## 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**

### 2011/05/18

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

On the linear independence of values of some Dirichlet series (JAPANESE)

**Masaki Nishimoto**(University of Tokyo)On the linear independence of values of some Dirichlet series (JAPANESE)

### 2011/05/11

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

Permanence following Temkin (ENGLISH)

**Michel Raynaud**(Universite Paris-Sud)Permanence following Temkin (ENGLISH)

[ Abstract ]

When one proceeds to a specialization, the good properties of algebraic equations may be destroyed. Starting with a bad specialization, one can try to improve it by performing modifications under control. If, at the end of the process, the initial good properties are preserved, one speaks of permanence. I shall give old and new examples of permanence. The new one concerns the relative semi-stable reduction of curves recently proved by Temkin.

When one proceeds to a specialization, the good properties of algebraic equations may be destroyed. Starting with a bad specialization, one can try to improve it by performing modifications under control. If, at the end of the process, the initial good properties are preserved, one speaks of permanence. I shall give old and new examples of permanence. The new one concerns the relative semi-stable reduction of curves recently proved by Temkin.

### 2011/04/27

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

An analogue of Sturm's theorem for Hilbert modular forms (JAPANESE)

**Yuuki Takai**(University of Tokyo)An analogue of Sturm's theorem for Hilbert modular forms (JAPANESE)

### 2011/02/10

11:00-12:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The motivic Galois group and periods of algebraic varieties (ENGLISH)

**Joseph Ayoub**(University of Zurich)The motivic Galois group and periods of algebraic varieties (ENGLISH)

[ Abstract ]

We give a construction of the motivic Galois group of $\\Q$ and explain the conjectural link with the ring of periods of algebraic varieties. Then we introduce the ring of formal periods and explain how the conjectural link with the motivic Galois group can be realized for them.

We give a construction of the motivic Galois group of $\\Q$ and explain the conjectural link with the ring of periods of algebraic varieties. Then we introduce the ring of formal periods and explain how the conjectural link with the motivic Galois group can be realized for them.

### 2011/01/26

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

The p-adic Gross-Zagier formula for elliptic curves at supersingular primes (JAPANESE)

**Shinichi Kobayashi**(Tohoku University)The p-adic Gross-Zagier formula for elliptic curves at supersingular primes (JAPANESE)

[ Abstract ]

The p-adic Gross-Zagier formula is a formula relating the derivative of the p-adic L-function of elliptic curves to the p-adic height of Heegner points. For a good ordinary prime p, the formula is proved by B. Perrin-Riou more than 20 years ago. Recently, the speaker proved it for a supersingular prime p. In this talk, he explains the proof.

The p-adic Gross-Zagier formula is a formula relating the derivative of the p-adic L-function of elliptic curves to the p-adic height of Heegner points. For a good ordinary prime p, the formula is proved by B. Perrin-Riou more than 20 years ago. Recently, the speaker proved it for a supersingular prime p. In this talk, he explains the proof.

### 2011/01/12

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

On regularized double shuffle relation for multiple zeta values (ENGLISH)

Spines with View Toward Modular Forms (ENGLISH)

**Zhonghua Li**(University of Tokyo) 16:30-17:30On regularized double shuffle relation for multiple zeta values (ENGLISH)

[ Abstract ]

Multiple zeta values(MZVs) are natural generalizations of Riemann zeta values. There are many rational relations among MZVs. It is conjectured that the regularized double shuffle relations contian all rational relations of MZVs. So other rational relations should be deduced from regularized dhouble shuffle relations. In this talk, we discuss some results on this problem. We define the gamma series accociated to elements satisfying regularized double shuffle relations and give some properties. Moreover we show that the Ohno-Zagier relations can be deduced from regularized double shuffle relations.

Multiple zeta values(MZVs) are natural generalizations of Riemann zeta values. There are many rational relations among MZVs. It is conjectured that the regularized double shuffle relations contian all rational relations of MZVs. So other rational relations should be deduced from regularized dhouble shuffle relations. In this talk, we discuss some results on this problem. We define the gamma series accociated to elements satisfying regularized double shuffle relations and give some properties. Moreover we show that the Ohno-Zagier relations can be deduced from regularized double shuffle relations.

