## 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.) |
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Organizer(s) | Naoki Imai, Yoichi Mieda |

**Seminar information archive**

### 2014/12/17

18:00-19:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Equivariant $\wideparen{\mathcal{D}}$ modules on rigid analytic spaces

(English)

**Konstantin Ardakov**(University of Oxford)Equivariant $\wideparen{\mathcal{D}}$ modules on rigid analytic spaces

(English)

[ Abstract ]

Locally analytic representations of p-adic Lie groups are of interest in several branches of arithmetic algebraic geometry, notably the p-adic local Langlands program. I will discuss some work in progress towards a Beilinson-Bernstein style localisation theorem for admissible locally analytic representations of semisimple compact p-adic Lie groups using equivariant formal models of rigid analytic flag varieties.

Locally analytic representations of p-adic Lie groups are of interest in several branches of arithmetic algebraic geometry, notably the p-adic local Langlands program. I will discuss some work in progress towards a Beilinson-Bernstein style localisation theorem for admissible locally analytic representations of semisimple compact p-adic Lie groups using equivariant formal models of rigid analytic flag varieties.

### 2014/11/19

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.

### 2014/11/12

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

Relative (φ, Γ)-modules (English)

**Ruochuan Liu**(BICMR)Relative (φ, Γ)-modules (English)

[ Abstract ]

In this talk, we will introduce the theory of (φ, Γ)-modules for general adic spaces. This is a joint work with Kedlaya.

In this talk, we will introduce the theory of (φ, Γ)-modules for general adic spaces. This is a joint work with Kedlaya.

### 2014/10/28

16:40-18:50 Room #002 (Graduate School of Math. Sci. Bldg.)

A p-adic Labesse-Langlands transfer (English)

Plectic cohomology (English)

**Judith Ludwig**(Imperial college) 16:40-17:40A p-adic Labesse-Langlands transfer (English)

[ Abstract ]

Let B be a definite quaternion algebra over the rationals, G the algebraic group defined by the units in B and H the subgroup of G of norm one elements. Then the classical transfer of automorphic representations from G to H is well understood thanks to Labesse and Langlands, who proved formulas for the multiplicity of irreducible admissible representations of H(adeles) in the discrete automorphic spectrum.

The goal of this talk is to prove a p-adic version of this transfer. By this we mean an extension of the classical transfer to p-adic families of automorphic forms as parametrized by certain rigid analytic spaces called eigenvarieties. We will prove the p-adic transfer by constructing a morphism between eigenvarieties, which agrees with the classical transfer on points corresponding to classical automorphic representations.

Let B be a definite quaternion algebra over the rationals, G the algebraic group defined by the units in B and H the subgroup of G of norm one elements. Then the classical transfer of automorphic representations from G to H is well understood thanks to Labesse and Langlands, who proved formulas for the multiplicity of irreducible admissible representations of H(adeles) in the discrete automorphic spectrum.

The goal of this talk is to prove a p-adic version of this transfer. By this we mean an extension of the classical transfer to p-adic families of automorphic forms as parametrized by certain rigid analytic spaces called eigenvarieties. We will prove the p-adic transfer by constructing a morphism between eigenvarieties, which agrees with the classical transfer on points corresponding to classical automorphic representations.

**Jan Nekovar**(Université Paris 6) 17:50-18:50Plectic cohomology (English)

### 2014/10/14

17:30-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

A p-adic criterion for good reduction of curves (ENGLISH)

**Fabrizio Andreatta**(Università Statale di Milano)A p-adic criterion for good reduction of curves (ENGLISH)

[ Abstract ]

Given a curve over a dvr of mixed characteristic 0-p with smooth generic fiber and with semistable reduction, I will present a criterion for good reduction in terms of the (unipotent) p-adic étale fundamental group of its generic fiber.

Given a curve over a dvr of mixed characteristic 0-p with smooth generic fiber and with semistable reduction, I will present a criterion for good reduction in terms of the (unipotent) p-adic étale fundamental group of its generic fiber.

### 2014/06/25

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

Periods of some two dimensional reducible p-adic representations and non-de Rham B-pairs (JAPANESE)

**Masahiko Takiguchi**(University of Tokyo)Periods of some two dimensional reducible p-adic representations and non-de Rham B-pairs (JAPANESE)

### 2014/06/17

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

Vinberg's monoid and automorphic L-functions (ENGLISH)

**Bao Châu Ngô**(University of Chicago, VIASM)Vinberg's monoid and automorphic L-functions (ENGLISH)

[ Abstract ]

We will explain a generalisation of the construction of the local factors of Godement-Jacquet's L-functions, based on Vinberg's monoid.

