## Number Theory Seminar

Seminar information archive ～04/21｜Next seminar｜Future seminars 04/22～

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**

### 2013/01/16

18:00-19:00 Room #002 (Graduate School of Math. Sci. Bldg.)

On differential modules associated to de Rham representations in the imperfect residue field case (ENGLISH)

**Shun Ohkubo**(University of Tokyo)On differential modules associated to de Rham representations in the imperfect residue field case (ENGLISH)

[ Abstract ]

Let K be a CDVF of mixed characteristic (0,p) and G the absolute Galois group of K. When the residue field of K is perfect, Laurent Berger constructed a p-adic differential equation N_dR(V) for any de Rham representation V of G. In this talk, we will generalize his construction when the residue field of K is not perfect. We also explain some ramification properties of our N_dR, which are due to Adriano Marmora in the perfect residue field case.

Let K be a CDVF of mixed characteristic (0,p) and G the absolute Galois group of K. When the residue field of K is perfect, Laurent Berger constructed a p-adic differential equation N_dR(V) for any de Rham representation V of G. In this talk, we will generalize his construction when the residue field of K is not perfect. We also explain some ramification properties of our N_dR, which are due to Adriano Marmora in the perfect residue field case.

### 2012/12/19

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

A generalization of Kato's local epsilon conjecture for

(φ, Γ)-modules over the Robba ring (JAPANESE)

**Kentarou Nakamura**(Hokkaido University)A generalization of Kato's local epsilon conjecture for

(φ, Γ)-modules over the Robba ring (JAPANESE)

[ Abstract ]

In his preprint “Lectures on the approach to Iwasawa theory of Hasse-Weil L-functions via B_dR, Part II ", Kazuya Kato proposed a conjecture called local epsilon conjecture. This conjecture predicts that the determinant of the Galois cohomology of a family of p-adic Galois representations has a canonical base whose specializations at de Rham points can be characterized by using Bloch-Kato exponential, L-factors and Deligne-Langlands epsilon constants of the associated Weil-Deligne representations.

In my talk, I generalize his conjecture for families of (φ, Γ)-modules over the Robba ring, and prove a part of this conjecture in the trianguline case. The two key ingredients are the recent result of Kedlaya-Pottharst-Xiao on the ﬁniteness of cohomologies of (φ, Γ)-modules and my result on Bloch-Kato exponential map for (φ, Γ)-modules.

In his preprint “Lectures on the approach to Iwasawa theory of Hasse-Weil L-functions via B_dR, Part II ", Kazuya Kato proposed a conjecture called local epsilon conjecture. This conjecture predicts that the determinant of the Galois cohomology of a family of p-adic Galois representations has a canonical base whose specializations at de Rham points can be characterized by using Bloch-Kato exponential, L-factors and Deligne-Langlands epsilon constants of the associated Weil-Deligne representations.

In my talk, I generalize his conjecture for families of (φ, Γ)-modules over the Robba ring, and prove a part of this conjecture in the trianguline case. The two key ingredients are the recent result of Kedlaya-Pottharst-Xiao on the ﬁniteness of cohomologies of (φ, Γ)-modules and my result on Bloch-Kato exponential map for (φ, Γ)-modules.

### 2012/12/12

18:00-19:00 Room #002 (Graduate School of Math. Sci. Bldg.)

The Tate conjecture for K3 surfaces and holomorphic symplectic varieties over finite fields (ENGLISH)

**François Charles**(CNRS & Université de Rennes 1)The Tate conjecture for K3 surfaces and holomorphic symplectic varieties over finite fields (ENGLISH)

[ Abstract ]

We prove the Tate conjecture for divisors on reductions of holomorphic symplectic varieties over finite fields -- with some restrictions on the characteristic of the base field. We will be concerned mostly with the supersingular case. As a special case, we prove the Tate conjecture, also known as Artin's conjecture in our case, for K3 surfaces over finite fields of characteristic at least 5 and for codimension 2 cycles on cubic fourfolds.

