## Number Theory / Arithmetic Geometry Seminar

Title

** Galois action on the etale fundamental group of the Fermat curve.
**

**
Organizer Akio Tamagawa, Naotake Takao (RIMS, Kyoto Univ.)
**

Date

October 7 (Mon), 2019, 16:30-17:30

Room

Room 110, RIMS

Speaker

Pries, Rachel (Colorado State University)

Abstract

If $X$ is a curve defined over a number field $K$, then we are motivated to understand the action of the absolute Galois group of $K$ on the etale fundamental group of $X$. When $X$ is the Fermat curve of degree $p$ and $K$ is the cyclotomic field generated by a pth root of unity, Anderson proved several theorems about this action on the etale homology of $X$. In earlier work, we made Anderson's results more explicit, when the homology has coefficients modulo $p$. More recently, we use a cup product in cohomology to determine the action on the lower central series of the fundamental group of the Fermat curve with coefficients modulo $p$. The proof involves some fun Galois theory and combinatorics. This is joint work with Davis and Wickelgren.

**
Monday, March 11,
16:00-17:00
Tuesday, March 12
10:30-11:30
Tuesday, March 12
13:30-15:00
Organizer Akio Tamagawa, Naotake Takao (RIMS, Kyoto Univ.)
Organizer Akio Tamagawa, Naotake Takao (RIMS, Kyoto Univ.)
Organizer Akio Tamagawa (RIMS, Kyoto Univ.)
Organizer Akio Tamagawa (RIMS, Kyoto Univ.)
Organizer Akio Tamagawa (RIMS, Kyoto Univ.)
Organizers Akio Tamagawa (RIMS, Kyoto Univ.)
連絡先： 松本眞（広島大）、玉川安騎男、望月新一（京大数理研）
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**

Date

March 11 and 12, 2019

Place

Speaker

** Rasmussen, Christopher (Wesleyan University) **

Title

** Improvements on Bounds for Heavenly Abelian Varieties **

Abstract

For a rational prime $\ell > 2$, a natural question arising
from the study of the arithmetic of certain Galois representations
attached to fundamental groups is to consider the collection of abelian
varieties over a fixed number field $K$ whose $\ell$-power torsion
generates an extension of $K(\mu_{\ell})$ which is both pro-$\ell$ and
unramified away from $\ell$. (We say such an abelian variety is heavenly
at $\ell$.) Conjecturally, the set of $K$-isomorphism classes of heavenly
varieties of fixed dimension $g$ is finite -- even when the prime $\ell$
is permitted to vary. In past joint work with Tamagawa, we have provided
a proof of the Conjecture under GRH, and settled the Conjecture
positively in a number of cases.

Such results imply the existence of a bound $L = L(K,g)$ such that $\ell
< L$ for heavenly abelian varieties over $K$ of dimension $g$. Explicit
formulas for such a bound $L$ exist in general, but are too weak to be
of practical use. In this talk, we review previous results and also
demonstrate a method for improving such bounds when one is willing to
restrict attention to a fixed field $K$ and dimension $g$.

Speaker

** Rasmussen, Christopher (Wesleyan University) **

Title

** Algorithms for Solving S-Unit Equations **

Abstract

Many enumerative problems in arithmetic geometry and number theory take the form ``Find, up to isomorphism, all objects $\mathcal{O}$ with arithmetic property $\mathcal{P}(S)$,'' where the data $S$ is a finite set of places of a fixed number field $K$. Such problems often encounter a common computational obstacle; namely the solution of the $S$-unit equation $x + y = 1$ over the ring of $S$-integers of $K$. In this talk, we describe joint work with several mathematicians to create an open-source implementation of such an algorithm for general choices of $K$ and $S$. (Emphasis of this discussion will be on the mathematical, rather than computational, aspects of the project.) In addition, we provide some new partial results to various problems, including Asymptotic Fermat's Last Theorem over certain cubic number fields.

