds F1 , F2 , then it is easy to obtain a “contradiction”
concerning the structure of the ﬁeld K. Such a “contradiction”
may, of course, be misinterpreted as an “internal contradiction” in the elementary theory of ﬁeld extensions. In fact, however, there is no “internal contradiction” in the elementary
theory of ﬁeld extensions; the apparent “internal contradiction”
is nothing more than a superﬁcial consequence of the erroneous
operation of omitting the labels used to distinguish the ﬁelds
F 1 , F2 .

(LbEx5) (This example may, roughly speaking, be thought of as a sort of generalization of the example discussed in (LbEx1).) Consider the theory of

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)(topological or diﬀerential) manifolds of dimension n, for n a positive integer. Manifolds (of dimension n) are, by deﬁnition, collections of distinct
copies — i.e., copies equipped with distinct labels — of (n-dimensional)
Euclidean space glued together by means of various gluing isomorphisms.
In particular, it is quite possible that the use of numerous distinct
copies of Euclidean space in this theory might impress a newcomer tothis theory (such as a student) as being entirely superﬂuous. Such a
newcomer might feel motivated to attempt to achieve a “dramatic simpliﬁcation” of this theory by identifying these copies of Euclidean spacewith one another and hence to assert that the geometry of arbitrary (topological or diﬀerential) manifolds (of dimension n) may be “reduced” tothe geometry of (n-dimensional) Euclidean space. Moreover:If one identiﬁes these distinct copies of Euclidean spacethat appear in the theory of (topological or diﬀerential) manifolds with one another, then it is easy to obtain a “contradiction”. Such a “contradiction” may, of course, be misinterpreted
as an “internal contradiction” in the theory of (topologicalor diﬀerential) manifolds. In fact, however, there is no “internal contradiction” in the theory of (topological or diﬀerential)
manifolds; the apparent “internal contradiction” is nothing morethan a superﬁcial consequence of the erroneous operation ofomitting the labels used to distinguish the distinct copies of Euclidean space.

(LbEx6) In the context of the mathematics discussed in the present report, one
fundamental example of the use of labels may be seen in the very definition of the “id-version” of (SSId). Indeed, the “−−” that appearsin (SSId) is a notational device that documents the understanding thatthe ring structures on opposite sides of the “−−” are to be regarded as
unrelated, i.e., in eﬀect, equipped with distinct labels, as in the discussion of (VUC2). Here, we recall that this understanding that the ring
structures on opposite sides of the “−−” with distinct labels — say, “”and “” — are to be regarded as unrelated is the crucial ingredient in theapproach discussed in (VUC2) to avoiding the incompatibility property —i.e., in eﬀect, internal contradiction — inherent in the modiﬁed versionof the Θ-link discussed in (VUC1). On the other hand,in order to apply the multiradial algorithms of IUTch, it is necessary to identify/confuse these distinct labels “” and “”[cf. the discussion of (ΘCR), († ΘCR), (SSIdFs)].
From a purely “label-theoretic” point of view, in order to obtain consistentdeﬁnitions of well-deﬁned mathematical objects that are immune, i.e.,
remain unaﬀected, by application of these two operations•→

− − −;



−−−


→

•

31

32

SHINICHI MOCHIZUKI[i.e., of introducing distinct labels for distinct copies “” and “” of asingle copy “•” and identifying/confusing distinct copies “” and “” toa single copy “•”], is to
identify the distinct labels once and for all and work only withmathematical objects that are independent of distinct labels.
On the other hand, as discussed above, this leads to an internal contradiction [cf. the discussion of (VUC1)] in the deﬁnition of the id-versionof (SSId), unless, of course, one regards
all exponents of “q” as being subject to the indeterminacy
given by possible multiplication by factors “j 2 ” — an indeterminacy which, as discussed in (AD), then renders all arithmetic
degree computations meaningless.

SS at times justiﬁed their opposition to the “apparently substantively meaningless” (cf. (T5-2)) use of “labels” by invoking the issue of simplicity (cf. (T7)).
At other times, SS asserted that the omission of labels could be justiﬁed simply
by “remembering” the fact that some arrow (corresponding to the “monodromy
operator” in (LbEx1), “f ·” in (LbEx2), “φ” in (LbEx3), “ψ” in (LbEx4), the “gluing isomorphisms” in (LbEx5), or the “modiﬁed version of the Θ-link” of (VUC1)
in the case of (LbEx6)) is not compatible with certain structures in its domain
and codomain. On the other hand, as one may see in the above examples (LbEx1),(LbEx2), (LbEx3), (LbEx4), (LbEx5), (LbEx6), omitting the labels may, dependingon the speciﬁcs of the situation, easily give rise to a “contradiction” (which is infact meaningless!) that does not occur if one respects the labels, i.e., if one respects
the distinct roles played by distinct copies. In particular, no matter “how good” aparticular mathematician’s memory may be,(DfLb) relying, in mathematical discussions, on declarations of “remembering”
that are not accompanied by precise, explicit documentation of the labeling apparatuses that are employed incurs the risk that diﬀerent people
will “remember” diﬀerent labeling apparatuses, which result in structurally non-equivalent mathematical structures.
That is to say, as one may see in the above examples (LbEx1), (LbEx2), (LbEx3),(LbEx4), (LbEx5), (LbEx6), the substantive mathematical content of a mathematical construction is only completely determined once one makes completely
explicit the labeling apparatus that is adopted in the construction.
§15. In light of the general considerations concerning the use of labels discussedin §14, it is of interest to review the way in which labels for distinct copies ofvarious familiar objects are employed in IUTch in order to construct apparatuses
that play various substantive roles in IUTch that cannot be achieved if the labels
are deleted. One fundamental example of this phenomenon is(LbHT) the bookkeeping apparatus for labels for evaluation points within asingle Hodge theater — an apparatus which plays the very substantive