We will explain a generalisation of the construction of the local factors of Godement-Jacquet's L-functions, based on Vinberg's monoid.

### 2014/05/28

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

On simultaneous approximation to powers of a real number by rational numbers (ENGLISH)

**Gantsooj Batzaya**(University of Tokyo)On simultaneous approximation to powers of a real number by rational numbers (ENGLISH)

### 2014/05/21

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

Parity of Betti numbers in étale cohomology (ENGLISH)

**Shenghao Sun**(Mathematical Sciences Center of Tsinghua University)Parity of Betti numbers in étale cohomology (ENGLISH)

[ Abstract ]

By Hodge symmetry, the Betti numbers of a complex projective smooth variety in odd degrees are even. When the base field has characteristic p, Deligne proved the hard Lefschetz theorem in etale cohomology, and the parity result follows from this. Suh has generalized this to proper smooth varieties in characteristic p, using crystalline cohomology.

The purity of intersection cohomology group of proper varieties suggests that the same parity property should hold for these groups in characteristic p. We proved this by investigating the symmetry in the categorical level.

In particular, we reproved Suh's result, using merely etale cohomology. Some related results will be discussed. This is joint work with Weizhe Zheng.

By Hodge symmetry, the Betti numbers of a complex projective smooth variety in odd degrees are even. When the base field has characteristic p, Deligne proved the hard Lefschetz theorem in etale cohomology, and the parity result follows from this. Suh has generalized this to proper smooth varieties in characteristic p, using crystalline cohomology.

The purity of intersection cohomology group of proper varieties suggests that the same parity property should hold for these groups in characteristic p. We proved this by investigating the symmetry in the categorical level.

In particular, we reproved Suh's result, using merely etale cohomology. Some related results will be discussed. This is joint work with Weizhe Zheng.

### 2014/04/30

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

An effective upper bound for the number of principally polarized Abelian schemes (JAPANESE)

**Takuya Maruyama**(University of Tokyo)An effective upper bound for the number of principally polarized Abelian schemes (JAPANESE)

### 2014/04/23

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

Non-tempered A-packets and the Rapoport-Zink spaces (JAPANESE)

**Yoichi Mieda**(University of Tokyo)Non-tempered A-packets and the Rapoport-Zink spaces (JAPANESE)

### 2014/04/16

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

On the cycle class map for zero-cycles over local fields (ENGLISH)

**Olivier Wittenberg**(ENS and CNRS)On the cycle class map for zero-cycles over local fields (ENGLISH)

[ Abstract ]

The Chow group of zero-cycles of a smooth and projective variety defined over a field k is an invariant of an arithmetic and geometric nature which is well understood only when k is a finite field (by higher-dimensional class field theory). In this talk, we will discuss the case of local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus over p-adic fields. The same statement holds for semistable K3 surfaces over C((t)), but does not hold in general for surfaces over C((t)) or over the maximal unramified extension of a p-adic field. This is a joint work with Hélène Esnault.

The Chow group of zero-cycles of a smooth and projective variety defined over a field k is an invariant of an arithmetic and geometric nature which is well understood only when k is a finite field (by higher-dimensional class field theory). In this talk, we will discuss the case of local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus over p-adic fields. The same statement holds for semistable K3 surfaces over C((t)), but does not hold in general for surfaces over C((t)) or over the maximal unramified extension of a p-adic field. This is a joint work with Hélène Esnault.

### 2014/02/05

17:10-18:10 Room #002 (Graduate School of Math. Sci. Bldg.)

The Franke filtration of spaces of automorphic forms (ENGLISH)

**Neven Grbac**(University of Rijeka)The Franke filtration of spaces of automorphic forms (ENGLISH)

[ Abstract ]

The Franke filtration is a filtration of the space of all adelic automorphic forms on a reductive group defined over a number field. The filtration steps can be described as certain induced representations, which has applications to the study of Eisenstein cohomology. In this talk, we shall describe the Franke filtration in general, give several examples, and explain its connection to cohomology.

The Franke filtration is a filtration of the space of all adelic automorphic forms on a reductive group defined over a number field. The filtration steps can be described as certain induced representations, which has applications to the study of Eisenstein cohomology. In this talk, we shall describe the Franke filtration in general, give several examples, and explain its connection to cohomology.

