## Number Theory Seminar

Seminar information archive ～12/05｜Next seminar｜Future seminars 12/06～

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

Organizer(s) | Naoki Imai, Yoichi Mieda |

**Seminar information archive**

### 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。

### 2007/10/10

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

Abel-Jacobi Maps Associated to Algebraic Cycles I

**James Lewis**(University of Alberta)Abel-Jacobi Maps Associated to Algebraic Cycles I

[ Abstract ]

This talk concerns the Bloch cycle class map from the higher Chow groups to Deligne cohomology of a projective algebraic manifold. We provide an explicit formula for this map in terms of polylogarithmic type currents.

This talk concerns the Bloch cycle class map from the higher Chow groups to Deligne cohomology of a projective algebraic manifold. We provide an explicit formula for this map in terms of polylogarithmic type currents.

### 2007/09/19

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

Etale cobordism

**Gereon Quick**(Universitaet Muenster)Etale cobordism

[ Abstract ]

We define and study a new candidate of etale topological cohomology theories for schemes over a field of abritrary characteristic: etale cobordism. As etale K-theory is related to algebraic K-theory, etale cobordism is related to algebraic cobordism of Voevodsky and Levine/Morel. It shares some nice properties of topological theories, e.g. it is equipped with an Atiyah-Hirzebruch spectral sequence from etale cohomology. We discuss in particular a comparison theorem between etale and algebraic cobordism after inverting a Bott element and, finally, we give an outlook to further possible applications of this theory.

We define and study a new candidate of etale topological cohomology theories for schemes over a field of abritrary characteristic: etale cobordism. As etale K-theory is related to algebraic K-theory, etale cobordism is related to algebraic cobordism of Voevodsky and Levine/Morel. It shares some nice properties of topological theories, e.g. it is equipped with an Atiyah-Hirzebruch spectral sequence from etale cohomology. We discuss in particular a comparison theorem between etale and algebraic cobordism after inverting a Bott element and, finally, we give an outlook to further possible applications of this theory.

### 2007/09/12

15:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Classification of p-divisible groups by displays and duality

Applications of the theory of displays

Presentation of mapping class groups from algebraic geometry

**E. Lau**(Univ. of Bielefeld) 15:00-15:45Classification of p-divisible groups by displays and duality

**T. Zink**(Univ. of Bielefeld) 16:00-16:45Applications of the theory of displays

**E. Looijenga**(Univ. of Utrecht) 17:00-18:00Presentation of mapping class groups from algebraic geometry

[ Abstract ]

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

### 2007/08/27

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

The reductive Borel-Serre motive

**Steven Zucker**(Johns Hopkins大学)The reductive Borel-Serre motive

### 2007/07/18

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

Tropical toric varieties

**梶原 健**(横浜国立大学)Tropical toric varieties

### 2007/07/11

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

Algebraic cycles on products of elliptic curves over p-adic fields

**Andreas Rosenschon**(University of Alberta)Algebraic cycles on products of elliptic curves over p-adic fields

### 2007/06/27

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

The conjecture of Birch and Swinnerton-Dyer is misleading

**Stephen Lichtenbaum**(Brown University)The conjecture of Birch and Swinnerton-Dyer is misleading

[ Abstract ]

All values of zeta and L-functions at integral points should be given in terms of products and quotients of Euler characteristics, and the order of the zeroes and poles at these

points should be given by the sum and difference of the ranks of

corresponding finitely generated abelian groups.

All values of zeta and L-functions at integral points should be given in terms of products and quotients of Euler characteristics, and the order of the zeroes and poles at these

points should be given by the sum and difference of the ranks of

corresponding finitely generated abelian groups.

### 2007/05/09

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

$(g,K)$-module structures of principal series representations

of $Sp(3,R)$

**宮崎 直**(東京大学大学院数理科学研究科)$(g,K)$-module structures of principal series representations

of $Sp(3,R)$

### 2007/05/02

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

Cohen-Eisenstein series and modular forms associated to imaginary quadratic fields

**長谷川 泰子**(東京大学大学院数理科学研究科)Cohen-Eisenstein series and modular forms associated to imaginary quadratic fields

### 2007/04/25

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

Localized Characteristic Class and Swan Class

**津嶋 貴弘**(東京大学大学院数理科学研究科)Localized Characteristic Class and Swan Class

### 2007/04/11

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

l進層の暴分岐と特性サイクル

**斎藤 毅**(東京大学大学院数理科学研究科)l進層の暴分岐と特性サイクル