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)33

role of allowing one to simulate global multiplicative subspaces andcanonical generators.
This phenomenon is discussed in detail in [IUTchI], §I1 (and indeed throughout[IUTchI]!). On the other hand, such labels within a single Hodge theater were only
mentioned very brieﬂy during the March discussions. The “label issues” that werediscussed in substantial detail during the March discussions concern the labels∈ Z × Z that correspond to the “•’s” in the log-theta-lattice. Here, we begin our
discussion of these labels by recalling the (highly noncommutative!) diagram that isused to denote the entire log-theta-lattice (i.e., the LHS of the “⊇”), together withthe portion of the log-theta-lattice (i.e., the RHS of the “⊇”, which consists of thevertical arrows in the 0- and 1-columns, together with the single horizontal arrow
between the •’s labeled (0, 0) and (1, 0)) that is actually used in the main results of
IUTch:
..
..
..
..
.
.
.
.
⏐log
⏐log
⏐log
⏐log
⏐
⏐
⏐
⏐
...

−→

Θ

• −→ • −→ . . .
⏐log
⏐log
⏐
⏐

Θ

Θ

...

−→

Θ

• −→ • −→ . . .
⏐log
⏐log
⏐
⏐

Θ

Θ

...

−→

Θ

• −→ • −→ . . .
⏐log
⏐log
⏐
⏐

Θ

Θ

•
⏐log
⏐

•
⏐log
⏐

..
.

..
.

..
.

•
⏐log
⏐
⊇

..
.

•
⏐log
⏐
Θ

• −→ •
⏐log
⏐log
⏐
⏐

We recall brieﬂy that each • denotes a single Hodge theater (labeled by an element∈ Z × Z), which one thinks of as a single model of the conventional ring/scheme
theory surrounding the elliptic curve over a number ﬁeld under consideration. One
then considers two types of gluing (denoted by the vertical and horizontal arrows in
the diagrams) between certain portions of the Hodge theaters in the domain andcodomain of each arrow. The vertical arrows denote log-links, while the horizontal
arrows denote Θ-links. The following example of the signiﬁcance of the labels∈ Z × Z for distinct •’s is already immediate from the structure of the above
diagrams:(LbLp) WriteΓZ
for the oriented graph given by the portion of the log-theta-lattice consisting of the vertical arrows in the 0-column and the horizontal arrows
originating from the 0-column. WriteΓ0for the oriented subgraph of ΓZ consisting of the vertical arrows in the 0column and the single horizontal arrow between the •’s labeled (0, 0) and34

SHINICHI MOCHIZUKI(1, 0). Then the operation of forgetting the vertical labels amounts to
replacing Γ0 by the oriented graphΓloopgiven by the single horizontal arrow of Γ0 , together with an oriented loop
originating from and terminating at the domain of this single horizontal
arrow. Alternatively, one may think of Γloop as the quotient “ΓZ /Z” ofΓZ by the action of Z on ΓZ via vertical translations. On the other hand,
purely at the level of oriented graphs, it is immediate that ΓZ , Γloop = ΓZ /Z
are structurally non-equivalent to Γ0 . For instance, (unlike ΓZ !) Γ0
fails to admit vertical symmetries; (unlike Γloop !) Γ0 (as well as ΓZ ) is
simply connected (cf. the discussion of (LbEx1)). These fundamental
structural diﬀerences between Γ0 , on the one hand, and ΓZ , Γloop , on
the other, have very substantive consequences in the context of thedeﬁnition of the Θ-link (i.e., the horizontal arrow of Γ0 ), which depends,
in an essential way, on ﬁxing the multiplicative structures of the rings
involved, i.e., is not invariant with respect to the “rotations/juggling”
of the additive and multiplicative structures that occur as one executesvarious iterates of the log-link (cf. the discussion, in the latter portion of[Alien], §3.3, (ii), of the importance, in the context of the deﬁnition of theΘ-link, of distinguishing the ring structures in the domain and codomainof the log-link).
In order to discuss further symmetry properties (cf. (SWE1), (SWE2)) of the log×
theta-lattice, it is useful to apply the notation “Π”, “G”, “O×μ ”, “k”, and “k ” of
(SWE1), (SWE2), (ΘNR), (ΘCR). Here, we recall that k is an algebraically closed
ﬁeld equipped with a p-adic valuation and a natural action by the topologicalgroup G, which, in turn, may be regarded as a quotient of the topological group×
Π, and that k ⊆ k is the multiplicative monoid of nonzero elements of k. Write
O× ⊆ k for the multiplicative group of units (with respect to the valuation) of×
k. (Thus, k, k , and O× are equipped with natural G-actions, and the object“O×μ ” may be regarded as the quotient of O× by its torsion subgroup.) Then
we observe the following non-symmetry properties of the log-theta-lattice (cf.
(SWE1), (SWE2)):(LbΘ) No horizontal arrow (i.e., Θ-link) of the log-theta-lattice admits asymmetry that permutes its domain and codomain. Indeed, this may beseen by considering the portion of a • (i.e., a Hodge theater) consistingof a single copy of k. That is to say, such a symmetry would imply theexistence of a symmetry that permutes the domain and codomain of the
diagram2
××k
qj → q ∈ k2(where j = 1, . . . , l ; q and the q j ’s are regarded up to multiplication
by possibly distinct roots of unity), i.e., the existence of a pair of isomor2∼∼
phisms of topological ﬁelds α : k → k, β : k → k such that α(q j ) = q,2β(q) = q j (up to multiplication by possibly distinct roots of unity). One
veriﬁes immediately that such a pair of isomorphisms does not exist.