Speaker

** Sakugawa, Kenji (RIMS) **

Title

** On Jannsen's conjecture for modular forms **

Abstract

Let M be a pure motive over Q and let M_p denotes its p-adic
etale realization for each prime number p. In 1987's paper, Jannsen
proposed a conjecture about a range of integers r such that the second
Galois cohomology of M(r)_p vanishes for any prime number p. Here, M(r)
denotes the rth Tate twist of M. When M is an Artin motive, Jannsen's
conjecture had already essentially proven a half by Soule aroud early
1980's. In this talk, we consider the case when M is a motive associated
to elliptic modular forms. I will explain my ongoing work about an
approach to the conjecture used the p-weighted fundamental groups of
modular curves.

(Sakugawa's lecture is given in Japanese.)

** We plan to have a meal for welcoming Prof. Rasmussen after the seminar of March 11. If you join the meal, could you inform Takao (e-mail: takao_at_kurims.kyoto-u.ac.jp) by March 4?

Date and Place

Lecture 1, October 1st (Mon), 10:00-12:00, Room 110

Lecture 2, October 1st (Mon), 16:30-18:30, Room 110

Lecture 3, October 5th (Fri), 13:30-15:30, Room 110

Lecture 4, October 5th (Fri), 16:30-18:30, Room 110

Lecture 5, October 12th (Fri), 16:30-18:30, Room 111, 2018

Speaker

** Collas, Benjamin (University of Bayreuth) **

Title

** Homotopical Arithmetic Geometry of Stacks **

Abstract

Moduli spaces of curves possess properties which make them ideal
spaces where to concretely study fundamental abstract theories of
arithmetic geometry: they give geometric Galois representations that can
be computed explicitly, furnish examples of anabelian spaces, and in genus
zero generate the category of mixed Tate motives. They also possess a dual
nature, being either considered as schemes or algebraic stacks.

The goal of this series of talks is to provide a basic introduction to
these aspects by covering various fundamental geometric and arithmetic
properties. It is intended for graduate students in algebraic geometry and
non-specialists researchers. Elementary notions will be either recalled or
illustrated with pictures/examples.

I- Algebraic & Deligne-Mumford Stacks (lectures 1 and 2)
Taking the functor of points for schemes as initial motivation, we
introduce the notion of stacks as lax functors in groupoids with descent
conditions and show how to recover Laumon-Moret-Bailly's original
definition. We present how the Artin and Deligne-Mumford algebraic
versions -- that admit topological coverings by schemes -- allow to
``push'' algebraic geometry properties in this context.
Keywords: diagrams of groupoids, Grothendieck topology and
etale/ffpf/smooth morphisms, examples of global quotient and inertia
stacks.

II - Moduli Problems & Moduli Spaces of Curves (lecture 3)
We present how the scheme-stack structures and the geometry of curves lead
to two solutions for building classifying spaces. Having introduced the
notion of functor of moduli, we present Gieseker and Deligne-Mumford
constructions of the moduli space of curves: the former follows Mumford
G.I.T-theory and give a quasi-projective scheme, the latter produces a
smooth algebraic Deligne-Mumford global stack with a nice stable
compactification.
Keywords: Hilbert scheme, explicit examples in low genus, stable
compactification, formal neighbourhood.

III - Fundamental Group & Arithmetic (lecture 4)
We follow Grothendieck construction of the etale fundamental group
that leads to Geometric Galois actions of the absolute Galois group of
rational on the geometric fundamental group of moduli stack of curves. We
adapt this approach in the case of Deligne-Mumford stacks and show how it
leads to a divisorial and a stack arithmetic of the spaces. Following the
seminal work of Ihara, Matsumoto and Nakamura, we present explicit results
and properties of the former, then recent similar results in the case of
cyclic inertia for the latter.
Keywords: properties of the etale fundamental group, explicit
computations in low dimensions, tangential base points and
representations.

IV - Motivic Theory for Moduli Stack of Curves (lecture 5)
We present recent progress on an ongoing project on the construction of a
category of motives for the moduli stacks of curves, whose main property
is to reflect the arithmetic properties of the cyclic stack inertia.
Having recalled briefly what the category of mixed motives should be, we
first present Morel-Voevodski stable/unstable motivic homotopy categories,
then how their homotopical-simplicial approach is well adapted to our
goal.
Keywords: Quillen model category, Artin-Mazur etale topological type,
Mixed Tate motives and loop space.