### 2014/01/24

16:40-18:50 Room #056 (Graduate School of Math. Sci. Bldg.)

An approach to p-adic Hodge theory over number fields (ENGLISH)

Canonical lifts of norm fields and applications (ENGLISH)

**Christopher Davis**(University of Copenhagen) 16:40-17:40An approach to p-adic Hodge theory over number fields (ENGLISH)

[ Abstract ]

As motivation from classical Hodge theory, we will first compare singular cohomology and (algebraic) de Rham cohomology for a complex analytic variety. We will also describe a sense in which this comparison does not have a natural analogue over the real numbers. We think of the complex numbers as a "big" ring which is necessary for the comparison isomorphism to work. In the p-adic setting, the analogous study is known as p-adic Hodge theory, and the "big" rings there are even bigger. There are many approaches to p-adic Hodge theory, and we will introduce one tool in particular: (phi, Gamma)-modules. The goal of this talk is to describe a preliminary attempt to find an analogue of this theory (and analogues of its "big" rings) which makes sense over number fields (rather than p-adic fields). This is joint work with Kiran Kedlaya.

As motivation from classical Hodge theory, we will first compare singular cohomology and (algebraic) de Rham cohomology for a complex analytic variety. We will also describe a sense in which this comparison does not have a natural analogue over the real numbers. We think of the complex numbers as a "big" ring which is necessary for the comparison isomorphism to work. In the p-adic setting, the analogous study is known as p-adic Hodge theory, and the "big" rings there are even bigger. There are many approaches to p-adic Hodge theory, and we will introduce one tool in particular: (phi, Gamma)-modules. The goal of this talk is to describe a preliminary attempt to find an analogue of this theory (and analogues of its "big" rings) which makes sense over number fields (rather than p-adic fields). This is joint work with Kiran Kedlaya.

**Bryden Cais**(University of Arizona) 17:50-18:50Canonical lifts of norm fields and applications (ENGLISH)

[ Abstract ]

In this talk, we begin by outlining the Fontaine-Wintenberger theory of norm fields and explain its application to the classification of p-adic Galois representations on F_p-vector spaces. In order to lift this to a classification of p-adic representations on Z_p-modules, it is necessary to lift the characteristic p norm field constructions of Fontaine-Wintenberger to characteristic zero. We will explain how to canonically perform such lifting in many interesting cases, as well as applications to generalizing a theorem of Kisin on the restriction of crystalline p-adic Galois representations. This is joint work with Christopher Davis.

In this talk, we begin by outlining the Fontaine-Wintenberger theory of norm fields and explain its application to the classification of p-adic Galois representations on F_p-vector spaces. In order to lift this to a classification of p-adic representations on Z_p-modules, it is necessary to lift the characteristic p norm field constructions of Fontaine-Wintenberger to characteristic zero. We will explain how to canonically perform such lifting in many interesting cases, as well as applications to generalizing a theorem of Kisin on the restriction of crystalline p-adic Galois representations. This is joint work with Christopher Davis.

### 2014/01/22

18:00-19:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Riemann-Hilbert correspondence for irregular holonomic D-modules (ENGLISH)

**Masaki Kashiwara**(RIMS, Kyoto University)Riemann-Hilbert correspondence for irregular holonomic D-modules (ENGLISH)

[ Abstract ]

The classical Riemann-Hilbert correspondence establishes an equivalence between the derived category of regular holonomic D-modules and the derived category of constructible sheaves. Recently, I, with Andrea D'Agnolo, proved a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular (arXiv:1311.2374). In this correspondence, we have to replace the derived category of constructible sheaves with a full subcategory of ind-sheaves on the product of the base space and the real projective line. The construction is therefore based on the theory of ind-sheaves by Kashiwara-Schapira, and also it is influenced by Tamarkin's work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Takuro Mochizuki and Kiran Kedlaya.

The classical Riemann-Hilbert correspondence establishes an equivalence between the derived category of regular holonomic D-modules and the derived category of constructible sheaves. Recently, I, with Andrea D'Agnolo, proved a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular (arXiv:1311.2374). In this correspondence, we have to replace the derived category of constructible sheaves with a full subcategory of ind-sheaves on the product of the base space and the real projective line. The construction is therefore based on the theory of ind-sheaves by Kashiwara-Schapira, and also it is influenced by Tamarkin's work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Takuro Mochizuki and Kiran Kedlaya.

