## Seminar information archive

Seminar information archive ～10/21｜Today's seminar 10/22 | Future seminars 10/23～

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Restricted Bergman kernel asymptotics (JAPANESE)

**Tomoyuki HISAMOTO**(Univ. of Tokyo)Restricted Bergman kernel asymptotics (JAPANESE)

### 2010/06/03

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Fixed Points in the Stone-Cech boundary of Groups (ENGLISH)

**Makoto Yamashita**(Univ. Tokyo)Fixed Points in the Stone-Cech boundary of Groups (ENGLISH)

[ Abstract ]

We discuss the class of discrete groups which admit fixed points under the adjoint action on the Stone-Cech boundary. Such groups have vanishing $L^2$-Betti numbers, and nonamenable ones fail to have property (AO).

We discuss the class of discrete groups which admit fixed points under the adjoint action on the Stone-Cech boundary. Such groups have vanishing $L^2$-Betti numbers, and nonamenable ones fail to have property (AO).

#### GCOE lecture series

16:30-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Introduction to the cohomology of locally symmetric spaces 2

(ENGLISH)

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**Birgit Speh**(Cornel University)Introduction to the cohomology of locally symmetric spaces 2

(ENGLISH)

[ Abstract ]

I will give an introduction to the cohomology of noncompact locally symmetric spaces $X_\\Gamma =K \\backslash G / \\Gamma $.

If $X_\\Gamma $ is cocompact this cohomology can be expressed as the $(\\bg,K) $-cohomology of automorphic representations. I will explain how representation theory and automorphic forms can be used to study the cohomology in this case.

[ Reference URL ]I will give an introduction to the cohomology of noncompact locally symmetric spaces $X_\\Gamma =K \\backslash G / \\Gamma $.

If $X_\\Gamma $ is cocompact this cohomology can be expressed as the $(\\bg,K) $-cohomology of automorphic representations. I will explain how representation theory and automorphic forms can be used to study the cohomology in this case.

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2010/06/02

#### Number Theory Seminar

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.

### 2010/06/01

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On Fatou-Julia decompositions (JAPANESE)

**Taro Asuke**(The University of Tokyo)On Fatou-Julia decompositions (JAPANESE)

[ Abstract ]

We will explain that Fatou-Julia decompositions can be

introduced in a unified manner to several kinds of one-dimensional

complex dynamical systems, which include the action of Kleinian groups,

iteration of holomorphic mappings and complex codimension-one foliations.

In this talk we will restrict ourselves mostly to the cases where the

dynamical systems have a certain compactness, however, we will mention

how to deal with dynamical systems without compactness.

We will explain that Fatou-Julia decompositions can be

introduced in a unified manner to several kinds of one-dimensional

complex dynamical systems, which include the action of Kleinian groups,

iteration of holomorphic mappings and complex codimension-one foliations.

In this talk we will restrict ourselves mostly to the cases where the

dynamical systems have a certain compactness, however, we will mention

how to deal with dynamical systems without compactness.

#### GCOE lecture series

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Introduction to the cohomology of locally symmetric spaces

(ENGLISH)

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**Birgit Speh**(Cornel University)Introduction to the cohomology of locally symmetric spaces

(ENGLISH)

[ Abstract ]

I will give an introduction to the cohomology of noncompact locally symmetric spaces $X_\\Gamma =K \\backslash G / \\Gamma $.

If $X_\\Gamma $ is cocompact this cohomology can be expressed as the $(g,K)$-cohomology of automorphic representations. I will explain how representation theory and automorphic forms can be used to study the cohomology in this case.

[ Reference URL ]I will give an introduction to the cohomology of noncompact locally symmetric spaces $X_\\Gamma =K \\backslash G / \\Gamma $.

If $X_\\Gamma $ is cocompact this cohomology can be expressed as the $(g,K)$-cohomology of automorphic representations. I will explain how representation theory and automorphic forms can be used to study the cohomology in this case.