** Emiliano Ambrosi's successive lectures **

Date

Lecture 1, August 24th (Thu), 13:30-15:00

Lecture 2, August 24th (Thu), 15:30-17:00

Lecture 3, August 28th (Mon), 13:30-15:00

Lecture 4, August 28th (Mon), 15:30-17:00

Lecture 5, September 1st (Fri), 13:30-15:00

Lecture 6, September 1st (Fri), 15:30-17:00, 2017

Room

Room 006, RIMS

Speaker

Emiliano Ambrosi 氏 (Ecole Polytechinique)

**Chieh-Yu Chang 氏連続講義**

Title

** On Hilbert's seventh problem and transcendence theory **

Date

October 9th (Wed), 23rd (Wed) and 28th (Mon), 2013, 10:30-12:00

[change in the date]

October 9th (Wed) 10:30-12:00,

October 23rd (Wed) 10:30-12:00, 13:30-15:00, 2013

Room

Room 006, RIMS

Speaker

Chieh-Yu Chang 氏 (National Tsing Hua University)

Abstract

Hilbert's seventh problem is about the linear independence question of two logarithms of algebraic numbers, which was solved by Gelfond and Schneider in the 1930s. Later on, it was generalized to several logarithms of algebraic numbers by Baker in the 1960s and generalized to general abelian logarithms of algebraic points by Wuestholz in the 1980s. This phenomenon can be also asked for multiple zeta values, but it is still open. In the first talk, we will give a survey on the classical theory and report recent progress on the parallel questions for function fields in positive characteristic. Current methods and tools of transcendence theory using t-motives will be discussed in the second and third talks.

** Mini-Workshop on Number Theory / Arithmetic Geometry **

Date

Thursday, January 31, 2013

Room

Room 206, RIMS, Kyoto University

10:00 -- 10:20

Arata Minamide (RIMS, M2)

Elementary Anabelian Properties of Graphs

10:30 -- 10:50

Yang Yu (RIMS, M2)

Arithmetic Fundamental Groups and Geometry of Curves
over a Discrete Valuation Ring

11:00 -- 11:20

Takeshi Okada (RIMS, M2)

On Finiteness of Twists of Abelian Varieties

13:00 -- 13:30

Yu Iijima (RIMS, D1)

Galois Action on Mapping Class Groups

13:45 -- 14:45

Chia-Fu Yu (Academia Sinica)

Density of the Ordinary Locus in the Hilbert-Siegel Moduli Spaces

Title

** Resolution of nonsingularities for Mumford curves. **

Date

December 15 (Thu), 2011, 14:15-15:45

Room

Room 206, RIMS

Speaker

Emmanuel Lepage 氏 (Institut Mathematique de Jussieu)

Abstract

Let $X$ be a hyperbolic curve over $\overline Q_p$. I am interested in the following property: for every semistable model $\mathcal X$ of $X$ and every closed point $x$ of the special fiber there exists a finite covering $Y$ of $X$ such that the minimal semistable model $\mathcal Y$ of $Y$ above $\mathcal X$ has a vertical component above $x$. I will try to explain why hyperbolic Mumford curves satisfy this property. I will give anabelian appplications of this to the tempered fundamental group.

** Mini-Workshop ``Rational Points on Modular Curves and Shimura Curves'' **

Date

Monday, October 26th, 2009

13:30--14:30

**Keisuke Arai (Univ. Tokyo)
Points on $X_0^+(N)$ over quadratic fields
(joint work with F. Momose)**

Abstract: Momose (1987) studied the rational points on the modular curve $X_0^+(N)$ for a composite number $N$. He showed that the rational points on $X_0^+(N)$ consist of cusps and CM points under certain conditions on a prime divisor $p$ of $N$. But $p=37$ was excluded. For $37$ is peculiar because $X_0(37)$ is a hyperelliptic curve and $w_{37}$ is not the hyperelliptic involution. We show that the rational points on $X_0^+(37M)$ consist of cusps and CM points. We also show that the $K$-rational points on $X_0^+(N)$ consist of cusps and CM points for a quadratic field $K$ under certain conditions (both $p=37$ and $p\ne 37$ allowed).

14:45--15:45

**Fumio Sairaiji (Hiroshima International Univ.)**

Takuya Yamauchi (Osaka Prefecture Univ.)

On rational torsion points of central $\mathbb{Q}$-curves

Takuya Yamauchi (Osaka Prefecture Univ.)

