## Number Theory Seminar

Seminar information archive ～02/07｜Next seminar｜Future seminars 02/08～

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**

### 2011/12/08

18:30-19:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Nonabelian p-adic Hodge theory and Frobenius (ENGLISH)

**Gerd Faltings**(Max Planck Institute for Mathematics, Bonn)Nonabelian p-adic Hodge theory and Frobenius (ENGLISH)

[ Abstract ]

Some time ago, I constructed a relation between Higgs-bundles and p-adic etale sheaves, on curves over a p-adic field. This corresponds (say in the abelian case) to a Hodge-Tate picture. In the lecture I try to explain one way to introduce Frobenius into the theory. We do not get a complete theory but at least can treat p-adic sheaves close to trivial.

(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。)

Some time ago, I constructed a relation between Higgs-bundles and p-adic etale sheaves, on curves over a p-adic field. This corresponds (say in the abelian case) to a Hodge-Tate picture. In the lecture I try to explain one way to introduce Frobenius into the theory. We do not get a complete theory but at least can treat p-adic sheaves close to trivial.

(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。)

### 2011/11/09

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On extension and restriction of overconvergent isocrystals (ENGLISH)

**Atsushi Shiho**(University of Tokyo)On extension and restriction of overconvergent isocrystals (ENGLISH)

[ Abstract ]

First we explain two theorems concerning (log) extension of overconvergent isocrystals. One is a p-adic analogue of the theorem of logarithmic extension of regular integrable connections, and the other is a p-adic analogue of Zariski-Nagata purity. Next we explain a theorem which says that we can check certain property of overconvergent isocrystals by restricting them to curves.

(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。)

First we explain two theorems concerning (log) extension of overconvergent isocrystals. One is a p-adic analogue of the theorem of logarithmic extension of regular integrable connections, and the other is a p-adic analogue of Zariski-Nagata purity. Next we explain a theorem which says that we can check certain property of overconvergent isocrystals by restricting them to curves.

(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。)

### 2011/11/02

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

Hypergeometric series and arithmetic-geometric mean over 2-adic fields (JAPANESE)

**Kensaku Kinjo**(University of Tokyo)Hypergeometric series and arithmetic-geometric mean over 2-adic fields (JAPANESE)

[ Abstract ]

Dwork proved that the Gaussian hypergeometric function on p-adic numbers

can be extended to a function which takes values of the unit roots of

ordinary elliptic curves over a finite field of characteristic p>2.

We present an analogous theory in the case p=2.

As an application, we give a relation between the canonical lift

and the unit root of an elliptic curve over a finite field of

characteristic 2

by using the 2-adic arithmetic-geometric mean.

Dwork proved that the Gaussian hypergeometric function on p-adic numbers

can be extended to a function which takes values of the unit roots of

ordinary elliptic curves over a finite field of characteristic p>2.

We present an analogous theory in the case p=2.

As an application, we give a relation between the canonical lift

and the unit root of an elliptic curve over a finite field of

characteristic 2

by using the 2-adic arithmetic-geometric mean.

### 2011/10/19

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

K_2 of the biquaternion algebra (ENGLISH)

[ Reference URL ]

http://www.ihes.fr/~abbes/SGA/suslin.pdf

**Andrei Suslin**(Northwestern University)K_2 of the biquaternion algebra (ENGLISH)

[ Reference URL ]

http://www.ihes.fr/~abbes/SGA/suslin.pdf

### 2011/07/27

16:00-18:15 Room #123 (Graduate School of Math. Sci. Bldg.)

Discriminants and determinant of a hypersurface of even dimension (ENGLISH)

Multiplicities of discriminants (ENGLISH)

**Takeshi Saito**(University of Tokyo) 16:00-17:00Discriminants and determinant of a hypersurface of even dimension (ENGLISH)

[ Abstract ]

The determinant of the cohomology of a smooth hypersurface

of even dimension as a quadratic character of the absolute

Galois group is computed by the discriminant of the de Rham

cohomology. They are also computed by the discriminant of a

defining polynomial. We determine the sign involved by testing

the formula for the Fermat hypersurfaces.

This is a joint work with J-P. Serre.

The determinant of the cohomology of a smooth hypersurface

of even dimension as a quadratic character of the absolute

Galois group is computed by the discriminant of the de Rham

cohomology. They are also computed by the discriminant of a

defining polynomial. We determine the sign involved by testing

the formula for the Fermat hypersurfaces.

This is a joint work with J-P. Serre.

**Dennis Eriksson**(University of Gothenburg) 17:15-18:15Multiplicities of discriminants (ENGLISH)

[ Abstract ]

The discriminant of a homogenous polynomial is another homogenous

polynomial in the coefficients of the polynomial, which is zero

if and only if the corresponding hypersurface is singular. In

case the coefficients are in a discrete valuation ring, the

order of the discriminant (if non-zero) measures the bad

reduction. We give some new results on this order, and in

particular tie it to Bloch's conjecture/the Kato-T.Saito formula

on equality of localized Chern classes and Artin conductors. We

can precisely compute all the numbers in the case of ternary

forms, giving a partial generalization of Ogg's formula for

elliptic curves.

The discriminant of a homogenous polynomial is another homogenous

polynomial in the coefficients of the polynomial, which is zero

if and only if the corresponding hypersurface is singular. In

case the coefficients are in a discrete valuation ring, the

order of the discriminant (if non-zero) measures the bad

reduction. We give some new results on this order, and in

particular tie it to Bloch's conjecture/the Kato-T.Saito formula

on equality of localized Chern classes and Artin conductors. We

can precisely compute all the numbers in the case of ternary

forms, giving a partial generalization of Ogg's formula for

elliptic curves.

### 2011/06/15

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

Product formula for $p$-adic epsilon factors (ENGLISH)

**Tomoyuki Abe**(IPMU)Product formula for $p$-adic epsilon factors (ENGLISH)

[ Abstract ]

I would like to talk about my recent work jointly with A. Marmora on a product formula for $p$-adic epsilon factors. In 80's Deligne conjectured that a constant appearing in the functional equation of $L$-function of $\\ell$-adic lisse sheaf can be written by means of local contributions, and proved some particular cases. This conjecture was proven later by Laumon, and was used in the Lafforgue's proof of the Langlands' program for functional filed case. In my talk, I would like to prove a $p$-adic analog of this product formula.

I would like to talk about my recent work jointly with A. Marmora on a product formula for $p$-adic epsilon factors. In 80's Deligne conjectured that a constant appearing in the functional equation of $L$-function of $\\ell$-adic lisse sheaf can be written by means of local contributions, and proved some particular cases. This conjecture was proven later by Laumon, and was used in the Lafforgue's proof of the Langlands' program for functional filed case. In my talk, I would like to prove a $p$-adic analog of this product formula.

### 2011/06/08

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

Congruences of modular forms and the Iwasawa λ-invariants (JAPANESE)

**Yuichi Hirano**(University of Tokyo)Congruences of modular forms and the Iwasawa λ-invariants (JAPANESE)

### 2011/05/25

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On good reduction of some K3 surfaces (JAPANESE)

**Yuya Matsumoto**(University of Tokyo)On good reduction of some K3 surfaces (JAPANESE)

### 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。)