## Number Theory Seminar

Date, time & place Wednesday 17:00 - 18:00 056Room #056 (Graduate School of Math. Sci. Bldg.) Naoki Imai, Yoichi Mieda

Seminar information archive

### 2009/11/18

16:30-18:45   Room #056 (Graduate School of Math. Sci. Bldg.)

Elementary computation of ramified component of the Jacobi sum
[ Abstract ]
R. Coleman and W. McCallum calculated the Jacobi sum Hecke characters using their computation of the stable reduction of the Fermat curve in 1988. In my talk, we give an elementary proof of the main result of them without using rigid geometry or the stable model of the Fermat curve.
Christopher Deninger (Universität Münster) 17:45-18:45
P-divisible groups and the p-adic Corona problem

### 2009/10/21

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Bernard Le Stum (Université de Rennes 1)
The local Simpson correspondence in positive characteristic
[ Abstract ]
A Simpson correspondance should relate Higgs bundles to differential modules (or local systems). We stick here to positive characteristic and recall some old and recent results : Cartier isomorphism, Van der Put's classification, Kaneda's theorem and Ogus-Vologodsky local theory. We'll try to explain how the notion of Azumaya algebra is a convenient tool to unify these results. Our main theorem is the equivalence between quasi-nilpotent differential modules of level m and quasi-nilpotent Higgs Bundles (depending on a lifting of Frobenius mod p-squared). This result is a direct generalization of the previous ones. The main point is to understand the Azumaya nature of the ring of differential operators of level m. Following Berthelot, we actually use the dual theory and study the partial divided power neighborhood of the diagonal.

### 2009/10/07

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Ahmed Abbes (Université de Rennes 1)
On GAGA theorems for the rigide-étale topology
[ Abstract ]
Last year, I finished my course in Todai on "Rigide Geometry following M. Raynaud" by stating a GAGA theorem for the rigide-étale topology, due to Gabber and Fujiwara. I will give a new proof of this theorem, inspired by another theorem of Gabber, namely the Affine analog of the proper base change theorem.

### 2009/09/14

11:00-12:00   Room #123 (Graduate School of Math. Sci. Bldg.)
Dinakar Ramakrishnan (カリフォルニア工科大学)
Modular forms and Calabi-Yau varieties

### 2009/08/07

16:30-17:30   Room #117 (Graduate School of Math. Sci. Bldg.)
Fabien Trihan (Nottingham大学)
On the $p$-parity conjecture in the function field case
[ Abstract ]
Let $F$ be a function field in one variable with field of constant a finite field of characteristic $p>0$. Let $E/F$ be an elliptic curve over $F$. We show that the order of the Hasse-Weil $L$-function of $E/F$ at $s=1$ and the corank of the $p$-Selmer group of $E/F$ have the same parity (joint work with C. Wuthrich).

### 2009/06/24

16:30-18:45   Room #056 (Graduate School of Math. Sci. Bldg.)
Vincent Maillot (Paris第7大学) 16:30-17:30
New algebraicity results for analytic torsion
Richard Hain (Duke大学) 17:45-18:45
On the Section Conjecture for the universal curve over function fields

### 2009/06/10

16:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Bruno Kahn (Paris第7大学)
On the classifying space of a linear algebraic group

### 2009/06/03

16:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Bruno Kahn (Paris第7大学)

### 2009/05/27

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Gombodorj Bayarmagnai (東京大学大学院数理科学研究科)
The (g,K)-module structure of principal series and related Whittaker functions of SU(2,2)

### 2009/05/20

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)

Generalized Whittaker functions for degenerate principal series of GL(4,R)

### 2009/05/13

16:30-18:45   Room #056 (Graduate School of Math. Sci. Bldg.)

A counterexample of Bloch-Kato conjecture over a local field and infinite torsion in algebraic cycles of codimension two

### 2009/01/28

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Pierre Colmez (École polytechnique)
On the p-adic local Langlands correspondence

### 2008/12/03

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)

Mean-periodicity and analytic properties of zeta-functions
[ Abstract ]
Mean-periodicityというのは周期性の概念のひとつの一般化である。最近、I. Fesenko, G. Ricottaとの共同研究により、数論的スキームのゼータ関数を含むある複素関数のクラスと、mean-periodicityとの関連性が新しく見出された。
これはHecke-Weilによる, 解析接続と関数等式を持つDirichlet級数と保型形式との対応の一つの拡張ともみなせる. この背景には, I. Fesenkoの高次元アデール上のゼータ積分の理論があり、数論的スキームのHasseゼータ関数の解析接続を高次元アデール上の調和解析から導こうというプログラムの一環となっている。
この講演ではそのような背景にも若干触れた上、ゼータ関数の解析的性質とmean-periodicityの関連、特に解析接続と関数等式との関連について解説する。

### 2008/11/26

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)

Lang's Observation in Diophantine Problems
[ Abstract ]
In 1964, Serge Lang suggested the following problem, which reads now as follows:
Let $E$ be an elliptic curve defined over a number field $K$, and $\\varphi$ be a rational function on $E$. Then, for every point $P\\in E(K)$ where $\\varphi$ does not vanish at $P$, the logarithms of a norm of $\\varphi(P)$ is at worst linear in the logarithms of the Neron-Tate height of the point $P$.
We give a simultaneous Diophantine approximation for linear forms in elliptic logarithms which actually implies this conjecture. We also present Lang's observations in Diophantine problems.

