RIMS/SYMMETRIES AND CORRESPONDENCES WORKSHOP:
INTER-UNIVERSAL TEICHMÜLLER THEORY SUMMIT 2016

Organizers: Ivan Fesenko (The University of Nottingham)
Shinichi Mochizuki (RIMS, Kyoto University)
Yuichiro Taguchi (Tokyo Institute of Technology)
Dates: July 18 – 27, 2016
Place: RIMS, Kyoto University, Room 420 (July 18 – 22),
Room 111 (July 25 – 27)
July 18 (Monday)
09:30 – 09:40 Opening address
Part I: Preparatory talks
09:45 – 10:45 Koichiro Sawada (RIMS, Kyoto University)
Uchida’s theorem for one-dimensional function ﬁelds over ﬁnite ﬁelds
11:00 – 12:00 Kazumi Higashiyama (RIMS, Kyoto University)
Mono-anabelian geometry I: Reconstruction of function ﬁelds via Belyi
cuspidalization
12:15 – 13:15 Arata Minamide (RIMS, Kyoto University)
Mono-anabelian geometry II: Mono-anabelian geometry over mixed
characteristic local ﬁelds
14:45 – 15:45 Ippei Nagamachi (The University of Tokyo)
Log-shell, log-volume, and log-link I
16:00 – 17:00 Ippei Nagamachi (The University of Tokyo)
Log-shell, log-volume, and log-link II
17:15 – 18:15 Weronika Czerniawska (The University of Nottingham)
Frobenioids 1
July 19 (Tuesday)
09:30 – 10:30 Weronika Czerniawska (The University of Nottingham)
Frobenioids 2
10:45 – 11:45 Arata Minamide (RIMS, Kyoto University)Étale theta functions and mono-theta environments I
12:00 – 13:00 Arata Minamide (RIMS, Kyoto University)Étale theta functions and mono-theta environments II
14:30 – 15:30 Seidai Yasuda (Osaka University)Étale theta functions, mono-theta environments, and [IUTchI] §1 – §3, I

15:45 – 16:45 Seidai Yasuda (Osaka University)Étale theta functions, mono-theta environments, and [IUTchI] §1 – §3, II
Part II: Overall survey
17:00 – 18:00 Shinichi Mochizuki (RIMS, Kyoto University)
The mathematics of mutually alien copies: from Gaussian integrals tointer-universal Teichmüller theory I

July 20 (Wednesday)
09:30 – 10:30 Shinichi Mochizuki (RIMS, Kyoto University)
The mathematics of mutually alien copies: from Gaussian integrals tointer-universal Teichmüller theory II
10:45 – 11:45 Shinichi Mochizuki (RIMS, Kyoto University)
The mathematics of mutually alien copies: from Gaussian integrals tointer-universal Teichmüller theory III
12:00 – 13:00 Shinichi Mochizuki (RIMS, Kyoto University)
The mathematics of mutually alien copies: from Gaussian integrals tointer-universal Teichmüller theory IV
14:30 – 15:30 Shinichi Mochizuki (RIMS, Kyoto University)
The mathematics of mutually alien copies: from Gaussian integrals tointer-universal Teichmüller theory V
15:45 – 16:45 Shinichi Mochizuki (RIMS, Kyoto University)
The mathematics of mutually alien copies: from Gaussian integrals tointer-universal Teichmüller theory VI
Part III: [IUTchI-II]
17:00 – 18:00 Seidai Yasuda (Osaka University)Étale theta functions, mono-theta environments, and [IUTchI] §1 – §3, III

July 21 (Thursday)
Part II: Overall survey (continued)
09:30 – 10:30 Shinichi Mochizuki (RIMS, Kyoto University)
The mathematics of mutually alien copies: from Gaussian integrals tointer-universal Teichmüller theory VII
10:45 – 11:45 Shinichi Mochizuki (RIMS, Kyoto University)
The mathematics of mutually alien copies: from Gaussian integrals tointer-universal Teichmüller theory VIIIPart III: [IUTchI-II] (continued)
12:00 – 13:00 Fucheng Tan (Shanghai Center for Mathematical Sciences andShanghai Jiao Tong University)
IUT-I: Hodge theaters and label classes of cusps 1

