## Number Theory Seminar

Seminar information archive ～11/02｜Next seminar｜Future seminars 11/03～

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**

### 2010/06/02

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

On some algebraic properties of CM-types of CM-fields and their

reflex fields (JAPANESE)

**Ryoko Tomiyasu**(KEK)On some algebraic properties of CM-types of CM-fields and their

reflex fields (JAPANESE)

[ Abstract ]

Shimura and Taniyama proved in their theory of complex

multiplication that the moduli of abelian varieties of a CM-type and their

torsion points generate an abelian extension, not of the field of complex

multiplication, but of a reflex field of the field. In this talk, I

introduce some algebraic properties of CM-types, half norm maps that might

shed new light on reflex fields.

For a CM-field $K$ and its Galois closure $K^c$ over the rational field $Q$,

there is a canonical embedding of $Gal (K^c/Q)$ into $(Z/2Z)^n \\rtimes S_n$.

Using properties of the embedding, a set of CM-types $\\Phi$ of $K$ and their

dual CM-types $(K, \\Phi)$ is equipped with a combinatorial structure. This

makes it much easier to handle a whole set of CM-types than an individual

CM-type.

I present a theorem that shows the combinatorial structure of the dual

CM-types is isomorphic to that of a Pfister form.

Shimura and Taniyama proved in their theory of complex

multiplication that the moduli of abelian varieties of a CM-type and their

torsion points generate an abelian extension, not of the field of complex

multiplication, but of a reflex field of the field. In this talk, I

introduce some algebraic properties of CM-types, half norm maps that might

shed new light on reflex fields.

For a CM-field $K$ and its Galois closure $K^c$ over the rational field $Q$,

there is a canonical embedding of $Gal (K^c/Q)$ into $(Z/2Z)^n \\rtimes S_n$.

Using properties of the embedding, a set of CM-types $\\Phi$ of $K$ and their

dual CM-types $(K, \\Phi)$ is equipped with a combinatorial structure. This

makes it much easier to handle a whole set of CM-types than an individual

CM-type.

I present a theorem that shows the combinatorial structure of the dual

CM-types is isomorphic to that of a Pfister form.

### 2010/05/12

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Differences between

Galois representations in outer-automorphisms

of the fundamental groups and those in automorphisms, implied by

topology of moduli spaces (ENGLISH)

**Makoto Matsumoto**(University of Tokyo)Differences between

Galois representations in outer-automorphisms

of the fundamental groups and those in automorphisms, implied by

topology of moduli spaces (ENGLISH)

[ Abstract ]

Fix a prime l. Let C be a proper smooth geometrically connected curve over a number ﬁeld K, and x be its closed point. Let Π denote the pro-l completion of the geometric fundamental group of C with geometric base point over x. We have two non-abelian Galois representations:

ρA : Galk(x) → Aut(Π),ρO : GalK → Out(Π).

Our question is: in the natural inclusion Ker(ρA) ⊂ Ker(ρO) ∩ Galk(x), whether the equality holds or not. Theorem: Assume that g ≥ 3, l divides 2g -2. Then, there are inﬁnitely many pairs (C, K) with the following property. If l does not divide the extension degree [k(x): K], then Ker(ρA) = (Ker(ρO) ∩ Galk(x)) holds.

This is in contrast to the case of the projective line minus three points and its canonical tangential base points, where the equality holds (a result of Deligne and Ihara).

There are two ingredients in the proof: (1) Galois representations often contain the image of the geometric monodromy (namely, the mapping class group) [M-Tamagawa 2000] (2) A topological result [S. Morita 98] [Hain-Reed 2000] on the cohomological obstruction of lifting the outer action of the mapping class group to automorphisms.

(This lecture is held as `Arithmetic Geometry Seminar Tokyo-Paris' and it is transmitted to IHES by the internet.)

Fix a prime l. Let C be a proper smooth geometrically connected curve over a number ﬁeld K, and x be its closed point. Let Π denote the pro-l completion of the geometric fundamental group of C with geometric base point over x. We have two non-abelian Galois representations:

ρA : Galk(x) → Aut(Π),ρO : GalK → Out(Π).

Our question is: in the natural inclusion Ker(ρA) ⊂ Ker(ρO) ∩ Galk(x), whether the equality holds or not. Theorem: Assume that g ≥ 3, l divides 2g -2. Then, there are inﬁnitely many pairs (C, K) with the following property. If l does not divide the extension degree [k(x): K], then Ker(ρA) = (Ker(ρO) ∩ Galk(x)) holds.

This is in contrast to the case of the projective line minus three points and its canonical tangential base points, where the equality holds (a result of Deligne and Ihara).

There are two ingredients in the proof: (1) Galois representations often contain the image of the geometric monodromy (namely, the mapping class group) [M-Tamagawa 2000] (2) A topological result [S. Morita 98] [Hain-Reed 2000] on the cohomological obstruction of lifting the outer action of the mapping class group to automorphisms.

(This lecture is held as `Arithmetic Geometry Seminar Tokyo-Paris' and it is transmitted to IHES by the internet.)

### 2010/04/14

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The cohomological weighted fundamental lemma

**Gerard Laumon**(CNRS, Universite Paris XI - Orsay)The cohomological weighted fundamental lemma

[ Abstract ]

Using the Hitchin fibration, Ngo Bao Chau has proved the Langlands-Shelstad fundamental lemma. In a joint work with Pierre-Henri Chaudouard, we have extended Ngo's proof to obtain the weighted fundamental lemma which had been conjectured by Arthur. In the talk, I would like to present our main cohomological result.

(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)

Using the Hitchin fibration, Ngo Bao Chau has proved the Langlands-Shelstad fundamental lemma. In a joint work with Pierre-Henri Chaudouard, we have extended Ngo's proof to obtain the weighted fundamental lemma which had been conjectured by Arthur. In the talk, I would like to present our main cohomological result.

(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)

### 2009/11/18

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

Elementary computation of ramified component of the Jacobi sum

P-divisible groups and the p-adic Corona problem

**津嶋 貴弘**(東京大学大学院数理科学研究科) 16:30-17:30Elementary 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.

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) 17:45-18:45P-divisible groups and the p-adic Corona problem

### 2009/10/21

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

The local Simpson correspondence in positive characteristic

**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.

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.)

On GAGA theorems for the rigide-étale topology

**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.

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.)

Modular forms and Calabi-Yau varieties

**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.)

On the $p$-parity conjecture in the function field case

**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).

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.)

New algebraicity results for analytic torsion

On the Section Conjecture for the universal curve over function fields

**Vincent Maillot**(Paris第7大学) 16:30-17:30New algebraicity results for analytic torsion

**Richard Hain**(Duke大学) 17:45-18:45On 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.)

On the classifying space of a linear algebraic group

**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.)

Motives and adjoints

**Bruno Kahn**(Paris第7大学)Motives and adjoints

### 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)