### 2008/11/19

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Olivier Fouquet (大阪大学)
Dihedral Iwasawa theory of ordinary modular forms
[ Abstract ]
According to Hida theory, the Galois representation attached to a nearly-ordinary Hilbert eigencuspform belongs to a p-adic analytic family of Galois representations parametrized by varying weights. After restricting it to the absolute Galois group of a quadratic totally complex extension, it also belongs to a p-adic family coming from classical dihedral Iwasawa theory. We will explain the proofs of part of the main conjecture in Iwasawa theory in these situations, i.e divisibilities of characteristic ideals when equalities are actually expected.

### 2008/10/29

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Daniel Caro (Université de Caen)
Overholonomicity of overconvergence $F$-isocrystals on smooth varieties
[ Abstract ]
Let $¥mathcal{V}$ be a complete discrete valuation ring
of characteristic $0$, with perfect residue field $k$ of
characteristic $p>0$. In order to construct $p$-adic coefficients
over $k$-varieties, Berthelot introduced the theory of
overconvergent $F$-isocrystals, i.e overconvergent isocrystals with
Frobenius structure. Moreover, to get a $p$-adic cohomology over
$k$-varieties stable under cohomological operations, Berthelot built
the theory of arithmetic $F$-$¥mathcal{D}$-modules. In this talk,
after recalling some elements of these theories, we introduce the
notion of overholonomicity with is a property as stable as the
holonomicity in the classical theory of $¥mathcal{D}$-modules. The
goal of the talk is to prove the overholonomicity of arithmetic
$¥mathcal{D}$-modules associated to overconvergent $F$-isocrystals
over smooth $k$-varieties. In the proof we need Christol's transfert
theorem, a comparison theorem between relative log rigid cohomology
and relative rigid cohomology and last but not least Kedlaya's
semistable reduction theorem. This is a joint work with Nobuo
Tsuzuki.

### 2008/10/22

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Pierre Parent (Universite Bordeaux 1)
Serre's uniformity in the split Cartan case
[ Abstract ]
We show that, for large enough prime number p, the modular curve
X_{split}(p) has no other point with values in Q than CM points and the rational cusp. This gives a partial answer to an old question of J.-P. Serre concerning the uniform surjectivity of Galois representations associated to torsion points on elliptic curves without complex multiplication.
(Joint work with Yuri Bilu.)

### 2008/09/29

16:30-17:30   Room #117 (Graduate School of Math. Sci. Bldg.)
Christopher Deninger (Munster大学)
A determinant for p-adic group algebras
[ Abstract ]
For a discrete countable group G there is a classical determinant on the units of the L^1-convolution algebra of G. It is defined using functional analysis and can be used for example to calculate the entropy of certain G-actions. We will discuss a p-adic analogue of this theory. Instead of functional analysis the definition of the p-adic determinant uses algebraic K-theory. It has an application to the study of the p-adic distribution of periodic G-orbits in certain G-action.

### 2008/08/27

16:30-17:30   Room #117 (Graduate School of Math. Sci. Bldg.)
Don Zagier (Max Planck研究所)
$q$-series and modularity

### 2008/08/01

13:00-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Olivier Brinon (Paris北大学) 13:00-14:00
B_dR-representations and Higgs bundles
Henrik Russell (Duisburg-Essen大学) 14:15-15:15
Generalized Albanese and duality
Thomas Geisser (南California大学) 15:45-16:45
Negative K-theory, homotopy invariance and regularity
[ Abstract ]
The topic of my talk are two classical conjectures in K-theory:
Weibel's conjecture states that a scheme of dimension d
has no K-groups below degree -d, and Vorst's conjecture
states that homotopy invariance of the K-theory of rings
implies that the ring must be regular.
I will give an easy introduction to the conjectures, and discuss
recent progress.
Fabien Trihan (Nottingham大学) 17:00-18:00
On Iwasawa theory for abelian varieties over function fields of positive characteristic

### 2008/07/16

16:30-17:30   Room #117 (Graduate School of Math. Sci. Bldg.)
Valentina Di Proietto (Padova大学)
On p-adic differential equation on semi-stable varieties

### 2008/07/02

16:30-17:30   Room #117 (Graduate School of Math. Sci. Bldg.)

(安田正大氏との共同研究)

### 2008/06/18

16:30-18:45   Room #117 (Graduate School of Math. Sci. Bldg.)

On a ramification bound of semi-stable torsion representations over a local field

Beilinson-Tate予想と楕円曲面のK_1の不分解元
[ Abstract ]
(佐藤周友氏との共同研究)

これは、p進レギュレーターの非消滅と関係しており、

### 2008/06/04

16:30-17:30   Room #117 (Graduate School of Math. Sci. Bldg.)

$p$-adic elliptic polylogarithm, $p$-adic Eisenstein series and Katz measure
(joint work with G. Kings)

[ Abstract ]
The Eisenstein classes are important elements in the motivic cohomology
of a modular curve, defined as the specializations of the motivic elliptic
polylogarithm by torsion sections. The syntomic Eisenstein classes are
defined as the image by the syntomic regulator of the motivic Eisenstein
classes. In this talk, we explain our result concerning the relation between
syntomic Eisenstein classes restricted to the ordinary locus and