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2010/05/31

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

On Pfaffian Calabi-Yau Varieties and Mirror Symmetry (JAPANESE)

**Atsushi Kanazawa**(The University of Tokyo)On Pfaffian Calabi-Yau Varieties and Mirror Symmetry (JAPANESE)

[ Abstract ]

We construct new smooth CY 3-folds with 1-dimensional Kaehler moduli and

determine their fundamental topological invariants. The existence of CY

3-folds with the computed invariants was previously conjectured. We then

report mirror symmetry for these non-complete intersection CY 3-folds.

We explicitly build their mirror partners, some of which have 2 LCSLs,

and carry out instanton computations for g=0,1.

We construct new smooth CY 3-folds with 1-dimensional Kaehler moduli and

determine their fundamental topological invariants. The existence of CY

3-folds with the computed invariants was previously conjectured. We then

report mirror symmetry for these non-complete intersection CY 3-folds.

We explicitly build their mirror partners, some of which have 2 LCSLs,

and carry out instanton computations for g=0,1.

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Singularities and analytic torsion (JAPANESE)

**Ken-ichi YOSHIKAWA**(Kyoto Univ.)Singularities and analytic torsion (JAPANESE)

### 2010/05/28

#### Classical Analysis

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On solutions of uniformization equations (JAPANESE)

**Jiro Sekiguchi**(Tokyo University of Agriculture and Technology)On solutions of uniformization equations (JAPANESE)

### 2010/05/27

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

The C*-algebra of codimension one foliations which are almost without holonomy (ENGLISH)

**Catherine Oikonomides**(Univ. Tokyo)The C*-algebra of codimension one foliations which are almost without holonomy (ENGLISH)

### 2010/05/26

#### Seminar on Probability and Statistics

16:20-17:30 Room #000 (Graduate School of Math. Sci. Bldg.)

Financial data analysis with R-YUIMA (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2010/02.html

**FUKASAWA, Masaaki**(CSFI, Osaka Univ.)Financial data analysis with R-YUIMA (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2010/02.html

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Hokkaido University)

A SELECTION CRITERION FOR SOLUTIONS OF A

SYSTEM OF EIKONAL EQUATIONS

(ENGLISH)

**Giovanni Pisante**(Department of MathematicsHokkaido University)

A SELECTION CRITERION FOR SOLUTIONS OF A

SYSTEM OF EIKONAL EQUATIONS

(ENGLISH)

[ Abstract ]

We deal with the system of eikonal equations |ðu/ðx1|=1, |ðu/ðx2|=1 in a planar Lipschitz domain with zero boundary condition. Exploiting the classical pyramidal construction introduced by Cellina, it is easy to prove that there exist infinitely many Lipschitz solutions. Then, the natural problem that has arisen in this framework is to find a way to select and characterize a particular meaningful class of solutions.

We propose a variational method to select the class of solutions which minimize the discontinuity set of the gradient. More precisely we select an optimal weighted measure for the jump set of the second derivatives of a given solution v of the system and we prove the existence of minimizers of the corresponding variational problem.

We deal with the system of eikonal equations |ðu/ðx1|=1, |ðu/ðx2|=1 in a planar Lipschitz domain with zero boundary condition. Exploiting the classical pyramidal construction introduced by Cellina, it is easy to prove that there exist infinitely many Lipschitz solutions. Then, the natural problem that has arisen in this framework is to find a way to select and characterize a particular meaningful class of solutions.

We propose a variational method to select the class of solutions which minimize the discontinuity set of the gradient. More precisely we select an optimal weighted measure for the jump set of the second derivatives of a given solution v of the system and we prove the existence of minimizers of the corresponding variational problem.

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Existence and non-existence of global solutions for a discrete semilinear heat equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Keisuke Matsuya**(University of Tokyo)Existence and non-existence of global solutions for a discrete semilinear heat equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

#### GCOE Seminars

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Existence and non-existence of global solutions for a discrete semilinear heat equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**松家 敬介**(東京大学大学院数理科学研究科)Existence and non-existence of global solutions for a discrete semilinear heat equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

### 2010/05/25

#### Lie Groups and Representation Theory

17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On endoscopy, packets, and invariants (JAPANESE)

**Kaoru Hiraga**(Kyoto University)On endoscopy, packets, and invariants (JAPANESE)

[ Abstract ]

The theory of endoscopy came out of the Langlands functoriality and the trace formula.

