## Seminar information archive

Seminar information archive ～12/11｜Today's seminar 12/12 | Future seminars 12/13～

### 2014/12/09

#### Tuesday Seminar on Topology

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

Stable commutator length on mapping class groups (JAPANESE)

**Koji Fujiwara**(Kyoto University)Stable commutator length on mapping class groups (JAPANESE)

[ Abstract ]

Let MCG(S) be the mapping class group of a closed orientable surface S.

We give a precise condition (in terms of the Nielsen-Thurston

decomposition) when an element

in MCG(S) has positive stable commutator length.

Stable commutator length tends to be positive if there is "negative

curvature".

The proofs use our earlier construction in the paper "Constructing group

actions on quasi-trees and applications to mapping class groups" of

group actions on quasi-trees.

This is a joint work with Bestvina and Bromberg.

Let MCG(S) be the mapping class group of a closed orientable surface S.

We give a precise condition (in terms of the Nielsen-Thurston

decomposition) when an element

in MCG(S) has positive stable commutator length.

Stable commutator length tends to be positive if there is "negative

curvature".

The proofs use our earlier construction in the paper "Constructing group

actions on quasi-trees and applications to mapping class groups" of

group actions on quasi-trees.

This is a joint work with Bestvina and Bromberg.

### 2014/12/08

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On maximal ideal cycle and fundamental cycle of normal two-dimensional quasi-homogeneous singularities, and singularities with star-shaped resolution (JAPANESE)

**Masatake Tomari**(Nihon University)On maximal ideal cycle and fundamental cycle of normal two-dimensional quasi-homogeneous singularities, and singularities with star-shaped resolution (JAPANESE)

### 2014/12/04

#### Geometry Colloquium

17:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Deformations and the moduli spaces of generalized complex manifolds (JAPANESE)

**Ryushi Goto**(Osaka University)Deformations and the moduli spaces of generalized complex manifolds (JAPANESE)

### 2014/12/03

#### Operator Algebra Seminars

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

Central property (T) for $SU_q(2n+1)$

(English)

**Yuki Arano**(Univ. Tokyo)Central property (T) for $SU_q(2n+1)$

(English)

#### Lectures

16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

New isoperimetric inequalities with densities, part II: Detailed proofs and related works (ENGLISH)

**Xavier Cabre**(ICREA and UPC, Barcelona)New isoperimetric inequalities with densities, part II: Detailed proofs and related works (ENGLISH)

[ Abstract ]

This is a sequel to the Tuesday Analysis Seminar on December 2 by the same speaker.

In joint works with X. Ros-Oton and J. Serra, the study of the regularity of stable solutions to reaction-diffusion problems has led us to certain Sobolev and isoperimetric inequalities with weights. We will present our results in these new isoperimetric inequalities with the best constant, that we establish via the ABP method.

More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (or densities) in open convex cones of R^n. Our results apply to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Surprisingly, even that our weights are not radially symmetric, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient. As a particular case of our results, we provide with new proofs of classical results such as the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella. Furthermore, we also study the anisotropic isoperimetric problem for the same class of weights and we prove that the Wulff shape always minimizes the anisotropic weighted perimeter under the weighted volume constraint.

This is a sequel to the Tuesday Analysis Seminar on December 2 by the same speaker.

In joint works with X. Ros-Oton and J. Serra, the study of the regularity of stable solutions to reaction-diffusion problems has led us to certain Sobolev and isoperimetric inequalities with weights. We will present our results in these new isoperimetric inequalities with the best constant, that we establish via the ABP method.

More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (or densities) in open convex cones of R^n. Our results apply to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Surprisingly, even that our weights are not radially symmetric, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient. As a particular case of our results, we provide with new proofs of classical results such as the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella. Furthermore, we also study the anisotropic isoperimetric problem for the same class of weights and we prove that the Wulff shape always minimizes the anisotropic weighted perimeter under the weighted volume constraint.

#### Mathematical Biology Seminar

14:50-16:20 Room #122 (Graduate School of Math. Sci. Bldg.)