**Dan Yasaki**(North Carolina University) 17:45-18:45Spines with View Toward Modular Forms (ENGLISH)

[ Abstract ]

The study of an arithmetic group is often aided by the fact that it acts naturally on a nice topological object. One can then use topological or geometric techniques to try to recover arithmetic data. For example, one often studies SL_2(Z) in terms of

its action on the upper half plane. In this talk, we will examine spines, which are the ``smallest" such spaces for a given arithmetic group. On overview of some known theoretical results and explicit computations will be given.

The study of an arithmetic group is often aided by the fact that it acts naturally on a nice topological object. One can then use topological or geometric techniques to try to recover arithmetic data. For example, one often studies SL_2(Z) in terms of

its action on the upper half plane. In this talk, we will examine spines, which are the ``smallest" such spaces for a given arithmetic group. On overview of some known theoretical results and explicit computations will be given.

### 2010/12/22

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

Inductive construction of the p-adic zeta functions for non-commutative

p-extensions of totally real fields with exponent p (JAPANESE)

**Takashi Hara**(University of Tokyo)Inductive construction of the p-adic zeta functions for non-commutative

p-extensions of totally real fields with exponent p (JAPANESE)

[ Abstract ]

We will discuss how to construct p-adic zeta functions and verify

the main conjecture in special cases in non-commutative Iwasawa theory

for totally real number fields.

The non-commutative Iwasawa main conjecture for totally real number

fields has been verified in special cases by Kazuya Kato,

Mahesh Kakde and the speaker by `patching method of p-adic zeta functions'

introduced by David Burns and Kazuya Kato (Jurgen Ritter and Alfred Weiss

have also constructed the successful example of the main conjecture

under somewhat different formulations).

In this talk we will explain that we can prove the main conjecture

for cases where the Galois group is isomorphic

to the direct product of the ring of p-adic integer and a finite p-group

of exponent p by utilizing Burns-Kato's method and inductive arguments.

Finally we remark that in 2010 Ritter-Weiss and Kakde independently

justified the non-commutative main conjecture

for totally real number fields under general settings.

We will discuss how to construct p-adic zeta functions and verify

the main conjecture in special cases in non-commutative Iwasawa theory

for totally real number fields.

The non-commutative Iwasawa main conjecture for totally real number

fields has been verified in special cases by Kazuya Kato,

Mahesh Kakde and the speaker by `patching method of p-adic zeta functions'

introduced by David Burns and Kazuya Kato (Jurgen Ritter and Alfred Weiss

have also constructed the successful example of the main conjecture

under somewhat different formulations).

In this talk we will explain that we can prove the main conjecture

for cases where the Galois group is isomorphic

to the direct product of the ring of p-adic integer and a finite p-group

of exponent p by utilizing Burns-Kato's method and inductive arguments.

Finally we remark that in 2010 Ritter-Weiss and Kakde independently

justified the non-commutative main conjecture

for totally real number fields under general settings.

### 2010/12/01

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

On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves (JAPANESE)

Galois theory for schemes (ENGLISH)

**Yuichiro Hoshi**(RIMS, Kyoto University) 16:30-17:30On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves (JAPANESE)

[ Abstract ]

In this talk, we will discuss the following problem posed by Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness of hyperbolic curves.

For a hyperbolic curve X over a number field, are the following three conditions equivalent?

(A) For any prime number l, X is quasi-l-monodromically full.

(B) There exists a prime number l such that X is l-monodromically full.

(C) X is l-monodromically full for all but finitely many prime numbers l.

The property of being (quasi-)monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves, and the above equivalence may be regarded as an analogue for hyperbolic curves of the following result concerning the Galois representation on the Tate module of an elliptic curve over a number field proven by Jean-Pierre Serre.

For an elliptic curve E over a number field, the following four conditions are equivalent:

(0) E does not admit complex multiplication.

(1) For any prime number l, the image of the l-adic Galois representation associated to E is open.

(2) There exists a prime number l such that the l-adic Galois representation associated to E is surjective.

(3) The l-adic Galois representation associated to E is surjective for all but finitely many prime numbers l.