In this talk, I will briefly explain what the endoscopy is, and talk about packet, formal degree and Whittaker normalization of transfer.

I would like to talk about the connection between these topics and the endoscopy.

The theory of endoscopy came out of the Langlands functoriality and the trace formula.

In this talk, I will briefly explain what the endoscopy is, and talk about packet, formal degree and Whittaker normalization of transfer.

I would like to talk about the connection between these topics and the endoscopy.

### 2010/05/24

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

A counterexample of the birational Torelli problem via Fourier--Mukai transforms (JAPANESE)

**Hokuto Uehara**(Tokyo Metropolitan University)A counterexample of the birational Torelli problem via Fourier--Mukai transforms (JAPANESE)

[ Abstract ]

We study the Fourier--Mukai numbers of rational elliptic surfaces. As

its application, we give an example of a pair of minimal 3-folds $X$

with Kodaira dimensions 1, $h^1(O_X)=h^2(O_X)=0$ such that they are

mutually derived equivalent, deformation equivalent, but not

birationally equivalent. It also supplies a counterexample of the

birational Torelli problem.

We study the Fourier--Mukai numbers of rational elliptic surfaces. As

its application, we give an example of a pair of minimal 3-folds $X$

with Kodaira dimensions 1, $h^1(O_X)=h^2(O_X)=0$ such that they are

mutually derived equivalent, deformation equivalent, but not

birationally equivalent. It also supplies a counterexample of the

birational Torelli problem.

### 2010/05/19

#### Seminar on Mathematics for various disciplines

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Mathematical sciences collaborating with clinical medicine (JAPANESE)

**Hiroshi Suito**(Okayama University)Mathematical sciences collaborating with clinical medicine (JAPANESE)

#### Seminar on Probability and Statistics

15:00-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Estimation of the variance-covariance structure for stochastic processes and applications of YUIMA (JAPANESE)

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2010/01.html

**YOSHIDA, Nakahiro**(University of Tokyo)Estimation of the variance-covariance structure for stochastic processes and applications of YUIMA (JAPANESE)

[ Abstract ]

We discuss limit theorems and asymptotic expansions in estimation of the variance-covariance structure for Ito processes. We will show some numerical examples by YUIMA, a package for statistical analysis and simulation of stochastic differential equations.

[ Reference URL ]We discuss limit theorems and asymptotic expansions in estimation of the variance-covariance structure for Ito processes. We will show some numerical examples by YUIMA, a package for statistical analysis and simulation of stochastic differential equations.

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2010/01.html

### 2010/05/18

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On roots of Dehn twists (JAPANESE)

**Naoyuki Monden**(Osaka University)On roots of Dehn twists (JAPANESE)

[ Abstract ]

Let $t_{c}$ be the Dehn twist about a nonseparating simple closed curve

$c$ in a closed orientable surface. If a mapping class $f$ satisfies

$t_{c}=f^{n}$ in mapping class group, we call $f$ a root of $t_{c}$ of

degree $n$. In 2009, Margalit and Schleimer constructed roots of $t_{c}$.

In this talk, I will explain the data set which determine a root of

$t_{c}$ up to conjugacy. Moreover, I will explain the minimal and the

maximal degree.

Let $t_{c}$ be the Dehn twist about a nonseparating simple closed curve

$c$ in a closed orientable surface. If a mapping class $f$ satisfies

$t_{c}=f^{n}$ in mapping class group, we call $f$ a root of $t_{c}$ of

degree $n$. In 2009, Margalit and Schleimer constructed roots of $t_{c}$.

In this talk, I will explain the data set which determine a root of

$t_{c}$ up to conjugacy. Moreover, I will explain the minimal and the

maximal degree.

#### Lie Groups and Representation Theory

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On the eigenvalues of the Laplacian on locally symmetric hyperbolic spaces (ENGLISH)

**B. Speh**(Cornel University)On the eigenvalues of the Laplacian on locally symmetric hyperbolic spaces (ENGLISH)

[ Abstract ]

A famous Theorem of Selberg says that the non-zero eigenvalues of the Laplacian acting on functions on quotients of the upper half plane H by congruence subgroups of the integral modular group, are bounded away from zero, as the congruence subgroup varies. Analogous questions on Laplacians acting on differential forms of higher degree on locally symmetric spaces (functions may be thought of as differential forms of degree zero) have geometric implications for the cohomology of the locally symmetric space.