Global analysis of age-structured SIS epidemic models with spatial

heterogeneity (JAPANESE)

**Toshikazu Kuniya**(Graduate School of System Informatics, Kobe University)Global analysis of age-structured SIS epidemic models with spatial

heterogeneity (JAPANESE)

### 2014/12/02

#### Tuesday Seminar on Topology

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

The Atiyah-Segal completion theorem in noncommutative topology (JAPANESE)

**Yosuke Kubota**(The University of Tokyo)The Atiyah-Segal completion theorem in noncommutative topology (JAPANESE)

[ Abstract ]

We introduce a new perspevtive on the Atiyah-Segal completion

theorem applying the "noncommutative" topology, which deals with

topological properties of C*-algebras. The homological algebra of the

Kasparov category as a triangulated category, which is developed by R.

Meyer and R. Nest, plays a central role. It contains the Atiyah-Segal

type completion theorems for equivariant K-homology and twisted K-theory.

This is a joint work with Yuki Arano.

We introduce a new perspevtive on the Atiyah-Segal completion

theorem applying the "noncommutative" topology, which deals with

topological properties of C*-algebras. The homological algebra of the

Kasparov category as a triangulated category, which is developed by R.

Meyer and R. Nest, plays a central role. It contains the Atiyah-Segal

type completion theorems for equivariant K-homology and twisted K-theory.

This is a joint work with Yuki Arano.

#### PDE Real Analysis Seminar

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

Remark on single exponential bound of the vorticity gradient for the two-dimensional Euler flow around a corner (JAPANESE)

**Tsubasa Itoh**(Tokyo Institute of Technology)Remark on single exponential bound of the vorticity gradient for the two-dimensional Euler flow around a corner (JAPANESE)

[ Abstract ]

In this talk, the two dimensional Euler flow under a simple symmetry condition with hyperbolic structure in a unit square $D=\{(x_{1}, x_{2}): 0 < x_{1} + x_{2} < \sqrt{2},\ 0<-x_{1} + x_{2} < \sqrt{2}\}$ is considered.

It is shown that the Lipschitz estimate of the vorticity on the boundary is at most single exponential growth near the stagnation point.

(Joint work with Tsuyoshi Yoneda and Hideyuki Miura.)

In this talk, the two dimensional Euler flow under a simple symmetry condition with hyperbolic structure in a unit square $D=\{(x_{1}, x_{2}): 0 < x_{1} + x_{2} < \sqrt{2},\ 0<-x_{1} + x_{2} < \sqrt{2}\}$ is considered.

It is shown that the Lipschitz estimate of the vorticity on the boundary is at most single exponential growth near the stagnation point.

(Joint work with Tsuyoshi Yoneda and Hideyuki Miura.)

#### Tuesday Seminar of Analysis

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

New isoperimetric inequalities with densities arising in reaction-diffusion problems (English)

**Xavier Cabre**(ICREA and UPC, Barcelona)New isoperimetric inequalities with densities arising in reaction-diffusion problems (English)

[ Abstract ]

In joint works with X. Ros-Oton and J. Serra, the study of the

regularity of stable solutions to reaction-diffusion problems

has led us to certain Sobolev and isoperimetric inequalities

with weights. We will present our results in these new

isoperimetric inequalities with the best constant, that we

establish via the ABP method. More precisely, we obtain

a new family of sharp isoperimetric inequalities with weights

(or densities) in open convex cones of R^n. Our results apply

to all nonnegative homogeneous weights satisfying a concavity

condition in the cone. Surprisingly, even that our weights are

not radially symmetric, Euclidean balls centered at the origin

(intersected with the cone) minimize the weighted isoperimetric

quotient. As a particular case of our results, we provide with

new proofs of classical results such as the Wulff inequality and

the isoperimetric inequality in convex cones of Lions and Pacella.

Furthermore, we also study the anisotropic isoperimetric problem

for the same class of weights and we prove that the Wulff shape

always minimizes the anisotropic weighted perimeter under the

weighted volume constraint.