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)35(Lblog) No vertical arrow (i.e., log-link) of the log-theta-lattice admits a symmetry that permutes its domain and codomain. Indeed, this may be seen
by considering the portion of a • (i.e., a Hodge theater) consisting of a single copy of k. That is to say, such a symmetry would imply the existenceof a symmetry that permutes the domain and codomain of the diagramlogk ⊇ O× −→ k(where log denotes the p-adic logarithm deﬁned on O× ), i.e., the existence∼∼of a pair of isomorphisms of topological ﬁelds α : k → k, β : k → k suchdef
that, if we write M = log−1 (O× ) ⊆ O× , then α|M = log◦(β|O× )◦(log|M ).
One veriﬁes immediately that such a pair of isomorphisms does not exist.
(LbMn) WriteΓmainfor the oriented graph given by the portion of the log-theta-lattice that is
actually used in the main results of IUTch, i.e., the portion consisting of
the vertical arrows in the 0- and 1-columns and the single horizontal arrow
between the •’s labeled (0, 0) and (1, 0). Then one veriﬁes immediately (cf.
(LbΘ), (Lblog)!) that the portion determined by Γmain of the log-thetalattice does not admit a symmetry that permutes the 0- and 1-columns.
It is precisely because of these non-symmetry properties (LbΘ), (Lblog), (LbMn)
that (at least from an a priori point of view!)
it is of crucial importance to specify the labels ∈ Z × Z when discussing
×various portions (such as “k”, “k ”, “O× ”) of the •’s (i.e., Hodge theaters)in the log-theta-lattice.
The portions of the •’s (i.e., Hodge theaters) that (at least from an a priori pointof view!) depend on the labels ∈ Z × Z are referred to, in IUTch, as Frobeniuslike. On the other hand, objects arising from certain Galois groups (such as G)
or arithmetic fundamental groups (such as Π) are referred to as étale-like. If one
restricts oneself to considering isomorphisms between objects, for instance, in
situations that often arise in anabelian geometry, then there is not so much of adiﬀerence between Frobenius-like and étale-like versions of various objects. On theother hand, when one considers the relationship between objects on opposite sides ofthe Θ- and log-links, there is a very fundamental diﬀerence between Frobenius-likeand étale-like versions of various objects, namely:(EtFr) Whereas Frobenius-like objects on opposite sides of the Θ- andlog-links are (typically) related to one another by means of some sort ofrelationship that is far from being an isomorphism (e.g., far from being
compatible with the respective ring structures!), étale-like objects on
opposite sides of the Θ- and log-links are (typically) related to one anotherby means of an isomorphism.
Perhaps the most fundamental classical example of this sort of situation is thesituation that arises in the case of the anabelian geometry of one-dimensional
function ﬁelds over ﬁnite ﬁelds, in the context of the Frobenius morphism

36

SHINICHI MOCHIZUKI

in positive characteristic [cf. the discussion of [Alien], §2.6, §2.7, §2.8, for moredetails]:def(EtFrEx) Consider the ﬁeld F = Fp (t) (i.e., a purely transcendental extension of
the ﬁeld of cardinality p), for some prime number p. Then the Frobenius
morphism φ : F → F , i.e., the ﬁeld homomorphism that maps t → tp ,induces an isomorphism∼

