RIMS/Symmetries and Correspondences Workshop:
Inter-universal Teichmüller Theory Summit 2021

Organizers: Yuichiro Hoshi (RIMS, Kyoto University)
Shinichi Mochizuki (RIMS, Kyoto University)
Ivan Fesenko (The University of Nottingham)
Yuichiro Taguchi (Tokyo Institute of Technology)
Benjamin Collas (RIMS, Kyoto University/The University of Lille)

Dates: September 7 – September 10, 2021
Place: RIMS, online

September 7(Tues.)

September 8(Wed.)

September 9(Thu.)

September 10(Fri.)

16:30 –17:30

Ess. Lgc. Str.
(Mochizuki)

Multirad. & Est.
(Hoshi)

Galois orbit(Hoshi)

Overview(Porowski)

17:50 –18:50

Ess. Lgc. Str.
(Mochizuki)

Hodge-Arak.
(Yamashita)

Galois orbit(Hoshi)

BGT(Tsujimura)

19:10 – Expl. estimates
20:10(Minamide)

Siegel zeroes(Táfula)Mod. Nm. Th.
(Fesenko)

BGT(Tsujimura)

20:30 – Expl. estimates
21:30(Minamide)

Princ. of IUT(Collas)

Q&A(Mochizuki)

Q&A(Mochizuki)

1Program
September 7 (Tuesday)
16:30 – 17:30 Shinichi Mochizuki (RIMS, Kyoto University)
On the essential logical structure of inter-universal Teichmüller theory IV
17:50 – 18:50 Shinichi Mochizuki (RIMS, Kyoto University)
On the essential logical structure of inter-universal Teichmüller theory V
19:10 – 20:10 Arata Minamide (RIMS, Kyoto University)
Explicit Estimates in Inter-universal Teichmüller Theory I
20:30 – 21:30 Arata Minamide (RIMS, Kyoto University)
Explicit Estimates in Inter-universal Teichmüller Theory II
September 8 (Wednesday)
16:30 – 17:30 Yuichiro Hoshi (RIMS, Kyoto University)
Multiradial representations and log-volume estimates
17:50 – 18:50 Go Yamashita (RIMS, Kyoto University)
A Motivation of Θ-link from Hodge-Arakelov theory
19:10 – 20:10 Christian Táfula Santos (University of Montreal)
From ABC to L: On singular moduli and Siegel zeroes
20:30 – 21:30 Benjamin Collas (RIMS, Kyoto University)
Principles of Inter-universal Teichmüller Geometry
September 9 (Thursday)
16:30 – 17:30 Yuichiro Hoshi (RIMS, Kyoto University)
On the Galois orbit version of inter-universal Teichmüller theory I: a progress report
17:50 – 18:50 Yuichiro Hoshi (RIMS, Kyoto University)
On the Galois orbit version of inter-universal Teichmüller theory II: a progress report
19:10 – 20:10 Ivan Fesenko (The University of Nottingham)
IUT and modern number theory
20:30 – 21:30 Shinichi Mochizuki (RIMS, Kyoto University)
Q&A

2September 10 (Friday)
16:30 – 17:30 Wojciech Porowski (The University of Nottingham)
Overview of IUT theory
17:50 – 18:50 Shota Tsujimura (RIMS, Kyoto University)
Combinatorial construction of the absolute Galois group of the ﬁeld of rational numbers(I)
19:10 – 20:10 Shota Tsujimura (RIMS, Kyoto University)
Combinatorial construction of the absolute Galois group of the ﬁeld of rational numbers(II)
20:30 – 21:30 Shinichi Mochizuki (RIMS, Kyoto University)
Q&A