In joint works with X. Ros-Oton and J. Serra, the study of the

regularity of stable solutions to reaction-diffusion problems

has led us to certain Sobolev and isoperimetric inequalities

with weights. We will present our results in these new

isoperimetric inequalities with the best constant, that we

establish via the ABP method. More precisely, we obtain

a new family of sharp isoperimetric inequalities with weights

(or densities) in open convex cones of R^n. Our results apply

to all nonnegative homogeneous weights satisfying a concavity

condition in the cone. Surprisingly, even that our weights are

not radially symmetric, Euclidean balls centered at the origin

(intersected with the cone) minimize the weighted isoperimetric

quotient. As a particular case of our results, we provide with

new proofs of classical results such as the Wulff inequality and

the isoperimetric inequality in convex cones of Lions and Pacella.

Furthermore, we also study the anisotropic isoperimetric problem

for the same class of weights and we prove that the Wulff shape

always minimizes the anisotropic weighted perimeter under the

weighted volume constraint.

### 2014/12/01

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Effective and noneffective extension theorems (Japanese)

**Takeo Ohsawa**(Nagoya University)Effective and noneffective extension theorems (Japanese)

[ Abstract ]

As an effective extension theorem, I will review the sharp $L^2$ extension theorem explaining the ideas of its proofs due to Blocki and Guan-Zhou. A new proof using the Poincare metric with be given, too. As a noneffective extension theorem, I will talk about an extension theorem from semipositive divisors. It is obtained as an application of an isomorphism theorem which is essentially contained in my master thesis.

As an effective extension theorem, I will review the sharp $L^2$ extension theorem explaining the ideas of its proofs due to Blocki and Guan-Zhou. A new proof using the Poincare metric with be given, too. As a noneffective extension theorem, I will talk about an extension theorem from semipositive divisors. It is obtained as an application of an isomorphism theorem which is essentially contained in my master thesis.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Induced Automorphisms on Hyperkaehler Manifolds (ENGLISH)

**Malte Wandel**(RIMS)Induced Automorphisms on Hyperkaehler Manifolds (ENGLISH)

[ Abstract ]

in this talk I want to report on a joint project with Giovanni Mongardi (Milano). We study automorphisms of hyperkaehler manifolds. All known deformation classes of these manifolds contain moduli spaces of stable sheaves on surfaces. If the underlying surface admits a non-trivial automorphism, it is often possible to transfer this automorphism to a moduli space of sheaves. In this way we obtain a big class of interesting examples of automorphisms of hyperkaehler manifolds. I will present a criterion to 'detect' automorphisms in this class and discuss several applications for the classification of automorphisms of manifolds of K3^[n]- and kummer n-type. If time permits I will try to talk about generalisations to O'Grady's sporadic examples.

in this talk I want to report on a joint project with Giovanni Mongardi (Milano). We study automorphisms of hyperkaehler manifolds. All known deformation classes of these manifolds contain moduli spaces of stable sheaves on surfaces. If the underlying surface admits a non-trivial automorphism, it is often possible to transfer this automorphism to a moduli space of sheaves. In this way we obtain a big class of interesting examples of automorphisms of hyperkaehler manifolds. I will present a criterion to 'detect' automorphisms in this class and discuss several applications for the classification of automorphisms of manifolds of K3^[n]- and kummer n-type. If time permits I will try to talk about generalisations to O'Grady's sporadic examples.

#### Numerical Analysis Seminar

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

An error estimate of a generalized particle method for Poisson equations

(日本語)

[ Reference URL ]

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

**Yusuke Imoto**(Kyushu University)An error estimate of a generalized particle method for Poisson equations

(日本語)

[ Reference URL ]

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

### 2014/11/28

#### Colloquium

16:30-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Estimating the reproduction numbers of emerging infectious diseases: Case studies of Ebola and dengue

(JAPANESE)

http://www.ghp.m.u-tokyo.ac.jp/profile/staff/hnishiura/

**Hiroshi Nishiura**(Graduate School of Medicine, The University of Tokyo)Estimating the reproduction numbers of emerging infectious diseases: Case studies of Ebola and dengue

(JAPANESE)

[ Abstract ]

The basic and effective reproduction numbers offer epidemiological

insights into the growth of generations of infectious disease cases,

informing the required control effort. Recently, the renewal process

model has appeared to be a usefu tool for quantifying the reproduction

numbers in real-time using only case data. Here I present methods,

results and pitfalls of the use of renewal process model, presenting

recent case studies of Ebola virus disease epidemic in West Africa and a

massive epidemic of dengue fever in the summer of Japan 2014.