Aut(F ) → Aut(F )between the respective groups of ﬁeld automorphisms, despite the fact thatφ is far from being an isomorphism.
(Indeed, it is precisely this fundamental classical example that gave rise to theterminology “Frobenius-like” and “étale-like”.) Unlike the various Frobenius-likeobjects discussed in (LbΘ), (Lblog), (LbMn), étale-like objects admit important
symmetry properties as follows (cf. (SWE1), (SWE2)):(EtΘ) It is a tautological consequence of the deﬁnitions that the “G’s” in thedomain and codomain of the Θ-link — i.e., which act on the copies of O× ⊆×k ⊆ k in the domain and codomain of the Θ-link — map isomorphicallyto one another via the Θ-link and, moreover, when regarded as abstract
topological groups (i.e., which are not equipped with any data arising×
from the usual actions of G on some copy of O× ⊆ k ⊆ k!), may be
permuted with one another.
(Etlog) It is a tautological consequence of the deﬁnitions that the “Π’s” in thedomain and codomain of the log-link — i.e., which act on the copies of×
O× ⊆ k ⊆ k in the domain and codomain of the log-link — map isomorphically to one another via the log-link and, moreover, when regarded
as abstract topological groups (i.e., which are not equipped with any
×data arising from the usual actions of Π on some copy of O× ⊆ k ⊆ k!),may be permuted with one another.
(EtMn) It is a tautological consequence of the deﬁnitions that the diagramΠ  G  Π
— where the Π’s and G are regarded as abstract topological groups(which are only known up to isomorphisms of topological groups); the , are the poly-morphisms (i.e., sets of morphisms) given by the compositesof the natural surjection Π  G with arbitrary automorphisms of thedomain and codomain; the Π on the LHS (respectively, RHS) arises fromthe 0- (respectively, 1-)column of Γmain (cf. (LbMn), (Etlog)); the G arises
from the unique horizontal arrow of Γmain (cf. (LbMn), (EtΘ)) — admits
a symmetry that permutes the Π’s on the LHS and RHS and ﬁxes theunique G of the diagram.
As discussed in §8, these symmetry properties (EtΘ), (Etlog), (EtMn) are thefundamental properties that underlie the multiradial algorithms of IUTch (i.e.,[IUTchIII], Theorem 3.11) and, in particular, play a central role in IUTch.

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)37§16. One sort of undercurrent theme that underlies the various misunderstandings(cf. §2) discussed so far in the present report is(DsGl) the deep sense of discomfort that some mathematicians feel withregard to the gluing/identiﬁcation given by the value group portion ofthe Θ-link (cf. §9)2∼{q j }N → q N— a sense of discomfort that is perhaps partially related to the discussion
surrounding (HstMod).
One possible approach to overcoming this sense of discomfort is to recall various
classical examples of closely related phenomena.
Perhaps the most fundamental such “classical example” is(GlMt) the special case of the property of invariance of heights with respect toisogeny (established by Faltings) in the case of global multiplicative
subspaces of modules of torsion points of elliptic curves over number
ﬁelds, i.e., subspaces that coincide, at all nonarchimedean primes of bad
multiplicative reduction, with the subspace arising from the copy of themultiplicative group scheme “Gm ” that occurs in the Tate uniformization(cf. [Alien], §2.3, for more details).
That is to say, such isogenies arising from global multiplicative subspaces correspond (say, in the case of l-torsion points, for l a prime number) locally at nonarchimedean primes of bad reduction to the operation of raising q-parameters to thel-th powerq → ql(cf. (DsGl)). In this context, it is useful to recall that:(DFL) As discussed in [Alien], §2.3, §2.4, §2.5, this example (GlMt) is closely
related to the technique of taking derivatives of Frobenius liftings,
which, as discussed in [Alien], §2.6, §2.7, motivated the development of
the technique of mono-anabelian transport, which plays a central role
in IUTch.
One way to summarize the essential mechanism involved here is as follows:(DiIsm) On the one hand, the assignment
q

→

ql

(cf. (DsGl)) is, in essence, a “dilation”, i.e., multiplies heights/degrees/logvolumes by a factor of l. On the other hand, this assignment inducesdlog(q)→l · dlog(q)(essentially, since the factor of l preceding the “dlog(q)” can typically, forvarious technical reasons, be ignored!) an “isometry”. Moreover, since
the arithmetic line bundle “ω” of invariant diﬀerentials on each of theelliptic curves under consideration arises from an ample line bundle on
the moduli stack of elliptic curves, its arithmetic degree (i.e., the height of38

