## Number Theory / Arithmetic Geometry Seminar

**
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
**