[ Reference URL ]The basic and effective reproduction numbers offer epidemiological

insights into the growth of generations of infectious disease cases,

informing the required control effort. Recently, the renewal process

model has appeared to be a usefu tool for quantifying the reproduction

numbers in real-time using only case data. Here I present methods,

results and pitfalls of the use of renewal process model, presenting

recent case studies of Ebola virus disease epidemic in West Africa and a

massive epidemic of dengue fever in the summer of Japan 2014.

http://www.ghp.m.u-tokyo.ac.jp/profile/staff/hnishiura/

### 2014/11/26

#### Operator Algebra Seminars

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

A program to construct and study conformal field theories (ENGLISH)

**Yi-Zhi Huang**(Rutgers Univ.)A program to construct and study conformal field theories (ENGLISH)

#### Lectures

16:00-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Intertwinings, wave equations and beta ensembles (ENGLISH)

**Mykhaylo Shkolnikov**(Princeton University)Intertwinings, wave equations and beta ensembles (ENGLISH)

[ Abstract ]

We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to wave equations and more general hyperbolic partial differential equations. The talk will be devoted to this recent development, as well as an algebraic perspective on intertwinings which, in particular, gives rise to a novel intertwining in beta random matrix theory. Based on joint works with Vadim Gorin and Soumik Pal.

We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to wave equations and more general hyperbolic partial differential equations. The talk will be devoted to this recent development, as well as an algebraic perspective on intertwinings which, in particular, gives rise to a novel intertwining in beta random matrix theory. Based on joint works with Vadim Gorin and Soumik Pal.

#### Seminar on Probability and Statistics

16:30-17:40 Room #052 (Graduate School of Math. Sci. Bldg.)

Sparse and robust linear regression: Iterative algorithm and its statistical convergence

**KATAYAMA, Shota**(Tokyo Institute of Technology)Sparse and robust linear regression: Iterative algorithm and its statistical convergence

### 2014/11/25

#### Tuesday Seminar of Analysis

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

Stationary scattering theory on manifold with ends (JAPANESE)

**Kenichi Ito**(Department of Mathematics, Graduate School of Science, Kobe University)Stationary scattering theory on manifold with ends (JAPANESE)

#### Tuesday Seminar on Topology

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

Quandle knot invariants and applications (JAPANESE)

**Masahico Saito**(University of South Florida)Quandle knot invariants and applications (JAPANESE)

[ Abstract ]

A quandles is an algebraic structure closely related to knots. Homology theories of

quandles have been defined, and their cocycles are used to construct invariants for

classical knots, spatial graphs and knotted surfaces. In this talk, an overview is given

for quandle cocycle invariants and their applications to geometric properties of knots.

The current status of computations, recent developments and open problems will also

be discussed.

A quandles is an algebraic structure closely related to knots. Homology theories of

quandles have been defined, and their cocycles are used to construct invariants for

classical knots, spatial graphs and knotted surfaces. In this talk, an overview is given

for quandle cocycle invariants and their applications to geometric properties of knots.

The current status of computations, recent developments and open problems will also

be discussed.

#### Kavli IPMU Komaba Seminar

10:30-11:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Donaldson-Thomas theory for Calabi-Yau fourfolds.

(ENGLISH)

**Naichung Conan Leung**(The Chinese University of Hong Kong)Donaldson-Thomas theory for Calabi-Yau fourfolds.

(ENGLISH)

[ Abstract ]

Donaldson-Thomas theory for Calabi-Yau threefolds is a

complexification of Chern-Simons theory. In this talk I will discuss

my joint work with Cao on the complexification of Donaldson theory.