SHINICHI MOCHIZUKI

the elliptic curve) is suﬃcient to control the moduli of the elliptic curve (upto ﬁnitely many possibilities). It is precisely this “tension” between the“dilation” and “isometry” aspects of the assignment q → q l , together withthe ampleness just mentioned, that gives rise to nontrivial consequences
concerning the elliptic curves involved.
Relative to the analogy with the technique of mono-anabelian transport (andhence with IUTch!):(MAT1) the assignment “q → q l ” corresponds to the link involved, i.e., such as
the Θ-link of IUTch;(MAT2) the dilation aspect of this assignment corresponds to the eﬀect of thelink on Frobenius-like objects;(MAT3) the isometry aspect of this assignment corresponds to the eﬀect of thelink on étale-like objects (cf., e.g., the switching symmetry discussedin (EtMn), which plays a crucial role in establishing the multiradial algorithms of IUTch!);(MAT4) the ampleness of “ω”, which may thought of as a sort of manifestationof hyperbolicity, corresponds to the mono-anabelian properties thatare applied in the technique of mono-anabelian transport, i.e., properties
that ensure that the isometry (e.g., étale-like) aspect is “suﬃciently rich”
to capture the objects of interest in their entirety.
Another fundamental “classical example” is the analogy between the multiradial representation algorithms developed in IUTch (i.e., [IUTchIII], Theorem 3.11) and the classical theory of connections, which we consider here fromthe point of view of Grothendieck’s deﬁnition of the notion of a connection, i.e.,
the point of view that gave rise to the notion of a crystal. A detailed discussionof this analogy — which includes a discussion of the analogy between the main
results of IUTch (i.e., [IUTchIII], Theorem 3.11; [IUTchIII], Corollary 3.12) andthe classical argument used to prove the geometric version of the Szpiro Conjecture
by considering the Kodaira-Spencer morphism — may be found in [Alien], §3.1, (v).
That is to say,
(MlCr1) just as the structure of a crystal allows one to relate liftings of an object
to distinct nilpotent thickenings of a given (say, reduced) scheme thatcoincide over the given scheme,(MlCr2) the multiradial representation algorithms developed in IUTch (i.e.,[IUTchIII], Theorem 3.11 — cf. also the discussion of [IUTchIII], Remark
3.11.1) allow one to relate distinct ring-theoretic realizations, i.e., via
2{q j } or q (which correspond to distinct thickenings), of the gluing datathat appears in the Θ-link, i.e., “prime-strips” of a certain type (which correspond to the object associated by a crystal to the given reduced scheme,i.e., for instance, a positive characteristic scheme).
Here, we recall (cf. the discussion at the beginning of [Alien], §3.1, (v)) that thisanalogy — which involves a comparison between degrees of ample line bundles (i.e.,“heights”) and “0” (i.e., as opposed to some large positive multiple of the “height”)— may be thought of as a sort of limiting version of a transformation of the form
q

→

qN

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)39(cf. (DsGl)) as N → +∞. The analogy between (MlCr1) and (MlCr2) is also ofinterest from a historical point of view:(MlCrH1) just as the classical scheme-theoretic argument of [Alien], §3.1, (v), which
presents the Kodaira-Spencer morphism from the point of view of (log)crystals and nilpotent thickenings of (log) schemes, only makes sense if
one considers distinct non-reduced schemes, i.e., distinct thickenings of a given reduced scheme — notions that were unfamiliar to many
algebraic geometers in the 1960’s, who were accustomed only to working with more classical notions of “varieties”, e.g., notions that revolvearound considering collections of subrings of some ﬁxed ﬁeld,(MlCrH2) the theory surrounding the main results of IUTch (i.e., [IUTchIII], Theorem 3.11; [IUTchIII], Corollary 3.12) only makes sense if one considers
distinct copies of ring/scheme theory (cf. (Smm), (GLR2), (IUAD)!)
glued together along data of the sort that appears in the Θ-link (i.e.,“prime-strips” of a certain type) — notions that are unfamiliar to manyarithmetic geometers today, who are accustomed only to working within
a ﬁxed copy of scheme theory.
Yet another fundamental “classical example” may be seen in Bogomolov’s
proof of the geometric version of the Szpiro Conjecture. We refer to [BogIUT], as
well as [Alien], §4.3, (iii), for a detailed discussion of the analogy between Bogomolov’s proof and IUTch. One way to understand the essential mathematicalphenomenon that underlies Bogomolov’s proof is to consider the following elementary observation:(SLZR) The unipotent matrices
1

m

0 1
— where m ranges over the positive integers — are conjugate to one
another in SL2 (R), but are not conjugate to one another in SL2 (Z).
Here, we recall that, from the point of view of the classical geometry of the upper
half-plane, the unipotent matrix in (SLZR) may be thought of as the monodromy
operator associated to the m-th power q m of the q-parameter q = exp(2πiz) (wherez denotes the standard complex coordinate on the upper half-plane). Bogomolov’s
proof centers around a(BgCp) comparison between a certain collection of 2 × 2 matrices regarded
as elements of SL2 (Z) and the same collection of matrices regarded as
elements of SL2 (R), i.e., a comparison (cf. (SLZR)!) between
some ﬁxed/rigid (positive integer) power “q m0 ” of qandan indeterminate power “q m ” of q(i.e., where m ranges over the positive integers) — cf. (DsGl).
More precisely, Bogomolov’s proof involves liftings of elements in SL2 (R) (or SL2 (Z))
to the canonical extension
1

