## Number Theory Seminar

Seminar information archive ～04/16｜Next seminar｜Future seminars 04/17～

Date, time & place | Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | Naoki Imai, Shane Kelly |

**Seminar information archive**

### 2009/05/27

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

The (g,K)-module structure of principal series and related Whittaker functions of SU(2,2)

**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)

**廣江 一希**(東京大学大学院数理科学研究科)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.)

剰余体が非完全な場合のB_dR^+のGalois理論

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

**大久保 俊**(東京大学大学院数理科学研究科) 16:30-17:30剰余体が非完全な場合のB_dR^+のGalois理論

**斎藤 秀司**(東京大学大学院数理科学研究科) 17:45-18:45A 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.)

On the p-adic local Langlands correspondence

**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

**鈴木正俊**(東京大学大学院数理科学研究科)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。

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

**平田典子**(日本大学理工学部)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.

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.)

Dihedral Iwasawa theory of ordinary modular forms

**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.

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.)

Overholonomicity of overconvergence $F$-isocrystals on smooth varieties

**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.

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.)

Serre's uniformity in the split Cartan case

**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.)

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.)

A determinant for p-adic group algebras

**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.

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.)

$q$-series and modularity

**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.)

B_dR-representations and Higgs bundles

Generalized Albanese and duality

Negative K-theory, homotopy invariance and regularity

On Iwasawa theory for abelian varieties over function fields of positive characteristic

**Olivier Brinon**(Paris北大学) 13:00-14:00B_dR-representations and Higgs bundles

**Henrik Russell**(Duisburg-Essen大学) 14:15-15:15Generalized Albanese and duality

**Thomas Geisser**(南California大学) 15:45-16:45Negative 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.

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) 17:00-18:00On 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.)

On p-adic differential equation on semi-stable varieties

**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.)

有限体上のスキームのふたつのモチビックコホモロジー群の計算

(安田正大氏との共同研究)

**近藤 智**(東京大学数物連携宇宙研究機構)有限体上のスキームのふたつのモチビックコホモロジー群の計算

(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

**服部 新**(北海道大学大学院理学研究院) 16:30-17:30On a ramification bound of semi-stable torsion representations over a local field

**朝倉 政典**(北海道大学大学院理学研究院) 17:45-18:45Beilinson-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)

代数サイクルのTate予想のK理論における類似であるBeilinson-Tate予想について、

楕円曲面の場合にそれが成り立つ非自明な例を構成する。

これは、p進レギュレーターの非消滅と関係しており、

応用としてK_1の不分解元であって整数環上のモデルからくるようなものを構成する。

(佐藤周友氏との共同研究)

代数サイクルのTate予想のK理論における類似であるBeilinson-Tate予想について、

楕円曲面の場合にそれが成り立つ非自明な例を構成する。

これは、p進レギュレーターの非消滅と関係しており、

応用としてK_1の不分解元であって整数環上のモデルからくるようなものを構成する。

### 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)

**坂内 健一**(慶應義塾大学理工学部 )$p$-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

p-adic Eisenstein series.

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

p-adic Eisenstein series.

### 2008/05/07

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

On the connected components of moduli spaces of finite flat models

**今井 直毅**

(東京大学大学院数理科学研究科)On the connected components of moduli spaces of finite flat models

### 2008/04/30

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

Iwasawa theory of totally real fields for certain non-commutative $p$-extensions

**原 隆**(東京大学大学院数理科学研究科)Iwasawa theory of totally real fields for certain non-commutative $p$-extensions

[ Abstract ]

Recently, Kazuya Kato has proven the non-commutative Iwasawa main

conjecture (in the sense of Coates, Fukaya, Kato, Sujatha and Venjakob) for

non-commutative Galois extensions of "Heisenberg type" of totally real fields,

using integral logarithmic homomorphisms. In this talk, we apply Kato's method

to certain non-commutative $p$-extensions which are more complicated than those

of Heisenberg type, and prove the main conjecture for them.

