FOUNDATIONS AND PERSPECTIVES OF ANABELIAN GEOMETRY,RIMS WORKSHOP, JUNE 28–JULY 2 2021, BY ZOOMSCHEDULE
Organisers: Benjamin Collas (RIMS), Ivan Fesenko (Univ. of Nottingham), Arata Minamide (RIMS),
Fucheng Tan (RIMS)
June 28 2021 (Monday)
17:30-18:00 (Kyoto time) Opening
18:00-19:00 Akio Tamagawa
Developments of anabelian geometry of curves over finite fields
19:15-20:15 Shinichi Mochizuki The technique of tripodal transport in arithmetic geometry
20:30-21:30 Arata Minamide
The Grothendieck–Teichmüller group and the outer automorphismgroups of the profinite braid groups

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June 29 2021 (Tuesday)
17:30-18:00 Q&A
18:00-19:00 Hiroaki Nakamura On arithmetic and geometry around the adelic Eisenstein function
19:15-20:15 Shinichi Mochizuki The technique of tripodal transport in arithmetic geometry
20:30-21:30 Fedor Bogomolov Birational geometry and group theory86
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June 30 2021 (Wednesday)
17:30-18:00 Q&A
18:00-19:00 Yuichiro Hoshi
19:15-20:15 Emmanuel Lepage
20:30-21:30 Yu Yang

The absolute anabelian geometry of quasi-tripods
The absolute anabelian conjecture for curves with resolutionof non-singularities
Moduli spaces of fundamental groups of curves in positive characteristic

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July 1 2021 (Thursday)
17:30-18:00 Q&A
18:00-19:00 Takahiro Murotani A p-adic analytic approach to the absolute Grothendieck conjecture 7
19:15-20:15 Wojciech Porowski Pro-p anabelian geometry of punctured elliptic curves9
20:30-21:30 Shota Tsujimura
Combinatorial Belyi cuspidalization and its applications
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July 2 2021 (Friday)
17:30-18:00 Q&A
18:00-19:00 Koichiro Sawada

On surjective homomorphisms from a configuration space group
to a surface group
19:15-20:15 Shota Tsujimura Anabelian group-theoretic properties of the absolute Galois groupsof higher local fields
20:30-21:30 Kazumi Higashiyama The mono-anabelian geometry of geometrically pro-p arithmetic
fundamental groups of second configuration spaces
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2SCHEDULE OF ANABELIAN GEOMETRY WORKSHOP, RIMS, JUNE 28–JULY 2 2 2021

T ITLES AND ABSTRACTS

1. Fedor Bogomolov. Birational geometry and group theory
I discuss birational invariants of algebraic varieties which can be derived from the structure of the corresponding Galois groups. I also discuss some related conjectures concerning the structure of the Galois groups
and their subgroups.

2. Kazumi Higashiyama. The mono-anabelian geometry of geometrically pro-p arithmetic fundamental groupsof second configuration spaces
The n-th configuration space of a hyperbolic curve is the scheme which parametrizes n-tuples of pairwise
distinct points in the hyperbolic curve. Mochizuki proved the Grothendieck conjecture for hyperbolic curves.
We discuss a certain pro-p version of the Grothendieck conjecture for hyperbolic curves. In this talk, wereconstruct group-theoretically the function field of a hyperbolic curve of type (0, 3) from the pro-p fundamentalgroup of the associated second configuration space equipped with the collection of decomposition groups.

3. Yuichiro Hoshi. The absolute anabelian geometry of quasi-tripods
A quasi-tripod is defined to be a hyperbolic orbicurve such that there exists a chain consisting of elementary operations (i.e., the operations of passing to a finite etale covering, passing to a finite etale quotient,de-cuspidalization, and de-orbification) whose initial orbicurve is the given orbicurve and whose terminal orbicurve is a tripod. In particular, a hyperbolic orbicurve isogenous to a hyperbolic curve of genus zero gives anexample of a quasi-tripod. In this talk, we discuss the absolute anabelian geometry of quasi-tripods. Moreover,
we also discuss some applications to the absolute anabelian geometry of higher dimensional varieties.

4. Emmanuel Lepage. The absolute anabelian conjecture for curves with resolution of non-singularities
The absolute anabelian conjecture asks whether every isomorphism between the étale fundamental groupsof two hyperbolic curves over p-adic fields comes from an isomorphism of curves. This was proved by
S. Mochizuki for curves of quasi-Belyi type. In this talk, I will discuss an extension of this result to a wider
family of curves.

5. Arata Minamide. The Grothendieck–Teichmüller group and the outer automorphism groups of the profinite
braid groups
The Grothendieck–Teichmüller group GT is a profinite group closely related to (and possibly isomorphic to)the absolute Galois group of the field of rational numbers. In this talk, we will determine the outer automorphism groups of the profinite braid groups with four or more strings in terms of GT. This is joint work withHiroaki Nakamura.