→

Z

→

SL2 (R)∼→

SL2 (R)→

1

40

SHINICHI MOCHIZUKI(or the restriction SL2 (Z)∼ of this extension to SL2 (Z) ⊆ SL2 (R)). From amore scheme-theoretic point of view, the comparison in Bogomolov’s proof may bethought of as a comparison between various “realizations of the ﬁrst Chern class”
of the Hodge bundle ω (i.e., arising from the invariant diﬀerentials on the tautological elliptic curve) on the moduli stack of elliptic curves Mell (and its canonicalcompactiﬁcation Mell ). That is to say:
(Chrn) The “SL2 (Z)∼ side” of the comparison in Bogolomov’s proof corresponds
to a sort of geometric version of the complex holomorphic realization of the ﬁrst Chern class of ω (with compact supports relative to thecompactiﬁcation Mell ⊆ Mell ), while the “SL2 (R)∼ side” of the comparison in Bogolomov’s proof corresponds to a sort of metric/diﬀerentialgeometric realization of the ﬁrst Chern class of ω, via various canonicalmetrics arising from the hyperbolic geometry of the upper half-plane.
At a more concrete level, the comparison — between the point of view of working
inside SL2 (R)∼ and the point of view of working inside SL2 (Z)∼ — in Bogomolov’s
proof centers around consideration of a certain relation
αg , βg ] · γ
1 · . . . · γ
r = (τ || )2n[
α1 , β1 ] · . . . · [(cf. the notation of the discussion of [BogIUT] preceding (B4)) between elements∈ SL2 (Z)∼ ⊆ SL2 (R)∼ . This “linking data” (i.e., consisting of the elementsthat appear in the above relation, together with the relation itself) between the“SL2 (R)∼ side” and “SL2 (Z)∼ side” of the comparison in Bogolomov’s proof isinteresting, relative to the analogy with IUTch, in that the “
γi ’s” (for i = 1, . . . , r)correspond to the local q-parameters of the family of elliptic curves under consid
eration, while (the integer power that appears in) “(τ || )2n ” corresponds to the(global) height of the family of elliptic curves under consideration. That is to say,
(LkDt) The local “γi ’s” (for i = 1, . . . , r) may be understood as corresponding
to the local value group portion (at, say, bad primes) of the gluing
data (i.e., “prime-strips” of a certain type) that appears in the Θ-link,
while the global “(τ || )2n ” may be understood as corresponding to theglobal realiﬁed Frobenioid portion (i.e., global value group portion)of the gluing data (i.e., “prime-strips” of a certain type) that appears inthe Θ-link. (Here, we note in passing that one may also interpret the“[αj , βj ]’s” (for j = 1, . . . , g) in this context as corresponding to the local
unit group portion (at, say, good primes) of the gluing data (i.e., “primestrips” of a certain type) that appears in the Θ-link, where we recall that
this local unit group portion gives rise, via log-volume computations of logshells, to the log-diﬀerent term (i.e., an arithmetic analogue of the Euler
characteristic 2 − 2g) in the bound for the height of an elliptic curve overa number ﬁeld obtained in [IUTchIV], Theorem 1.10.)
Bogomolov’s proof then proceeds by comparing(Bg1) a certain displacement estimate — which is invariant with respect toindependent SL2 (R)∼ -conjugation indeterminacies acting on each of thelinking data elements — that arises from considering the above linking
data in SL2 (R)∼ (cf. the discussion preceding [BogIUT], (B5))

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)

41

with
(Bg2) a certain computation of the height of the family of elliptic curves
under consideration in terms of the local q-parameters (cf. [BogIUT], (B4),
together with the discussion immediately preceding [BogIUT], (B4)) thatarises from considering the above linking data in SL2 (Z)∼ .
Relative to the analogy with IUTch,(BgIUT1) the displacement estimate in Bogomolov’s proof — which arises fromworking with indeterminate powers of q-parameters (cf. the independent SL2 (R)∼ -conjugation indeterminacies!), relative to SL2 (R)∼ — corresponds to the multiradial representation of IUTch (i.e., [IUTchIII],
Theorem 3.11), while(BgIUT2) the computation of the height in terms of local q-parameters in
Bogomolov’s proof — which arises from working with rigid powers of
q-parameters, relative to SL2 (Z)∼ — corresponds to the argument of[IUTchIII], Corollary 3.12.
Moreover, in this context, it is of interest to observe, relative to the analogy with
IUTch, that:(DsInd) This computation via displacement estimates in Bogomolov’s proof
furnishes a fascinating and relatively elementary example of the phenomenon of computing the “size” of a region subject to indeterminacies (cf.
(Bg1), (BgIUT1), i.e., the displacement estimates in Bogomolov’s proof,
which are invariant with respect to the independent SL2 (R)∼ -conjugationindeterminacies acting on each of the linking data elements; the multiradial representation of the Θ-pilot object in IUTch) that approximates(say, from above) some region of interest that is “rigid” and not subject
to indeterminacies (cf. (Bg2), (BgIUT2), i.e., the linking data in Bogomolov’s proof considered in SL2 (Z)∼ ; the Θ-pilot relative to the Θ-hol.
str. in IUTch).
Finally, we observe that the partition of Bogomolov’s proof mentioned above into(Bg1)/(BgIUT1) and (Bg2)/(BgIUT2) is of interest not only from the point of viewof the correspondence with [IUTchIII], Theorem 3.11, and [IUTchIII], Corollary
3.12, but also from a historical point of view. Indeed, (Bg1) (though apparently
rediscovered by Bogomolov around the year 2000) was already known to Milnor in
the 1950’s, while (Bg2) was, apparently, not known until Bogomolov’s work aroundthe year 2000.
§17. The fundamental misunderstandings of IUTch discussed in the present report
may be summarized as a failure to understand the following central aspects of
IUTch:(FM1) the signiﬁcance of the operation of re-initialization of histories (cf.
(T1), (H2)), the explicit speciﬁcation of types of mathematicalobjects (cf. (T2), (H2)), and the use of poly-morphisms (cf. (T4),(PMInd), (PMEss)) in the construction of mathematical structures (e.g.,
in IUTch) that admit switching symmetries (cf. (SW), (Sym), (SWC1),