We prove the Tate conjecture for divisors on reductions of holomorphic symplectic varieties over finite fields -- with some restrictions on the characteristic of the base field. We will be concerned mostly with the supersingular case. As a special case, we prove the Tate conjecture, also known as Artin's conjecture in our case, for K3 surfaces over finite fields of characteristic at least 5 and for codimension 2 cycles on cubic fourfolds.

### 2012/11/14

18:00-19:00 Room #002 (Graduate School of Math. Sci. Bldg.)

De Rham-Witt complexes with coefficients and rigid cohomology

(ENGLISH)

**Pierre Berthelot**(Université de Rennes 1)De Rham-Witt complexes with coefficients and rigid cohomology

(ENGLISH)

[ Abstract ]

For a smooth scheme over a perfect field of characteristic p, we will explain a generalization of the classical comparison theorem between crystalline cohomology and de Rham-Witt cohomology to the case of cohomologies with coefficients in a p-torsion free crystal. This provides in particular a description of the rigid cohomology of a proper singular scheme in terms of a de Rham-Witt complex built from a closed immersion into a smooth scheme.

For a smooth scheme over a perfect field of characteristic p, we will explain a generalization of the classical comparison theorem between crystalline cohomology and de Rham-Witt cohomology to the case of cohomologies with coefficients in a p-torsion free crystal. This provides in particular a description of the rigid cohomology of a proper singular scheme in terms of a de Rham-Witt complex built from a closed immersion into a smooth scheme.

### 2012/07/18

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Voevodsky motives and a theorem of Gabber (ENGLISH)

**Shane Kelly**(Australian National University)Voevodsky motives and a theorem of Gabber (ENGLISH)

[ Abstract ]

The assumption that the base field satisfies resolution of singularities litters Voevodsky's work on motives. While we don't have resolution of singularities in positive characteristic p, there is a theorem of Gabber on alterations which may be used as a substitute if we are willing to work with Z[1/p] coefficients. We will discuss how this theorem of Gabber may be applied in the context of Voevodsky's work and mention some consequences.

The assumption that the base field satisfies resolution of singularities litters Voevodsky's work on motives. While we don't have resolution of singularities in positive characteristic p, there is a theorem of Gabber on alterations which may be used as a substitute if we are willing to work with Z[1/p] coefficients. We will discuss how this theorem of Gabber may be applied in the context of Voevodsky's work and mention some consequences.

### 2012/07/04

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

A cohomological Hasse principle of varieties over higher local fields and applications to higher dimensional class field theory (ENGLISH)

**Patrick Forré**(University of Tokyo)A cohomological Hasse principle of varieties over higher local fields and applications to higher dimensional class field theory (ENGLISH)

[ Abstract ]

In this talk I will give an overview of the necessary tools for a description of the class field theory of varieties over higher local fields developed by sevaral mathematicians. On this I will motivate the importance of the proposal and verification of a cohomological Hasse principle for varieties over higher local fields, a generalization of Kato's conjectures, and sketch the recent progress on this.

In this talk I will give an overview of the necessary tools for a description of the class field theory of varieties over higher local fields developed by sevaral mathematicians. On this I will motivate the importance of the proposal and verification of a cohomological Hasse principle for varieties over higher local fields, a generalization of Kato's conjectures, and sketch the recent progress on this.

### 2012/06/20

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

On the reduction modulo p of representations of a quaternion

division algebra over a p-adic field (JAPANESE)

**Kazuki Tokimoto**(University of Tokyo)On the reduction modulo p of representations of a quaternion

division algebra over a p-adic field (JAPANESE)

[ Abstract ]

The p-adic Langlands correspondence and the mod p Langlands correspondence for GL_2(Q_p) are known to be compatible with the reduction modulo p in many cases.

In this talk, we examine whether a similar compatibility exists for the composition of the local Langlands correspondence and the local Jacquet-Langlands correspondence.

The simplest case has already been treated by Vign¥'eras. We deal with more cases.

The p-adic Langlands correspondence and the mod p Langlands correspondence for GL_2(Q_p) are known to be compatible with the reduction modulo p in many cases.

In this talk, we examine whether a similar compatibility exists for the composition of the local Langlands correspondence and the local Jacquet-Langlands correspondence.

The simplest case has already been treated by Vign¥'eras. We deal with more cases.