On rational torsion points of central $\mathbb{Q}$-curves

Abstract: Let $E$ be a central $\mathbb{Q}$-curve over a polyquadratic field $k$. In this talk we give an upper bound for prime divisors $p$ of the order of the $k$-rational torsion subgroup of $E$. For example, $p$ is less than or equal to 13, if the scalar restriction of $E$ from $k$ to $\mathbb{Q}$ is of GL$_2$-type with real multiplications. Our result is a generalization of the result of Mazur on elliptic curves over $\mathbb{Q}$, and it is a precision of the upper bounds of Merel and Oesterl\'{e}.

16:00--17:00

**Pierre Parent (Univ. Bordeaux 1)**

Rational points on Shimura curves

Rational points on Shimura curves

Abstract: For $B$ a rational quaternion algebra, the Shimura curve associated with $B$ (or more precisely its quotient by certain Atkin-Lehner involutions) is a moduli space, in a certain sense, for abelian surfaces with potential multiplication by $B$. Proving that those curves almost never have rational points would therefore allow a small step towards the conjecture, attributed to Coleman and Mazur, which predicts the scarcity of endomorphism algebras for abelian varieties of GL$_2$-type over $\mathbb{Q}$. We will present a method to study such rational points, developped by A. Yafaev and myself, and recently improved by F. Gillibert.

** Marco Boggi 氏連続講演 **

Title

** ``Profinite curve complexes and the congruence subgroup problem
for the mapping class group''**

Date

September 27th (Thu) and 28th (Fri), 2007

Room

at Room 206, RIMS, Kyoto University

27th (Thu)

10:00--12:00 Boggi

lunch

14:00--16:00 Boggi

16:00-- free discussion

dinner

28th (Fri)

10:00--12:00 Boggi

lunch

14:00--16:00 free discussion

Abstract

別紙のとおり

Title

** Arithmetic from Geometry on Elliptic Curves **

Date

June 2 (Fri), 2006, 16:30-17:30

Room

Room 202, RIMS

Speaker

Christopher Rasmussen (Rice Univ.)

Abstract

One of the philosophies of arithmetic geometry made popular by Grothendieck was the notion that the structure of the absolute Galois group of $\mathbf{Q}$, could be determined from geometric (or even combinatoric) data. In a related vein, one finds that the arithmetic properties of a curve are sometimes determined by its geometry. Specifically, the structure of a curve as a cover of the projective line can have arithmetic consequences for the Jacobian of the curve. We will discuss this situation in the case of elliptic curves, where this connection between arithmetic and geometry can be seen very clearly.

Title

** Arithmetic Algebraic Geometry Lecture（集中講義） **

Date

May 8 (Mon)- May 19, 2006.

Room

こちらをご覧ください

Speaker

加藤和也（京大 理）

Abstract

Weil が 1949 年に提出した Weil 予想は、 1970 年代のはじめに最終的に証明されるまで、 代数幾何学の大きな進展の原動力となり、 とくに Grothendieck が Weil 予想の証明をめざして導入した エタールコホモロジーは、現在の整数論の重要な道具となった。 その経緯をふり返りつつ、エタールコホモロジーの解説をおこなう。

Title

** Algebraic dynamical systems (preperiodic points, Mahler measures,
equidistribution of small points) **

Date

May 1 (Mon), 2006, 16:30-

Room

Room 202, RIMS

Speaker

Lucien Szpiro (City Univ. New York)

Abstract

Reference: (available at http://math.gc.cuny.edu/faculty/szpiro/People_Faculty_Szpiro.html)
--Joint papers with T. Tucker

--Joint paper with E. Ullmo and S. Zhang

Date

April 10 (Mon), 2006, 14:00-17:00

Room

Room 202, RIMS

(14:00-15:15)

Speaker

Michel Matignon (Univ. Bordeaux 1/Chuo Univ.)

Title

** Wild monodromy groups and automorphisms groups of curves **

(16:00-16:45)

Speaker

Barry Green (Univ. Stellenbosch/Chuo Univ.)

Title

** Selected results on liftings of Galois covers of smooth curves
from char. p to char. 0 **

Date

September 12 (Mon), 13(Tue), 2005