In this talk, I will present some results concerning the above problem in the case where the given hyperbolic curve is of genus zero. In particular, I will give an example of a hyperbolic curve of type (0,4) over a number field which satisfies condition (C) but does not satisfy condition (A).

In this talk, we will discuss the following problem posed by Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness of hyperbolic curves.

For a hyperbolic curve X over a number field, are the following three conditions equivalent?

(A) For any prime number l, X is quasi-l-monodromically full.

(B) There exists a prime number l such that X is l-monodromically full.

(C) X is l-monodromically full for all but finitely many prime numbers l.

The property of being (quasi-)monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves, and the above equivalence may be regarded as an analogue for hyperbolic curves of the following result concerning the Galois representation on the Tate module of an elliptic curve over a number field proven by Jean-Pierre Serre.

For an elliptic curve E over a number field, the following four conditions are equivalent:

(0) E does not admit complex multiplication.

(1) For any prime number l, the image of the l-adic Galois representation associated to E is open.

(2) There exists a prime number l such that the l-adic Galois representation associated to E is surjective.

(3) The l-adic Galois representation associated to E is surjective for all but finitely many prime numbers l.

In this talk, I will present some results concerning the above problem in the case where the given hyperbolic curve is of genus zero. In particular, I will give an example of a hyperbolic curve of type (0,4) over a number field which satisfies condition (C) but does not satisfy condition (A).

**Marco Garuti**(University of Padova) 17:45-18:45Galois theory for schemes (ENGLISH)

[ Abstract ]

We discuss some aspects of finite group scheme actions: the Galois correspondence and the notion of Galois closure.

We discuss some aspects of finite group scheme actions: the Galois correspondence and the notion of Galois closure.

### 2010/11/17

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

Fast lattice reduction for F_2-linear pseudorandom number generators (JAPANESE)

**Shin Harase**(University of Tokyo)Fast lattice reduction for F_2-linear pseudorandom number generators (JAPANESE)

### 2010/10/06

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

Finite group actions on the affine space (ENGLISH)

**Hélène Esnault**(Universität Duisburg-Essen)Finite group actions on the affine space (ENGLISH)

[ Abstract ]

If $G$ is a finite $\\ell$-group acting on an affine space $\\A^n$ over a

finite field $K$ of cardinality prime to $\\ell$, Serre shows that there

exists a rational fixed point. We generalize this to the case where $K$ is a

henselian discretely valued field of characteristic zero with algebraically

closed residue field and with residue characteristic different from $\\ell$.

We also treat the case where the residue field is finite of cardinality $q$

such that $\\ell$ divides $q-1$. To this aim, we study group actions on weak

N\\'eron models.

(Joint work with Johannes Nicaise)

If $G$ is a finite $\\ell$-group acting on an affine space $\\A^n$ over a

finite field $K$ of cardinality prime to $\\ell$, Serre shows that there

exists a rational fixed point. We generalize this to the case where $K$ is a

henselian discretely valued field of characteristic zero with algebraically

closed residue field and with residue characteristic different from $\\ell$.

We also treat the case where the residue field is finite of cardinality $q$

such that $\\ell$ divides $q-1$. To this aim, we study group actions on weak

N\\'eron models.

(Joint work with Johannes Nicaise)

### 2010/07/07

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

On the stable reduction of $X_0(p^4)$ (JAPANESE)

**Takahiro Tsushima**(University of Tokyo)On the stable reduction of $X_0(p^4)$ (JAPANESE)

### 2010/06/16

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

Vanishing theorems revisited, after K.-W. Lan and J. Suh (ENGLISH)

**Luc Illusie**(Universite de Paris-Sud)Vanishing theorems revisited, after K.-W. Lan and J. Suh (ENGLISH)

[ Abstract ]

Let k be an algebraically closed field of characteristic p and X,

Y proper, smooth k-schemes. J. Suh has proved a vanishing theorem of Kollar

type for certain nef and big line bundles L on Y and morphisms f : X -> Y

having semistable reduction along a divisor with simple normal crossings. It

holds both if p = 0 and if p > 0 modulo some additional liftability mod p^2

and dimension assumptions, and generalizes vanishing theorems of Esnault-

Viehweg and of mine. I'll give an outline of the proof and sketch some

applications, due to K.-W. Lan and J. Suh, to the cohomology of certain

automorphic bundles arising from PEL type Shimura varieties.