### 2012/06/13

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Singular homologies of non-Archimedean analytic spaces and integrals along cycles (JAPANESE)

**Tomoki Mihara**(University of Tokyo)Singular homologies of non-Archimedean analytic spaces and integrals along cycles (JAPANESE)

### 2012/05/30

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Kernel of the monodromy operator for semistable curves (ENGLISH)

**Valentina Di Proietto**(University of Tokyo)Kernel of the monodromy operator for semistable curves (ENGLISH)

[ Abstract ]

For a semistable curve, we study the action of the monodromy operator on the first log-crystalline cohomology group. In particular we examine the relation between the kernel of the monodromy operator and the first rigid cohomology group, in the case of trivial coefficients, giving a new proof of a theorem of B. Chiarellotto and in the case of certain unipotent F-isocrystals as coefficients.

This is a joint work in progress with B. Chiarellotto, R. Coleman and A. Iovita.

For a semistable curve, we study the action of the monodromy operator on the first log-crystalline cohomology group. In particular we examine the relation between the kernel of the monodromy operator and the first rigid cohomology group, in the case of trivial coefficients, giving a new proof of a theorem of B. Chiarellotto and in the case of certain unipotent F-isocrystals as coefficients.

This is a joint work in progress with B. Chiarellotto, R. Coleman and A. Iovita.

### 2012/05/23

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Simply connected elliptic surfaces (JAPANESE)

**Kentaro Mitsui**(University of Tokyo)Simply connected elliptic surfaces (JAPANESE)

[ Abstract ]

We characterize simply connected elliptic surfaces by their singular fibers in any characteristic case. To this end, we study orbifolds of curves, local canonical bundle formula, and resolutions of multiple fibers. The result was known for the complex analytic case. Our method can be applied to the rigid analytic case.

We characterize simply connected elliptic surfaces by their singular fibers in any characteristic case. To this end, we study orbifolds of curves, local canonical bundle formula, and resolutions of multiple fibers. The result was known for the complex analytic case. Our method can be applied to the rigid analytic case.

### 2012/05/16

16:40-17:40 Room #002 (Graduate School of Math. Sci. Bldg.)

On uniform bound of the maximal subgroup of the inertia group acting unipotently on $¥ell$-adic cohomology (JAPANESE)

**Naoya Umezaki**(University of Tokyo)On uniform bound of the maximal subgroup of the inertia group acting unipotently on $¥ell$-adic cohomology (JAPANESE)

[ Abstract ]

For a smooth projective variety over a local field,

the action of the inertia group on the $¥ell$-adic cohomology group is

unipotent if it is restricted to some open subgroup.

In this talk, we give a uniform bound of the index of the maximal open

subgroup satisfying this property.

This bound depends only on the Betti numbers of $X$ and certain Chern

numbers depending on a projective embedding of $X$.

For a smooth projective variety over a local field,

the action of the inertia group on the $¥ell$-adic cohomology group is

unipotent if it is restricted to some open subgroup.

In this talk, we give a uniform bound of the index of the maximal open

subgroup satisfying this property.

This bound depends only on the Betti numbers of $X$ and certain Chern

numbers depending on a projective embedding of $X$.

### 2012/04/18

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Explicit constructions of rational points on elliptic curves (ENGLISH)

**Alan Lauder**(University of Oxford)Explicit constructions of rational points on elliptic curves (ENGLISH)

[ Abstract ]

I will present an algorithm for computing certain special

values of p-adic L-functions, and discuss an application to

the efficient construction of rational points on elliptic curves.

I will present an algorithm for computing certain special

values of p-adic L-functions, and discuss an application to

the efficient construction of rational points on elliptic curves.