This work is supported by a RGC grant of HK Government.

Donaldson-Thomas theory for Calabi-Yau threefolds is a

complexification of Chern-Simons theory. In this talk I will discuss

my joint work with Cao on the complexification of Donaldson theory.

This work is supported by a RGC grant of HK Government.

### 2014/11/22

#### Harmonic Analysis Komaba Seminar

13:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On the Inclusion of Generalized Morrey Spaces and the Boundedness of the Generalized Fractional Maximal Operators (ENGLISH)

Hardy-type inequality for 0 < p < 1 and hypodecreasing functions (ENGLISH)

Sharp spectral stability estimate for uniformly elliptic differential operators (EMGLISH)

**Denny Hakim**(Tokyo Metropolitan University) 13:30-14:30On the Inclusion of Generalized Morrey Spaces and the Boundedness of the Generalized Fractional Maximal Operators (ENGLISH)

[ Abstract ]

In this talk, we shall prove a necessary and sufficient condition for an inclusion property of generalized Morrey spaces. We use this property in our proof of the boundedness of the generalized fractional maximal operators on these spaces. Our result also cover the generalized weak Morrey spaces.

This research is a joint work with Y. Sawano, H. Gunawan, K.M. Limanta and A.A. Masta.

In this talk, we shall prove a necessary and sufficient condition for an inclusion property of generalized Morrey spaces. We use this property in our proof of the boundedness of the generalized fractional maximal operators on these spaces. Our result also cover the generalized weak Morrey spaces.

This research is a joint work with Y. Sawano, H. Gunawan, K.M. Limanta and A.A. Masta.

**Tamara Tararykova**(Cardiff University / Eurasian National University) 14:45-15:45Hardy-type inequality for 0 < p < 1 and hypodecreasing functions (ENGLISH)

[ Abstract ]

T.B.A.

T.B.A.

**Victor Burenkov**(Cardift School of Mathematics / Peoples' Friendship University of Russia / Steklov Institute of Mathematics) 16:00-17:00Sharp spectral stability estimate for uniformly elliptic differential operators (EMGLISH)

[ Abstract ]

T.B.A.

T.B.A.

### 2014/11/21

#### Operator Algebra Seminars

15:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Introduction to Haagerup's bicentralizer paper (English)

**Reiji Tomatsu**(Hokkaido University)Introduction to Haagerup's bicentralizer paper (English)

### 2014/11/20

#### Operator Algebra Seminars

13:00-15:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Introduction to Haagerup's bicentralizer paper (English)

**Reiji Tomatsu**(Hokkaido University)Introduction to Haagerup's bicentralizer paper (English)

#### Infinite Analysis Seminar Tokyo

15:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

An explicit formula for the specialization of nonsymmetric

Macdonald polynomials at $t = \infty$ (JAPANESE)

divisor function and strict partition (JAPANESE)

**Fumihiko Nomoto**(Department of Mathematics, Tokyo Institute of Technology, Graduate school of science and Engineering) 15:00-16:30An explicit formula for the specialization of nonsymmetric

Macdonald polynomials at $t = \infty$ (JAPANESE)

[ Abstract ]

Orr-Shimozono obtained an explicit formula for nonsymmetric Macdonald polynomials with Hecke parameter $t$ set to $\infty$, which is described in terms of an affine root system

and an affine Weyl group. On the basis of this work, we give another explicit formula for the specialization above, which is described in terms of the quantum Bruhat graph associated with a finite root system and a finite Weyl group.

More precisely, we interpret the specialization above as the graded character of an explicitly specified set of quantum Lakshmibai-Seshadri (LS) paths. Here we note that the set of quantum LS paths (of a given shape) provides an explicit realization of the crystal basis of a quantum Weyl module over the quantum affine algebra.

In this talk, I will explain our explicit formula

by exhibiting a few examples.

Also, I will give an outline of the proof.