Let k be an algebraically closed field of characteristic p and X,

Y proper, smooth k-schemes. J. Suh has proved a vanishing theorem of Kollar

type for certain nef and big line bundles L on Y and morphisms f : X -> Y

having semistable reduction along a divisor with simple normal crossings. It

holds both if p = 0 and if p > 0 modulo some additional liftability mod p^2

and dimension assumptions, and generalizes vanishing theorems of Esnault-

Viehweg and of mine. I'll give an outline of the proof and sketch some

applications, due to K.-W. Lan and J. Suh, to the cohomology of certain

automorphic bundles arising from PEL type Shimura varieties.

### 2010/06/09

16:15-17:15 Room #052 (Graduate School of Math. Sci. Bldg.)

Universal mixed elliptic motives (ENGLISH)

**Richard Hain**(Duke University)Universal mixed elliptic motives (ENGLISH)

[ Abstract ]

This is joint work with Makoto Matsumoto. A mixed elliptic

motive is a mixed motive (MHS, Galois representation, etc) whose

weight graded quotients are Tate twists of symmetric powers of the the

motive of elliptic curve. A universal mixed elliptic motive is an

object that can be specialized to a mixed elliptic motive for any

elliptic curve and whose specialization to the nodal cubic is a mixed

Tate motive. Universal mixed elliptic motives form a tannakian

category. In this talk I will define universal mixed elliptic motives,

give some fundamental examples, and explain what we know about the

fundamental group of this category. The "geometric part" of this group

is an extension of SL_2 by a prounipotent group that is generated by

Eisenstein series and which has a family of relations for each cusp

form. Although these relations are not known, we have a very good idea

of what they are, thanks to work of Aaron Pollack, who determined

relations between the generators in a very large representation of

this group.

This is joint work with Makoto Matsumoto. A mixed elliptic

motive is a mixed motive (MHS, Galois representation, etc) whose

weight graded quotients are Tate twists of symmetric powers of the the

motive of elliptic curve. A universal mixed elliptic motive is an

object that can be specialized to a mixed elliptic motive for any

elliptic curve and whose specialization to the nodal cubic is a mixed

Tate motive. Universal mixed elliptic motives form a tannakian

category. In this talk I will define universal mixed elliptic motives,

give some fundamental examples, and explain what we know about the

fundamental group of this category. The "geometric part" of this group

is an extension of SL_2 by a prounipotent group that is generated by

Eisenstein series and which has a family of relations for each cusp

form. Although these relations are not known, we have a very good idea

of what they are, thanks to work of Aaron Pollack, who determined

relations between the generators in a very large representation of

this group.

### 2010/06/09

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

Constructibilité uniforme des images directes supérieures en

cohomologie étale

(ENGLISH)

**Fabrice Orgogozo**(CNRS, École polytechnique)Constructibilité uniforme des images directes supérieures en

cohomologie étale

(ENGLISH)

[ Abstract ]

Motivé par une remarque de N. Katz sur le lien entre la

torsion de la Z_ℓ-cohomologie étale et les ultraproduits de groupes de

F_ℓ-cohomologie, nous démontrons un théorème d'uniformité en ℓ pour la

constructibilité des images directes supérieures entre schémas de type fini

sur un trait excellent. (Un tel théorème avait été considéré par

O. Gabber il y a plusieurs années déjà.)

La méthode est maintenant classique : on utilise des

théorèmes de A. J. de Jong et un peu de log-géométrie.

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

Motivé par une remarque de N. Katz sur le lien entre la

torsion de la Z_ℓ-cohomologie étale et les ultraproduits de groupes de

F_ℓ-cohomologie, nous démontrons un théorème d'uniformité en ℓ pour la

constructibilité des images directes supérieures entre schémas de type fini

sur un trait excellent. (Un tel théorème avait été considéré par

O. Gabber il y a plusieurs années déjà.)

La méthode est maintenant classique : on utilise des

théorèmes de A. J. de Jong et un peu de log-géométrie.

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

### 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