### 2012/04/11

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

Around the Mordell-Lang conjecture in positive characteristic (ENGLISH)

**Damian Rossler**(CNRS, Universite de Toulouse)Around the Mordell-Lang conjecture in positive characteristic (ENGLISH)

[ Abstract ]

Let V be a subvariety of an abelian variety A over C and let G\\subseteq A(C) be a subgroup. The classical Mordell-Lang conjecture predicts that if V is of general type and G\\otimesQ is finite dimensional, then V\\cap G is not Zariski dense in V. This statement contains the Mordell conjecture as well as the Manin-Mumford conjecture (for curves). The positive characteristic analog of the Mordell-Lang conjecture makes sense, when A is supposed to have no subquotient, which is defined over a finite field. This positive characteristic analog was proven in 1996 by E. Hrushovski using model-theoretic methods. We shall discuss the prehistory and context of this proof. We shall also discuss the proof (due to the speaker) of the fact that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture (whereas this seems far from true in characteristic 0).

Let V be a subvariety of an abelian variety A over C and let G\\subseteq A(C) be a subgroup. The classical Mordell-Lang conjecture predicts that if V is of general type and G\\otimesQ is finite dimensional, then V\\cap G is not Zariski dense in V. This statement contains the Mordell conjecture as well as the Manin-Mumford conjecture (for curves). The positive characteristic analog of the Mordell-Lang conjecture makes sense, when A is supposed to have no subquotient, which is defined over a finite field. This positive characteristic analog was proven in 1996 by E. Hrushovski using model-theoretic methods. We shall discuss the prehistory and context of this proof. We shall also discuss the proof (due to the speaker) of the fact that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture (whereas this seems far from true in characteristic 0).

### 2012/02/22

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

Twistor $D$-module and harmonic bundle (ENGLISH)

**Takuro Mochizuki**(Research Institute for Mathematical Sciences, Kyoto University)Twistor $D$-module and harmonic bundle (ENGLISH)

[ Abstract ]

Abstract:

We shall overview the theory of twistor $D$-modules and

harmonic bundles. I am planning to survey the following topics,

motivated by the Hard Lefschetz Theorem for semisimple holonomic

$D$-modules:

1. What is a twistor $D$-module?

2. Local structure of meromorphic flat bundles

3. Wild harmonic bundles from local and global viewpoints

(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。)

Abstract:

We shall overview the theory of twistor $D$-modules and

harmonic bundles. I am planning to survey the following topics,

motivated by the Hard Lefschetz Theorem for semisimple holonomic

$D$-modules:

1. What is a twistor $D$-module?

2. Local structure of meromorphic flat bundles

3. Wild harmonic bundles from local and global viewpoints

(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。)

### 2012/01/12

18:15-19:15 Room #056 (Graduate School of Math. Sci. Bldg.)

New perspectives on the Breuil-Mézard conjecture (joint with M. Emerton)

(ENGLISH)

**Toby Gee**(Imperial College London)New perspectives on the Breuil-Mézard conjecture (joint with M. Emerton)

(ENGLISH)

[ Abstract ]

I will discuss joint work with Matthew Emerton on geometric approaches to the Breuil-Mézard conjecture, generalising a geometric approach of Breuil and Mézard. I will discuss a proof of the geometric version of the original conjecture, as well as work in progress on a geometric version of the conjecture which does not make use of a fixed residual representation.

(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。)

I will discuss joint work with Matthew Emerton on geometric approaches to the Breuil-Mézard conjecture, generalising a geometric approach of Breuil and Mézard. I will discuss a proof of the geometric version of the original conjecture, as well as work in progress on a geometric version of the conjecture which does not make use of a fixed residual representation.

(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。)

### 2011/12/21

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

On Sharifi's conjecture (JAPANESE)

**Kazuya Kato**(University of Chicago)On Sharifi's conjecture (JAPANESE)

### 2011/12/19

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

Galois Theory: Past and Present (ENGLISH)

**Tamas Szamuely**(Budapest)Galois Theory: Past and Present (ENGLISH)

### 2011/12/14

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

Good and bad reduction for algebraic dynamical systems (ENGLISH)

**Lucien Szpiro**(City University of New York)Good and bad reduction for algebraic dynamical systems (ENGLISH)

[ Abstract ]

We will report on a recent work with collaborators in New York on the

different ways to look at degenerations of a dynamical system in a one

parameter family. Resultants, conductors and isotriviality will be analyzed.

We will report on a recent work with collaborators in New York on the

different ways to look at degenerations of a dynamical system in a one

parameter family. Resultants, conductors and isotriviality will be analyzed.

### 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)