Orr-Shimozono obtained an explicit formula for nonsymmetric Macdonald polynomials with Hecke parameter $t$ set to $\infty$, which is described in terms of an affine root system

and an affine Weyl group. On the basis of this work, we give another explicit formula for the specialization above, which is described in terms of the quantum Bruhat graph associated with a finite root system and a finite Weyl group.

More precisely, we interpret the specialization above as the graded character of an explicitly specified set of quantum Lakshmibai-Seshadri (LS) paths. Here we note that the set of quantum LS paths (of a given shape) provides an explicit realization of the crystal basis of a quantum Weyl module over the quantum affine algebra.

In this talk, I will explain our explicit formula

by exhibiting a few examples.

Also, I will give an outline of the proof.

**Masanori Ando**(Wakkanai Hokusei Gakuen University) 17:00-18:30divisor function and strict partition (JAPANESE)

[ Abstract ]

We know that the q-series identity of Uchimura-type is related with the divisor function.

It is obtained also as a specialization of basic hypergeometric series.

In this seminar, we interprete this identity from the point of view of combinatorics of partitions of integers.

We give its proof by using the mock involution map.

We know that the q-series identity of Uchimura-type is related with the divisor function.

It is obtained also as a specialization of basic hypergeometric series.

In this seminar, we interprete this identity from the point of view of combinatorics of partitions of integers.

We give its proof by using the mock involution map.

### 2014/11/19

#### Operator Algebra Seminars

13:00-15:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Introduction to Haagerup's bicentralizer paper (English)

**Reiji Tomatsu**(Hokkaido University)Introduction to Haagerup's bicentralizer paper (English)

#### Number Theory Seminar

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

Bad reduction of curves with CM jacobians (English)

**Fabien Pazuki**(Univ Bordeaux and Univ Copenhagen)Bad reduction of curves with CM jacobians (English)

[ Abstract ]

An abelian variety defined over a number field and having complex multiplication (CM) has potentially good reduction everywhere. If a curve of positive genus which is defined over a number field has good reduction at a given finite place, then so does its jacobian variety. However, the converse statement is false already in the genus 2 case, as can be seen in the entry $[I_0-I_0-m]$ in Namikawa and Ueno's classification table of fibres in pencils of curves of genus 2. In this joint work with Philipp Habegger, our main result states that this phenomenon prevails for certain families of curves.

We prove the following result: Let F be a real quadratic number field. Up to isomorphisms there are only finitely many curves C of genus 2 defined over $\overline{\mathbb{Q}}$ with good reduction everywhere and such that the jacobian Jac(C) has CM by the maximal order of a quartic, cyclic, totally imaginary number field containing F. Hence such a curve will almost always have stable bad reduction at some prime whereas its jacobian has good reduction everywhere. A remark is that one can exhibit an infinite family of genus 2 curves with CM jacobian such that the endomorphism ring is the ring of algebraic integers in a cyclic extension of $\mathbb{Q}$ of degree 4 that contains $\mathbb{Q}(\sqrt{5})$, for example.

An abelian variety defined over a number field and having complex multiplication (CM) has potentially good reduction everywhere. If a curve of positive genus which is defined over a number field has good reduction at a given finite place, then so does its jacobian variety. However, the converse statement is false already in the genus 2 case, as can be seen in the entry $[I_0-I_0-m]$ in Namikawa and Ueno's classification table of fibres in pencils of curves of genus 2. In this joint work with Philipp Habegger, our main result states that this phenomenon prevails for certain families of curves.

We prove the following result: Let F be a real quadratic number field. Up to isomorphisms there are only finitely many curves C of genus 2 defined over $\overline{\mathbb{Q}}$ with good reduction everywhere and such that the jacobian Jac(C) has CM by the maximal order of a quartic, cyclic, totally imaginary number field containing F. Hence such a curve will almost always have stable bad reduction at some prime whereas its jacobian has good reduction everywhere. A remark is that one can exhibit an infinite family of genus 2 curves with CM jacobian such that the endomorphism ring is the ring of algebraic integers in a cyclic extension of $\mathbb{Q}$ of degree 4 that contains $\mathbb{Q}(\sqrt{5})$, for example.

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