Let $X$ be the real hyperbolic n-space and $\\Gamma \\subset $ SO(n, 1) a congruence arithmetic subgroup. Bergeron conjectured that the eigenvalues of the Laplacian acting on the differential forms on $ X / \\Gamma $ are bounded from the below by a constant independent of the congruence subgroup. In the lecture I will show how one can use representation theory to show that this conjecture is true provided that it is true in the middle degree.

This is joint work with T.N. Venkataramana

A famous Theorem of Selberg says that the non-zero eigenvalues of the Laplacian acting on functions on quotients of the upper half plane H by congruence subgroups of the integral modular group, are bounded away from zero, as the congruence subgroup varies. Analogous questions on Laplacians acting on differential forms of higher degree on locally symmetric spaces (functions may be thought of as differential forms of degree zero) have geometric implications for the cohomology of the locally symmetric space.

Let $X$ be the real hyperbolic n-space and $\\Gamma \\subset $ SO(n, 1) a congruence arithmetic subgroup. Bergeron conjectured that the eigenvalues of the Laplacian acting on the differential forms on $ X / \\Gamma $ are bounded from the below by a constant independent of the congruence subgroup. In the lecture I will show how one can use representation theory to show that this conjecture is true provided that it is true in the middle degree.

This is joint work with T.N. Venkataramana

### 2010/05/17

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

On the GIT stability of Polarized Varieties (JAPANESE)

**Yuji Odaka**(Research Institute for Mathematical Sciences)On the GIT stability of Polarized Varieties (JAPANESE)

[ Abstract ]

Background:

Original GIT-stability notion for polarized variety is

"asymptotic stability", studied by Mumford, Gieseker etc around 1970s.

Recently a version appeared, so-called "K-stability", introduced by

Tian(1997) and reformulated by Donaldson(2002), by the way of seeking

the analogue of Kobayashi-Hitchin correspondence, which gives

"differential geometric" interpretation of "stability". These two have

subtle but interesting differences in dimension higher than 1.

Contents:

(1*) Any semistable (in any sense) polarized variety should have only

"semi-log-canonical" singularities. (Partly observed around 1970s)

(2) On the other hand, we proved some stabilities, which corresponds to

"Calabi conjecture", also with admitting mild singularities.

As applications these yield

(3*) Compact moduli spaces with GIT interpretations.

(4) Many counterexamples (as orbifolds) to folklore conjecture:

"K-stability implies asymptotic stability".

(*: Some technical points are yet to be settled.

Some parts for (1)(2) are available on arXiv:0910.1794.)

Background:

Original GIT-stability notion for polarized variety is

"asymptotic stability", studied by Mumford, Gieseker etc around 1970s.

Recently a version appeared, so-called "K-stability", introduced by

Tian(1997) and reformulated by Donaldson(2002), by the way of seeking

the analogue of Kobayashi-Hitchin correspondence, which gives

"differential geometric" interpretation of "stability". These two have

subtle but interesting differences in dimension higher than 1.

Contents:

(1*) Any semistable (in any sense) polarized variety should have only

"semi-log-canonical" singularities. (Partly observed around 1970s)

(2) On the other hand, we proved some stabilities, which corresponds to

"Calabi conjecture", also with admitting mild singularities.

As applications these yield

(3*) Compact moduli spaces with GIT interpretations.

(4) Many counterexamples (as orbifolds) to folklore conjecture:

"K-stability implies asymptotic stability".

(*: Some technical points are yet to be settled.

Some parts for (1)(2) are available on arXiv:0910.1794.)