14:30 – 15:30 Fucheng Tan (Shanghai Center for Mathematical Sciences andShanghai Jiao Tong University)
IUT-I: Hodge theaters and label classes of cusps 2
15:45 – 16:45 Taylor Dupuy (Hebrew University and the University of Vermont)
Functors to and from mono-theta environments
17:00 – 18:00 Emmanuel Lepage (University Paris 6)
Hodge-Arakelov evaluation 1
19:00 – 21:00 Buﬀet style dinnerJuly 22 (Friday)
09:30 – 10:30 Taylor Dupuy (Hebrew University and the University of Vermont)
Multiradiality
10:45 – 11:45 Emmanuel Lepage (University Paris 6)
Hodge-Arakelov evaluation 2
Part IV: [IUTchIII-IV] from the point of view of mono-anabelian transport
12:00 – 13:00 Yuichiro Hoshi (RIMS, Kyoto University)[IUTchIII-IV] from the point of view of mono-anabelian transport I
14:30 – 15:30 Yuichiro Hoshi (RIMS, Kyoto University)[IUTchIII-IV] from the point of view of mono-anabelian transport II
15:45 – 16:45 Yuichiro Hoshi (RIMS, Kyoto University)[IUTchIII-IV] from the point of view of mono-anabelian transport III
17:00 – 18:00 Yuichiro Hoshi (RIMS, Kyoto University)[IUTchIII-IV] from the point of view of mono-anabelian transport IV

July 25 (Monday)
Part V: Related topics
09:30 – 10:30 Yuki Wada (RIMS, Kyoto University)
Near miss abc-triples in compactly bounded subsets
10:45 – 11:45 Vesselin Dimitrov (Yale University)
Notes on the epsilon term in the abc conjecture
12:00 – 13:00 Kobi Kremnitzer (The University of Oxford)
Milnor-Wood inequality and bounded cohomology
14:30 – 15:30 Ivan Fesenko (The University of Nottingham)
Reciprocity and IUT
15:45 – 16:45 Boris Zilber (The University of Oxford)
Model theory of anabelian geometry
17:00 – 18:00 Free discussion

July 26 (Tuesday)
Part VI: [IUTchIII-IV] with remarks on the function-theoretic roots of the theory
09:30 – 10:30 Go Yamashita (RIMS, Kyoto University)[IUTchIII-IV] with remarks on the function-theoretic roots of the theory I
10:45 – 11:45 Go Yamashita (RIMS, Kyoto University)[IUTchIII-IV] with remarks on the function-theoretic roots of the theory II
12:00 – 13:00 Go Yamashita (RIMS, Kyoto University)[IUTchIII-IV] with remarks on the function-theoretic roots of the theory III
14:30 – 15:30 Go Yamashita (RIMS, Kyoto University)[IUTchIII-IV] with remarks on the function-theoretic roots of the theory IV
15:45 – 16:45 Free discussion
17:00 – 18:00 Free discussion

July 27 (Wednesday)
Part VII: Overall discussion and impromptu talks
09:30 – 10:10 Adam Topaz (University of California, Berkeley)
Kummer theory and anabelian geometry over algebraically closed ﬁelds
10:25 – 10:45 Dinesh Thakur (Rochester University)
Diophantine approximation and deformation hierarchies in ﬁnite characteristic
11:00 – 12:00 Paul Vojta (University of California, Berkeley)
On changes to the Thue-Siegel method
12:15 – 13:00 Free discussion
14:30 – 15:30 Free discussion
15:45 – 16:45 Free discussion
17:00 – 18:00 Free discussion
18:00 – 18:15 Closing remarks