42

SHINICHI MOCHIZUKI(SWC2), (SWE1), (SWE2), (LbΘ), (Lblog), (LbMn), (EtΘ), (Etlog),(EtMn));(FM2) the signiﬁcance of explicit speciﬁcation of label apparatuses (cf.
(T3), (T5), (IUAD), (SSDLFs), (LbRl), (LbEx1), (LbEx2), (LbEx3),(LbEx4), (LbEx5), (LbEx6), (LbHT), (LbΘ), (Lblog), (LbMn)) to distinguish distinct copies of various familiar objects that play substantively
diﬀerent roles within a complicated mathematical structure;(FM3) the signiﬁcance of the requirement that the various data (i.e., value
group portion and unit group portion) that occurs in the domain of theΘ-link arise from a single ring str. (cf. (ΘCR), (VUSQ), (VUC)) — a
requirement that must be satisﬁed in order to apply the multiradial representation algorithms developed in IUTch, i.e., [IUTchIII], Theorem
3.11 (cf. (T3), (SSInd), (GIUT), (SSId), († ΘCR), (SSIdFs), (SSADFs)).
In this context, we observe that the point of view of (FM1), i.e., of “forgetting
histories”, may, at ﬁrst glance, appear to lie in a contradictory direction to thepoint of view of (FM2), i.e., of “remembering labels”. In fact, however, although, depending upon the content of various technical goals, it may be necessary
to “remember” some mathematical structures in some situations and to “forget”
other mathematical structures in other situations,the common thread that unites (FM1) and (FM2) is the principle that it isalways of fundamental importance to specify explicitly which structures
one wishes to remember or forget.
One way to summarize (FM1), (FM2), (FM3) is as follows:(MVEx) Each of (FM1), (FM2), (FM3) centers around the issue of experimenting
with various modiﬁed versions of IUTch to examine to what extent
the essential content of IUTch continues to hold in such modiﬁed versions.
Since, typically, research papers in mathematics (such as, for instance, the papers[IUTchI], [IUTchII], [IUTchIII], [IUTchIV]!) do not contain any exercises, experimenting with various modiﬁed versions of a mathematical theory, i.e., such as thosethat appear in (FM1), (FM2), (FM3), is an important step in acquiring a deep understanding of the theory. From this point of view, the experimental versions of
IUTch that appear (either explicitly or implicitly!) in (FM1), (FM2), (FM3) may
be regarded as a valuable pedagogical tool for (both present and future) students of IUTch and, in particular, as an important step forward (cf. the discussionof §3) towards the goal ofcompletely diagnosing and explicating the logical structure of all misunderstandings concerning IUTch.
In particular, one might hope (cf. the discussion of §3) that (FM1), (FM2), (FM3)
may serve to stimulate the further examination and development of such exercises/experimental versions of IUTch.
§18. In the context of the present report, it is important to recall that(Vrf1) IUTch has been checked, veriﬁed, read and reread, and orally exposedin detail in seminars in its entirety countless times since the release of

DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)43

preprints on IUTch in August 2012 by a collection of mathematicians (notincluding myself) involved in this line of research. (For instance, Fesenko
estimates, in the most recent updated version of §3.1 of his survey [Fsk],
that IUTch has been veriﬁed at least 30 times.) This collection of mathematicians has (together with me) also been actively involved in detailed
discussions and dialogues with mathematicians who have any questions
concerning IUTch.
Such discussions have been useful in enhancing the level of understanding of IUTchby many mathematicians, as well as in ﬁnding minor inessential inaccuracies, misprints, etc. in the preprints on IUTch. Such discussions have also served to conﬁrm
countless times that(Vrf2) there is no substantive mathematical reason whatsoever to suspect
the existence of any oversights, i.e., any ﬂaws or incompleteness ofa mathematically substantive nature, in the numerous veriﬁcations thathave been conducted so far concerning IUTch.
In light of this state of aﬀairs, it should not be surprising that my oral explanations,over the past few months, to various colleagues involved in (Vrf1), (Vrf2) of themisunderstandings summarized in §17 were met with a remarkably unanimous
response of utter astonishment and even disbelief (at times accompanied bybouts of laughter!) that such manifestly erroneous misunderstandings could haveoccurred. At times, this sort of response put me in the somewhat paradoxical
position of having to “justify” or “defend” the line of reasoning of SS that apparently
led to such misunderstandings. Put another way,
(CMA) the ﬂaws in IUTch that are alleged in (SSId), (SSAD) — i.e., whichcorrespond to the most central aspects of the misunderstandings summarized in §17 — are so utterly pronounced and conspicuous in theirabsurdity that it is very diﬃcult to believe that such ﬂaws could haveremained undetected by any competent mathematician for even a few minutes, let alone throughout the duration of the intensive veriﬁcation activities (cf. (Vrf1), (Vrf2)) by quite a number of talented mathematiciansthat have been carried out over the years since August 2012.
Careful reﬂection concerning this state of aﬀairs over the months since the March
discussions has led me to the following conclusion:(DfMth) Perhaps the most reasonable explanation for the state of aﬀairs summarized above in (Vrf1), (Vrf2), (CMA) is that the mathematics that SSperceive as being referred to by the term “IUTch” is simply substantiallydiﬀerent from the mathematics that HM (as well as other mathematicians
who have been deeply involved in the various activities concerning IUTch
described in (Vrf1), (Vrf2)) understand by the term “IUTch” (cf. (Smm),(GLR2), (T6), (T7), (T8), (SSId), (SSAD), (DfLb)).
Indeed, at numerous points in the March discussions, I was often tempted to issue
a response of the following form to various assertions of SS (but typically refrained
from doing so!):Yes! Yes! Of course, I completely agree that the theory that you are
discussing is completely absurd and meaningless, but that theory is completely diﬀerent from IUTch!

44

SHINICHI MOCHIZUKI

Nevertheless, the March discussions were productive in the sense that they yieldeda valuable ﬁrst glimpse at the mathematical content of the misunderstandings that
underlie criticism of IUTch (cf. the discussion of §3). In the present report, weconsidered various possible causes for these misunderstandings, namely:(PCM1) lack of suﬃcient time to reﬂect deeply on the mathematics underdiscussion (cf. the discussion in the ﬁnal portions of §2, §10);(PCM2) communication issues and related procedural irregularities (cf.
(T6), (T7), (T8));(PCM3) a deep sense of discomfort, or unfamiliarity, with new ways ofthinking about familiar mathematical objects (cf. the discussion of §16;
[Rpt2014], (T2); [Fsk], §3.3).
On the other hand, the March discussions were, unfortunately, by no means
suﬃcient to yield a complete elucidation of the logical structure of the causes
underlying the misunderstandings summarized in §17.

Bibliography[Rtv] Wikipedia article: Criticism of the theory of relativity, available at:https://en.wikipedia.org/wiki/Criticism of the theory of relativity[Fsk] I. Fesenko, Arithmetic deformation theory via arithmetic fundamental groups
and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki,
Eur. J. Math. 1 (2015), pp. 405-440; updated version available at:https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf[Gll] Wikipedia article: Galileo Galilei, available at:https://en.wikipedia.org/wiki/Galileo Galilei[FKvid] F. Kato, The abc Conjecture and New Mathematics - Prof. Fumiharu Kato,Oct 7, 2017 (with English subtitles), video (with English subtitles) available
at:https://www.youtube.com/watch?v=fNS7N04DLAQ&feature=youtu.be[IUTchI] S. Mochizuki, Inter-universal Teichmüller Theory I: Construction of Hodge
Theaters, RIMS Preprint 1756 (August 2012).
[IUTchII] S. Mochizuki, Inter-universal Teichmüller Theory II: Hodge-Arakelov-theoretic
Evaluation, RIMS Preprint 1757 (August 2012).
[IUTchIII] S. Mochizuki, Inter-universal Teichmüller Theory III: Canonical Splittings of
the Log-theta-lattice, RIMS Preprint 1758 (August 2012).
[IUTchIV] S. Mochizuki, Inter-universal Teichmüller Theory IV: Log-volume Computations and Set-theoretic Foundations, RIMS Preprint 1759 (August 2012).
[Rpt2014] S. Mochizuki, On the Veriﬁcation of Inter-universal Teichmüller theory: a
Progress Report (as of December 2014), December 2014, available at:DISCUSSIONS ON IUTCH (MARCH 15 – 20, 2018)45

http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202014-12.pdf[QC2016] S. Mochizuki, On Questions and Comments Concerning Inter-universal Teichmller Theory (IUT), July 2016, available at:
http://www.kurims.kyoto-u.ac.jp/~motizuki/Questions%20and%20
Comments%20on%20IUT.pdf[Alien] S. Mochizuki, The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmüller Theory, RIMS Preprint 1854 (July 2016).
[BogIUT] S. Mochizuki, Bogomolov’s proof of the geometric version of the Szpiro Conjecture from the point of view of inter-universal Teichmüller theory, Res. Math.
Sci. 3 (2016), 3:6.
[Ymgt] S. Yamagata, A counterexample for the local analogy of a theorem by Iwasawa
and Uchida, Proc. Japan Acad. 52 (1976), pp. 276-278.
Updated versions of preprints are available at the following webpage:
http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html