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On the norm defined on the holomorphic maps of compact Riemann surfaces (JAPANESE)

**Masaharu TANABE**(Tokyo Inst. Tech.)On the norm defined on the holomorphic maps of compact Riemann surfaces (JAPANESE)

### 2010/05/15

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Strict positivity of the central values of some Rankin-Selberg

L-functions (JAPANESE)

Calabi-Yau manifolds associated to hypergeometric sheaves and their application

Osaka Pref. Univ. (JAPANESE)

**NARITA, Hiroaki**(Kumamoto University, Fac. of Science) 13:30-14:30Strict positivity of the central values of some Rankin-Selberg

L-functions (JAPANESE)

[ Abstract ]

We consider the Arakawa lift which is an automprphic form on an inner twist of $GSp(2)$. We construct examples the case when the central values of the $L$-functions of Rankin-Selberg type with degree 8 Euler factors take positive values. ....

We consider the Arakawa lift which is an automprphic form on an inner twist of $GSp(2)$. We construct examples the case when the central values of the $L$-functions of Rankin-Selberg type with degree 8 Euler factors take positive values. ....

**YAMAUCHI, Takuya**(Osaka Pref. Univ. ) 15:00-16:00Calabi-Yau manifolds associated to hypergeometric sheaves and their application

Osaka Pref. Univ. (JAPANESE)

[ Abstract ]

Let U be the P_1 munus 3 points, and form hypergeometric sheaves on U, by iterative convolutions of certain local sysytem of rank 1 on U. We construct certain families of Calabi-Yau manifolds whose cohomology groups of middle degree are these hypergeometric sheaves. We discuss the potential-modularity of these varieties and unit root formula. This is a joint work with Michio Tsuzuki. (trans. by the organizer of the seminar)

Let U be the P_1 munus 3 points, and form hypergeometric sheaves on U, by iterative convolutions of certain local sysytem of rank 1 on U. We construct certain families of Calabi-Yau manifolds whose cohomology groups of middle degree are these hypergeometric sheaves. We discuss the potential-modularity of these varieties and unit root formula. This is a joint work with Michio Tsuzuki. (trans. by the organizer of the seminar)

### 2010/05/12

#### Lectures

15:30-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Independence of families of $\\ell$-adic representations and uniform constructibility (ENGLISH)

**Luc Illusie**(東京大学/Paris南大学)Independence of families of $\\ell$-adic representations and uniform constructibility (ENGLISH)

[ Abstract ]

Let $k$ be a number field, $\\overline{k}$ an algebraic closure of $k$, $\\Gamma_k = \\mathrm{Gal}(\\overline{k}/k)$. A family of continuous homomorphisms $\\rho_{\\ell} : \\Gamma_k \\rightarrow G_{\\ell}$, indexed by prime numbers $\\ell$, where $G_{\\ell}$ is a locally compact $\\ell$-adic Lie group, is said to be independent if $\\rho(\\Gamma_k) = \\prod \\rho_{\\ell}(\\Gamma_k)$, where $\\rho = (\\rho_{\\ell}) : \\Gamma_k \\rightarrow \\prod G_{\\ell}$. Serre gave a criterion for such a family to become independent after a finite extension of $k$. We will explain Serre's criterion and show that it applies to families coming from the $\\ell$-adic cohomology (or cohomology with compact support) of schemes separated and of finite type over $k$. This application uses a variant of Deligne's generic constructibility theorem with uniformity in $\\ell$.

Let $k$ be a number field, $\\overline{k}$ an algebraic closure of $k$, $\\Gamma_k = \\mathrm{Gal}(\\overline{k}/k)$. A family of continuous homomorphisms $\\rho_{\\ell} : \\Gamma_k \\rightarrow G_{\\ell}$, indexed by prime numbers $\\ell$, where $G_{\\ell}$ is a locally compact $\\ell$-adic Lie group, is said to be independent if $\\rho(\\Gamma_k) = \\prod \\rho_{\\ell}(\\Gamma_k)$, where $\\rho = (\\rho_{\\ell}) : \\Gamma_k \\rightarrow \\prod G_{\\ell}$. Serre gave a criterion for such a family to become independent after a finite extension of $k$. We will explain Serre's criterion and show that it applies to families coming from the $\\ell$-adic cohomology (or cohomology with compact support) of schemes separated and of finite type over $k$. This application uses a variant of Deligne's generic constructibility theorem with uniformity in $\\ell$.

#### Number Theory Seminar

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.)

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