Abstracts

Part I: Preparatory talks
Koichiro Sawada (RIMS, Kyoto University)
Uchida’s theorem for one-dimensional function ﬁelds over ﬁnite ﬁelds
Abstract:
Uchida proved that a one-dimensional function ﬁeld over a ﬁnite ﬁeld can be reconstructed from the absolute Galois group. In this talk, I will explain this theorem.
Kazumi Higashiyama (RIMS, Kyoto University)
Mono-anabelian geometry I: Reconstruction of function ﬁelds via Belyi
cuspidalization
Abstract:
In this talk, we study mono-anabelian geometry. In more concrete terms, we prove
the following assertion: Let k0 be a number ﬁeld, X0 /k0 a hyperbolic curve which
is isogenous to a hyperbolic curve of genus 0, k0 ⊆ k, where k is sub-p-adic. Then
we reconstruct group-theoretically the function ﬁeld k 0 (X0 ) of X0 ×k0 k 0 from
1 → π1 (X0 ×k0 k) → π1 (X0 ×k0 k) → Galk → 1(regarded as an exact sequence of abstract proﬁnite groups), via the technique of
Belyi cuspidalization.
Arata Minamide (RIMS, Kyoto University)
Mono-anabelian geometry II: Mono-anabelian geometry over mixed
characteristic local ﬁelds
Abstract:
Let k be a mixed characteristic local ﬁeld, X a hyperbolic curve of strictly Belyi
type over k. In the present talk, I will discuss:(i) group-theoretic algorithms for reconstructing various invariants, groups, andmonoids related to k from the absolute Galois group of k;(ii) cyclotomic rigidity via local class ﬁeld theory;(iii) a group-theoretic algorithm for reconstructing the function ﬁeld of X fromthe étale fundamental group of X.
Ippei Nagamachi (The University of Tokyo)
Log-shell, log-volume, and log-link I - II
Abstract:
In this talk, I will explain some construction algorithms and notions which are main
ingredients of “log-link” in IUT. First, I will give an algorithm for constructing a
new ﬁeld k ∼ from an algebraic closure k of a mixed characteristic local ﬁeld k by perfection of its unit group and p-adic logarithm. Then I deﬁne “log-volume” (notion
corresponding to Haar measure) and “log-shell” (the subgroup of k ∼ nomalizing
log-volume). These notions can be mono-analytically reconstructed. More precisely, we can deﬁne the (étale-like) log-shell and log-volume for the reconstructed(k ∼ )× from a proﬁnite group which is isomorphic to some absolute Galois group ofa mixed characteristic local ﬁeld, despite we can not reconstruct the ﬁeld structureof k ∼ . A key diagram in my talk is the following one
Ok

⊇

Ok×

perfection−→(Ok× )pf=:k∼ .

Here, Ok is the multiplicative monoid Ok \ {0}, which plays a crucial role foryielding cyclotomic rigidity.
Weronika Czerniawska (The University of Nottingham)
Frobenioids 1-2
Abstract:
Frobenioids are a categorical generalization of the line bundles or monoids of divisors over a Galois category. Almost all of frobenioids occuring in the IUT papers
are model frobenioids, among them crucial ones are associated to submonoids offamiliar abelian groups in in local and global class ﬁeld theories. In my talks I
will discuss their basic properties including a torsor-theoretic approach. I will thenpresent various results from the theory of Frobenioids which are important fromthe point of view of studying IUT.
Arata Minamide (RIMS, Kyoto University)Étale theta functions and mono-theta environments I
Abstract:
In this talk, ﬁrst, I will review basic properties of a theta function on a tempered
covering of a once-punctured elliptic curve over a p-adic local ﬁeld. By means of
the Kummer classes associated to theta functions, i.e., étale theta functions, we canconstruct mono-theta environments, which serve as a sort of bridge between tempered fundamental groups and tempered frobenioids. Finally, I will discuss various
anabelian properties related to étale theta functions and mono-theta environments.
Étale theta functions and mono-theta environments II
Abstract:
In this talk, ﬁrst, I will explain three rigidities — cyclotomic rigidity, constantmultiple rigidity, and discrete rigidity — of a mono-theta environment. Next, I will
introduce the notion of a multiradial functorial algorithm and discuss a multiradial
interpretation of étale theta functions.
Seidai Yasuda (Osaka University)Étale theta functions, mono-theta environments, and [IUTchI] §1 – §3, I - II
Abstract:
In the ﬁrst talk, I will explain the notion of initial Θ-data that provides the initial
setting where the IU Teichmüller theory is developed. Several types of Frobenioids
are associated with the initial Θ-data. Among those the Frobenioids F at the “bad”
primes are of especially strong importance. I will recall the construction in a local
situation, i.e, starting from a Tate elliptic curve over a p-adic ﬁeld K. The construction of F involves a lot of interesting knowledge of E. In the ﬁrst and the second
talk we will see that such background knowledge is weaved into the purely categorical structure of F. Precise statements are mostly given as solutions to categoricalreconstruction problems, i.e., problems of recovering once forgotten information on
objects around E only from the categorical structure of F. Especially we will seethat how canonical isomorphisms between cyclotomes and the mono-theta environments are recovered. In the second talk, the other kinds of Frobenioids associatedto a given initial Θ-data will be brieﬂy explained.