3AbstractsSpeaker: Benjamin Collas
Title: Principles of Inter-universal Teichmüller Geometry
Abstract: Grothendieck’s original functorial algebraic geometry of rings and schemes,
once exposed to further classical questions as can be seen in the examples of the sheaf
model category of motives and the group-theoretic anabelian geometry, crystallizes into
new ﬁelds which then thrive on new techniques and new questions.
We here review some key techniques and principles of IUT geometry, a categoricalmonoidal weakened version of scheme theory whose symmetries reﬂect the anabeliangeometric point-reconstruction of spaces in terms of Galois-rational data, which thenresults in a new kind of anabelian-diophantine geometry.
This talk should be considered as a modest and ﬁrst attempt to throw a transversal
light on IUT theory.
Speaker: Ivan Fesenko
Title: IUT and modern number theory
Abstract: I will discuss the potential of IUT and anabelian geometry to cause or inﬂuence
developments in parts of modern number theory including the Langlands program, higher
class ﬁeld theory and arithmetic of elliptic curves.
Speaker: Yuichiro Hoshi
Title: Multiradial representations and log-volume estimates
Abstract: In this talk, I ﬁrst give a brief review of the content of the main theoremof inter-universal Teichmüller theory. After the review, I also explain the relationshipbetween the main theorem and an inequality of log-volumes.
Speaker: Yuichiro Hoshi
Title: On the Galois orbit version of inter-universal Teichmüller theory I, II: a progress
reportAbstract: In these two talks, I explain some technical aspects of work in progress concerning the Galois orbit version of inter-universal Teichmüller theory, that may be regarded
as a reﬁnement of the theory, developed by Shinichi Mochizuki and the speaker.
Speaker: Arata Minamide
Title: Explicit Estimates in Inter-universal Teichmüller Theory I, II
Abstract: In the ﬁnal paper of a series of papers concerning inter-universal Teichmüller
theory, Mochizuki veriﬁed various numerically non-eﬀective versions of the Vojta, ABC,and Szpiro Conjectures over number ﬁelds. In this series of two talks, we will give various
numerically eﬀective versions of Mochizuki’s results. This is joint work with Shinichi
Mochizuki, Ivan Fesenko, Yuichiro Hoshi, and Wojciech Porowski.

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Speaker: Shinichi Mochizuki
Title: On the essential logical structure of inter-universal Teichmüller theory IV, V
Abstract: Introductory lectures and expositions on inter-universal Teichmüller theory —such as, for instance, [Alien] — have a tendency to concentrate on exposing the technicaldetails surrounding the various mathematical objects that appear in the theory. Toa certain extent, of course, this is unavoidable. On the other hand, concentrating onsuch technical details can lead to a situation where one is overwhelmed with seemingly
meaningless technicalities to such an extent that one loses sight of the essential logicalstructure of the theory. The purpose of this series of talks is to discuss this essential
logical structure of the theory, as exposed in [EssLgcIUT].
Speaker: Wojciech Porowski
Title: Overview of IUT theory
Abstract: In this talk we will give a brief overview of the structure of IUT theory andexplain a role played by individual papers. We will also discuss various notions such asétale-like and Frobenius-like structures, holomorphic vs. mono-analytic structures andindeterminacies.
Speaker: Christian Táfula SantosTitle: From ABC to L: On singular moduli and Siegel zeroes
Abstract: In 2000, using analytical, algebraic, and arithmetical ideas, Granville and Stark
showed that the “uniform” ABC for number ﬁelds implies that odd Dirichlet L-functions
have no “Siegel zeroes”, which are a severe type of (not yet unconditionally ruled out)
counterexample to the Generalized Riemann Hypothesis. In this talk we are going tofocus on the structure of their main argument, and discuss recent work that allows us toget more precise relations between the analysis (zero-free regions of L-functions) and thearithmetics (heights of singular moduli).
Speaker: Shota Tsujimura
Title: Combinatorial construction of the absolute Galois group of the ﬁeld of rational
numbers (I), (II)Abstract: In my talk in the workshop “Combinatorial Anabelian Geometry and Related Topics”, we gave the deﬁnition of a purely combinatorial/group-theoretic candidate
BGT for the absolute Galois group of the ﬁeld of rational numbers in the GrothendieckTeichmüller group and constructed an algebraic closure of the ﬁeld of rational numbers
associated to BGT. In this talk, by imposing further purely combinatorial/group-theoreticconditions (“AA-property”) on BGT, we discuss a purely combinatorial/group-theoretic
construction of an algebraic closure of the function ﬁeld of a tripod associated to such
BGT. As a consequence of this construction, we characterize the conjugacy class of theabsolute Galois group of the ﬁeld of rational numbers in the Grothendieck-Teichmüllergroup as the maximal closed subgroups satisfying AA-property. The construction of thefunction ﬁeld of a tripod associated to BGT is closely related to the construction of function ﬁelds of hyperbolic curves of strictly Belyi type that has appeared in a S. Mochizuki’s
previous work and played a key role in his inter-universal Teichmüller theory. This is joint
work with Y. Hoshi and S. Mochizuki.
5Speaker: Go Yamashita
Title: A Motivation of Θ-link from Hodge-Arakelov theory
Abstract: I will explain a motivation of Θ-link from the point of view of Hodge-Arakelov
theory.

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