Recently, Kazuya Kato has proven the non-commutative Iwasawa main

conjecture (in the sense of Coates, Fukaya, Kato, Sujatha and Venjakob) for

non-commutative Galois extensions of "Heisenberg type" of totally real fields,

using integral logarithmic homomorphisms. In this talk, we apply Kato's method

to certain non-commutative $p$-extensions which are more complicated than those

of Heisenberg type, and prove the main conjecture for them.

### 2008/01/30

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

Odds and ends on finite group actions and traces

**Luc Illusie**(Universite Paris-Sud 11)Odds and ends on finite group actions and traces

[ Abstract ]

Suppose a finite group G acts on a scheme X separated and of finite type over a field k. This raises several questions about the traces of elements s of G (or more generally products sg, for g in the Galois group of k) on cohomology groups of various types associated with X/k (with compact support or no support, Betti if k = C, l-adic, rigid). Some were considered and solved long ago, others only recently. I will in particular discuss an equivariant generalization of a theorem of Laumon on Euler-Poincar¥'e characteristics.

Suppose a finite group G acts on a scheme X separated and of finite type over a field k. This raises several questions about the traces of elements s of G (or more generally products sg, for g in the Galois group of k) on cohomology groups of various types associated with X/k (with compact support or no support, Betti if k = C, l-adic, rigid). Some were considered and solved long ago, others only recently. I will in particular discuss an equivariant generalization of a theorem of Laumon on Euler-Poincar¥'e characteristics.

### 2008/01/23

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

Integrality, Rationality, and Independence of l in l-adic Cohomology over Local Fields

**Weizhe Zheng**(Universite Paris-Sud 11)Integrality, Rationality, and Independence of l in l-adic Cohomology over Local Fields

[ Abstract ]

I will discuss two problems on traces in l-adic cohomology over local fields with finite residue field. In the first part, I will describe the behavior of integral complexes of l-adic sheaves under Grothendieck's six operations and the nearby cycle functor. In the second part, I will talk about rationality and independence of l. More precisely, I will introduce a notion of compatibility for systems of l-adic complexes and explain the proof of its stability by the above operations, in a slightly more general context (equivariant under finite groups). The main tool in this talk is a theorem of de Jong on

alterations.

I will discuss two problems on traces in l-adic cohomology over local fields with finite residue field. In the first part, I will describe the behavior of integral complexes of l-adic sheaves under Grothendieck's six operations and the nearby cycle functor. In the second part, I will talk about rationality and independence of l. More precisely, I will introduce a notion of compatibility for systems of l-adic complexes and explain the proof of its stability by the above operations, in a slightly more general context (equivariant under finite groups). The main tool in this talk is a theorem of de Jong on

alterations.

### 2008/01/16

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

Equidistribution theorems in Arakelov geometry

**Antoine Chambert-Loir**(Universite de Rennes 1)Equidistribution theorems in Arakelov geometry

[ Abstract ]

The proof of Bogomolov's conjecture by Zhang made a crucial use

of an equidistribution property for the Galois orbits of points of small

heights in Abelian varieties defined over number fields.

Such an equidistribution property is proved using a method invented

by Szpiro, Ullmo and Zhang, and makes use of Arakelov theory.

This equidistribution theorem takes place in the complex torus

associated to the Abelian variety. I will show how a similar

equidistribution theorem can be proven for the p-adic topology ;

we have to use Berkovich space. Thanks to recent results of Yuan

about `big line bundles' in Arakelov geometry, the situation

is now very well understood.

The proof of Bogomolov's conjecture by Zhang made a crucial use

of an equidistribution property for the Galois orbits of points of small

heights in Abelian varieties defined over number fields.

Such an equidistribution property is proved using a method invented

by Szpiro, Ullmo and Zhang, and makes use of Arakelov theory.

This equidistribution theorem takes place in the complex torus

associated to the Abelian variety. I will show how a similar

equidistribution theorem can be proven for the p-adic topology ;

we have to use Berkovich space. Thanks to recent results of Yuan

about `big line bundles' in Arakelov geometry, the situation

is now very well understood.