Part II: Overall survey
Shinichi Mochizuki (RIMS, Kyoto University)
The mathematics of mutually alien copies: from Gaussian integrals tointer-universal Teichmüller theory I - VIII
Abstract:
Inter-universal Teichmüller theory may be described as a construction of certain
canonical deformations of the ring structure of a number ﬁeld
equipped with certain auxiliary data, which includes an elliptic curve over thenumber ﬁeld and a prime number ≥ 5. In the present talk, we survey this theory
by focusing on the rich analogies between this theory and the classical computationof the Gaussian integral. The main common features that underlie these analogies
may be summarized as follows:· the introduction of two mutually alien copies of the object of interest;· the computation of the eﬀect — i.e., on the two mutually alien copies of theobject of interest — of two-dimensional changes of coordinates by considering
the eﬀect on inﬁnitesimals;· the passage from planar cartesian to polar coordinates and the resulting
splitting, or decoupling, into radial — i.e., in more abstract valuation-theoretic
terminology, “value group” — and angular — i.e., in more abstract valuationtheoretic terminology, “unit group” — portions;· the straightforward evaluation of the radial portion by applying the quadraticity of the exponent of the Gaussian distribution;· the straightforward evaluation of the angular portion by considering themetric geometry of the group of units determined by a suitable version of thenatural logarithm function.
[Here, the intended sense of the descriptive “alien” is that of its original Latin root,i.e., a sense of abstract, tautological “otherness”.] After reviewing the classicalcomputation of the Gaussian integral, we give a detailed survey of inter-universal
Teichmüller theory by concentrating on the common features listed above. If timeallows, we will also brieﬂy discuss various historical aspects of the mathematics thatappears in inter-universal Teichmüller theory.
Part III: [IUTchI-II]
Seidai Yasuda (Osaka University)Étale theta functions, mono-theta environments, and [IUTchI] §1 – §3, III
Abstract:
In the third talk, we discuss the notion of Θ-Hodge theater. In the ﬁnal part of
the third talk, I will explain some technical results in Section 2 of [IUTchI] concerning subgroups of tempered fundamental groups which are conjugate in proﬁnite
fundamental groups.
Fucheng Tan (Shanghai Center for Mathematical Sciences and Shanghai
Jiao Tong University)
IUT-I: Hodge theaters and label classes of cusps 1-2
Abstract:
In these two talks, I will explain the construction of the various prime-strips associated to the initial theta-data. They carry two diﬀerent kinds of label classesof cusps with multiplicative or additive symmetry, which are of arithmetic or geometric nature, respectively. Between these prime-strips, there are bridges arisingfrom the restrictions from (curves over) global ﬁelds to local ﬁelds and from theevaluations sections at cusps. The prime-strips and the bridges among them will
form the Hodge theaters. The natural maps from a Hodge theater to the underlying prime-strips and bridges will carry symmetries related to the symmetry on thecorresponding label classes of cusps.
Taylor Dupuy (Hebrew University and the University of Vermont)
Functors to and from mono-theta environments
Abstract:
I will review in a functorial way how mono-theta environments can be constructedfrom both groups and frobenioids, and compare the two constructions. I will also
discuss some auxiliary constructions needed in Emmanuel’s discussions on the evaluation of the theta classes and conjugacy synchronization.
Emmanuel Lepage (University Paris 6)
Hodge-Arakelov evaluation 1
Abstract:
I will discuss evaluation of theta monoids at bad places. These evaluation maps are
obtained by pulling back the cohomology classes deﬁning the theta monoids along
Galois sections coming from some torsion points of our elliptic curves (that canbe recovered group theoretically through elliptic cuspidalization) and will give riseto the Gaussian monoids. The main technicality will be to avoid indeterminacies
coming from the fact that the diﬀerent decomposition groups are only deﬁned up
to conjugacy: we will study how proﬁnite conjugacy acts on tempered decomposition groups and how the conjugacies on the various decomposition groups can besynchronized through symmetrizing isomorphisms.
Taylor Dupuy (Hebrew University and the University of Vermont)
Multiradiality
Abstract:
I am going to introduce the notion of multiradiality of a functor. I will explain how
and where this notion is used at various points in IUT.
Emmanuel Lepage (University Paris 6)
Hodge-Arakelov evaluation 2
Abstract:
We will keep on studying Gaussian Frobenioids and will quickly deﬁne Gaussian
frobenioids at good places. Finally, I will deﬁne global realiﬁed Gaussian frobenioids, to get modiﬁed versions of the theta-link deﬁned in IUT-I.
Part IV: [IUTchIII-IV] from the point of view of mono-anabelian transport
Yuichiro Hoshi (RIMS, Kyoto University)[IUTchIII-IV] from the point of view of mono-anabelian transport I - IV
Abstract:
The main purpose of the present talk is to give an explanation of multiradial representations — i.e., descriptions in terms that make sense from the point of view ofan alien arithmetic holomorphic structure — of log-shells, theta values, and number
ﬁelds from the point of view of mono-anabelian transport. We also discuss, in thepresent talk, a diophantine inequality which may be regarded as a main applicationof the multiradial representations.