SCHEDULE OF ANABELIAN GEOMETRY WORKSHOP, RIMS, JUNE 28–JULY 2 2 2021

36. Shinichi Mochizuki. The technique of tripodal transport in arithmetic geometry
The technique of "tripodal transport" allows one to "transport" various deep arithmetic properties of thetripod (i.e., the projective line minus three points) to more general hyperbolic curves over arithmetic fields.
This technique is closely related to various results in anabelian geometry. In this talk, we shall survey threeimportant examples of this technique (and its relation to anabelian geometry), namely, p-adic Teichmüller
theory, Belyi-type injectivity results for arbitrary hyperbolic curves, and inter-universal Teichmüller theory.

7. Takahiro Murotani. A p-adic analytic approach to the absolute Grothendieck conjecture
Consider a hyperbolic curve X over a finite extension of the field of p-adic numbers. Let π1 (X) be itsétale fundamental group. The p-adic absolute Grothendieck conjecture asks: Is it possible to recover X grouptheoretically, solely from π1 (X)? To consider this problem, we introduce a certain p-adic analytic invariant
defined by Serre (which we call i-invariant) and discuss the relation between the invariant and the problem.

8. Hiroaki Nakamura. On arithmetic and geometry around the adelic Eisenstein function
We introduce the adelic Eisenstein function arising from the monodromy representation in the universal
family of punctured elliptic curves, and illustrate its basic aspects. If time allows, we discuss a new relation tosome knot theoretic invariants.

9. Wojciech Porowski. Pro-p anabelian geometry of punctured elliptic curves
Let E be an elliptic curve over a local field of characteristic zero and let X be a hyperbolic curve of type(1, 1) obtained by removing the origin from E. Consider the geometrically pro-p étale fundamental group ofX, where p is the characteristic of the residue field. In this talk we are going to study the problem whether
this fundamental group determines the reduction type of the elliptic curve E. We will prove that under theassumption p > 3 the answer is positive.

10. Koichiro Sawada. On surjective homomorphisms from a configuration space group to a surface group
In this talk, we classify all surjective homomorphisms from the étale fundamental group of the configuration
space of a hyperbolic curve (over an algebraically closed field of characteristic zero) to the étale fundamentalgroup of a hyperbolic curve. More specifically, we show that such a surjective homomorphism factors throughone of the homomorphisms which arise from certain morphisms of schemes. Moreover, we discuss an application to anabelian geometry.

11. Akio Tamagawa. Developments of anabelian geometry of curves over finite fields
This is a survey talk on anabelian geometry of curves over finite fields.
It will cover various topics, from Uchida’s theorem for function fields in 1970s to several recent developments.

4SCHEDULE OF ANABELIAN GEOMETRY WORKSHOP, RIMS, JUNE 28–JULY 2 2 2021

12. Shota Tsujimura. Combinatorial Belyi cuspidalization and its applications
I will introduce the theory of combinatorial Belyi cuspidalization, which is a certain combinatorial versionof the theory of Belyi cuspidalization developed by S. Mochizuki in the context of absolute anabelian geometry. This combinatorial version is based on combinatorial anabelian geometry developed by Y. Hoshi andS. Mochizuki. I will also talk about the following applications of the theory of combinatorial Belyi cuspidalization:(I) The construction of a splitting of the natural inclusion of the absolute Galois group of the field of p-adic
numbers to the (largest) p-adic Grothendieck–Teichmüller group.
(II) Absolute version of the Grothendieck conjecture for higher dimensional (> 1) configuration spaces associated to hyperbolic curves of genus 0 over finite extensios of the maximal abelian extension of rationals. [Thisapplication is a consequence of recent joint work with Y. Hoshi and S. Mochizuki.]

13. Shota Tsujimura. Anabelian group-theoretic properties of the absolute Galois groups of higher local
fieldsClassically, it is well-known that the absolute Galois groups of various fields (e.g., p-adic local fields, number fields) appearing in anabelian geometry satisfy the slimness (i.e., every open subgroup is center-free) andelasticity (i.e., every nontrivial topologically finitely generated closed normal subgroup is open). In this talk,we will explain that the absolute Galois groups of higher local fields of mixed or positive characteristic alsosatisfy the slimness and elasticity. This is joint work with A. Minamide.

14. Yu Yang. Moduli spaces of fundamental groups of curves in positive characteristic
In this talk, I will explain some current developments in the theory of anabelian geometry of curves overalgebraically closed fields of positive characteristic from the point of view of "Homeomorphism Conjecture"
which was formulated by the speaker. This conjecture shows a new anabelian phenomenon which says thatthe moduli spaces of curves can be reconstructed group-theoretically as topological spaces from fundamentalgroups. In particular, I will give a sketch of my proof of Homeomorphism Conjecture when the dimension ofM g,n is one.