### 2007/12/05

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

Classification of two dimensional trianguline representations of p-adic fields

**中村健太郎**(東京大学大学院数理科学研究科)Classification of two dimensional trianguline representations of p-adic fields

[ Abstract ]

Trianguline representation is a class of p-adic Galois representations of p-adic fields. This was defined by P.Colmez by using ($\\varphi, \\Gamma$)-modules over Robba ring. In his study of p-adic local Langlands correspondence of GL_2(Q_p), he completely classified two dimensional trianguline representations of Q_p. On the other hand, L.Berger recently defined the category of B-pairs and established the equivalence between the category of B-pairs and the category of ($\\varphi,\\Gamma$)-modules over Robba ring. In this talk, we extend the Colmez's result by using B-pairs. We completely classify two dimensional trianguline representations of K for any finite extension of Q_p. We also talk about a relation between two dimensional trianguline representations and principal series or special series of GL_2(K).

Trianguline representation is a class of p-adic Galois representations of p-adic fields. This was defined by P.Colmez by using ($\\varphi, \\Gamma$)-modules over Robba ring. In his study of p-adic local Langlands correspondence of GL_2(Q_p), he completely classified two dimensional trianguline representations of Q_p. On the other hand, L.Berger recently defined the category of B-pairs and established the equivalence between the category of B-pairs and the category of ($\\varphi,\\Gamma$)-modules over Robba ring. In this talk, we extend the Colmez's result by using B-pairs. We completely classify two dimensional trianguline representations of K for any finite extension of Q_p. We also talk about a relation between two dimensional trianguline representations and principal series or special series of GL_2(K).

### 2007/11/21

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

Abelian varieties with constrained torsion

**Christopher Rasmussen**(京都大学数理解析研究所)Abelian varieties with constrained torsion

[ Abstract ]

The pro-$l$ Galois representation attached to the arithmetic fundamental group of a curve $X$ is heavily influenced by the arithmetic of certain classes of its branched covers. It is natural, therefore, to search for and classify these special covers in a meaningful way. When $X$ is the projective line minus three points, one finds that such covers are very scarce. In joint work with Akio Tamagawa, we formulate a conjecture to quanitify this scarcity, and present a proof for the conjecture in the case of genus one curves defined over $\\Q$.

The pro-$l$ Galois representation attached to the arithmetic fundamental group of a curve $X$ is heavily influenced by the arithmetic of certain classes of its branched covers. It is natural, therefore, to search for and classify these special covers in a meaningful way. When $X$ is the projective line minus three points, one finds that such covers are very scarce. In joint work with Akio Tamagawa, we formulate a conjecture to quanitify this scarcity, and present a proof for the conjecture in the case of genus one curves defined over $\\Q$.

### 2007/10/31

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

On the p-adic local Langlands correspondance for GL2(Qp)

**Pierre Colmez**(Ecole Polytechnique)On the p-adic local Langlands correspondance for GL2(Qp)

### 2007/10/24

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

l進層のSwan導手とunit-root

overconvergent F-isocrystalの特性サイクルについて

**阿部知行**(東京大学大学院数理科学研究科)l進層のSwan導手とunit-root

overconvergent F-isocrystalの特性サイクルについて

[ Abstract ]

今回の講演ではBerthelotによる数論的D加群の理論を用いることによってunit-root overconvergent F-isocrystalに対してSwan導手を定義し、Kato-Saitoにより幾何学的な手法を用いて定義されたSwan導手と比較する。応用として、特異点の解消の仮定のもとでKato-SaitoのSwan導手の整数性予想を導く。

今回の講演ではBerthelotによる数論的D加群の理論を用いることによってunit-root overconvergent F-isocrystalに対してSwan導手を定義し、Kato-Saitoにより幾何学的な手法を用いて定義されたSwan導手と比較する。応用として、特異点の解消の仮定のもとでKato-SaitoのSwan導手の整数性予想を導く。