Part V: Related topics
Yuki Wada (RIMS, Kyoto University)
Near miss abc-triples in compactly bounded subsets
Abstract:
In this talk, we study the existence of near miss abc-triples in compactly bounded
subsets. In more concrete terms, we prove that there exist inﬁnitely many abctriples such that: (1) the absolute value of abc exceeds a certain quantity determinedby the product of the distinct prime numbers of abc, and, moreover, (2) a certain
value determined by a, b, and c, which corresponds to the quantity “λ” in the
Legendre form of an elliptic curve, lies in a given compactly bounded subset.
Vesselin Dimitrov (Yale University)
Notes on the epsilon term in the abc conjecture
Abstract:
I will explain why Theorem 1.10 of IUT-IV implies an eﬀective abc inequality.
The resulting bound is however much too large to be computationally feasible,
until a more eﬃcient way of producing Belyi maps is found. I will also formulate
the abc conjecture on the modular curve Y (2) in a way that (a) manifests the
Granville-Stark relationship to the Siegel zero problem for quadratic imaginaryﬁelds (odd Dirichlet characters), and (b) suggests an extension of the abc conjecture
in a diﬀerent framework than the Vojta conjectures. As it turns out, if IUT theory
could be extended to give the same “Θ − q” bound in Corollary 3.12 with theArchimedean places included, this would forbid a Siegel zero for the odd Dirichlet
characters.
Kobi Kremnitzer (The University of Oxford)
Milnor-Wood inequality and bounded cohomology
Abstract:
I will explain the construction of the various prime-strips associated to the initial
theta-data. They carry two diﬀerent kinds of label classes of cusps with multiplicative or additive symmetry,I will explain the Milnor-Wood inequality in termof bounded cohomology and its relation to the abc conjecture in the function ﬁeldcase. I will then discuss possible applications of bounded cohomology in the number
ﬁeld setting.
Ivan Fesenko (The University of Nottingham)
Reciprocity and IUT
Abstract:
I will discuss links between IUT and class ﬁeld theory, which is not used in IUT,
as well as two-dimensional class ﬁeld theory. I will also discuss similarities between
IUT and two two-dimensional adelic structures on proper models of elliptic curves
over number ﬁelds and the computation of its two-dimensional zeta integral.
Boris Zilber (The University of Oxford)
Model theory of anabelian geometry
Abstract:
We deﬁne a category M of structures (in the model-theoretic sense) which is equivalent to the category of proﬁnite groups. The morphisms of the category M areinterpretations and the functor realising the equivalence takes the structure to its
automorphism group. The étale fundamental group of a variety X over k can beidentiﬁed as an automorphism group of a certain object (structure) Cov(X) of thecategory M . We then study the structure Cov(X) and identify a model-theoretic
property of Cov(X) which is suﬃcient (and probably necessary) for the section
conjecture to hold for X.
Part VI: [IUTchIII-IV] with remarks on the function-theoretic roots of the theory
Go Yamashita (RIMS, Kyoto University)[IUTchIII-IV] with remarks on the function-theoretic roots of the theory I - IV
Abstract:
We will explain IUTch-III and IUT-IV with a motivation of Θ-link from HodgeArakelov theory.
Part VII: Overall discussion and impromptu talks
Adam Topaz (University of California, Berkeley)
Kummer theory and anabelian geometry over algebraically closed ﬁelds
Abstract:
This talk will describe how Kummer theory and divisor-like groups are used in anabelian geometry over algebraically closed ﬁelds, speciﬁcally in the context of Bogomolov’s program in birational anabelian geometry. Along the way, I will highlightsome similarities and diﬀerences between this context and the anabelian geometry
used in IUT.
Dinesh Thakur (Rochester University)
Diophantine approximation and deformation hierarchies in ﬁnite characteristic
Paul Vojta (University of California, Berkeley)
On changes to the Thue-Siegel method
Abstract:
I will describe a potential change to the Thue-Siegel method.