### 2014/01/15

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

Special values of zeta-functions of schemes (ENGLISH)

**Stephen Lichtenbaum**(Brown University)Special values of zeta-functions of schemes (ENGLISH)

[ Abstract ]

We will give conjectured formulas giving the behavior of the

seta-function of regular schemes projective and flat over Spec Z at

non-positive integers in terms of Weil-etale cohomology. We will also

explain the conjectured relationship of Weil-etale cohomology to etale

cohomology, which makes it possible to express these formulas also in terms

of etale cohomology.

We will give conjectured formulas giving the behavior of the

seta-function of regular schemes projective and flat over Spec Z at

non-positive integers in terms of Weil-etale cohomology. We will also

explain the conjectured relationship of Weil-etale cohomology to etale

cohomology, which makes it possible to express these formulas also in terms

of etale cohomology.

### 2014/01/08

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

Roots of the discriminant of an elliptic curves and its torsion points (JAPANESE)

**Sho Yoshikawa**(University of Tokyo)Roots of the discriminant of an elliptic curves and its torsion points (JAPANESE)

[ Abstract ]

We give an explicit and intrinsic description of (the torsor defined by the 12th roots of) the discriminant of an elliptic curve using the group of its 12-torsion points and the Weil pairing. As an application, we extend a result of Coates (which deals with the characteristic 0 case) to the case where the characteristic of the base field is not 2 or 3. This is a joint work with Kohei Fukuda.

We give an explicit and intrinsic description of (the torsor defined by the 12th roots of) the discriminant of an elliptic curve using the group of its 12-torsion points and the Weil pairing. As an application, we extend a result of Coates (which deals with the characteristic 0 case) to the case where the characteristic of the base field is not 2 or 3. This is a joint work with Kohei Fukuda.

### 2013/12/18

18:00-19:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Heights of motives (ENGLISH)

**Kazuya Kato**(University of Chicago)Heights of motives (ENGLISH)

[ Abstract ]

The height of a rational number a/b (a, b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion height is so naive, height has played fundamental roles in number theory. There are important variants of this notion. In 1983, when Faltings proved Mordell conjecture, Faltings first proved Tate conjecture for abelian variaties by defining heights of abelian varieties, and then he deduced Mordell conjecture from the latter conjecture. I explain that his height of an abelian variety is generalized to the height of a motive. This generalization of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded heights, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.

The height of a rational number a/b (a, b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion height is so naive, height has played fundamental roles in number theory. There are important variants of this notion. In 1983, when Faltings proved Mordell conjecture, Faltings first proved Tate conjecture for abelian variaties by defining heights of abelian varieties, and then he deduced Mordell conjecture from the latter conjecture. I explain that his height of an abelian variety is generalized to the height of a motive. This generalization of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded heights, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.

### 2013/11/20

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

On the homotopy exact sequence for the logarithmic de Rham fundamental group (ENGLISH)

**Valentina Di Proietto**(The University of Tokyo)On the homotopy exact sequence for the logarithmic de Rham fundamental group (ENGLISH)

[ Abstract ]

Let K be a field of characteristic 0 and let X* be a quasi-projective simple normal crossing log variety over the log point K* associated to K. We construct a log de Rham version of the homotopy sequence \\pi_1(X*/K*)-->\\pi_1(X*/K)--\\pi_1(K*/K)-->1 and prove that it is exact. Moreover we show the injectivity of the first map for certain quotients of the groups. Our proofs are purely algebraic. This is a joint work with A. Shiho.

Let K be a field of characteristic 0 and let X* be a quasi-projective simple normal crossing log variety over the log point K* associated to K. We construct a log de Rham version of the homotopy sequence \\pi_1(X*/K*)-->\\pi_1(X*/K)--\\pi_1(K*/K)-->1 and prove that it is exact. Moreover we show the injectivity of the first map for certain quotients of the groups. Our proofs are purely algebraic. This is a joint work with A. Shiho.

### 2013/11/13

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

Goren-Oort stratification and Tate cycles on Hilbert modular varieties (ENGLISH)

**Yichao Tian**(Morningside Center for Mathematics)Goren-Oort stratification and Tate cycles on Hilbert modular varieties (ENGLISH)

[ Abstract ]

Let B be a quaternionic algebra over a totally real field F, and p be a prime at least 3 unramified in F. We consider a Shimura variety X associated to B^* of level prime to p. A generalization of Deligne-Carayol's "modèle étrange" allows us to define an integral model for X. We will then define a Goren-Oort stratification on the characteristic p fiber of X, and show that each closed Goren-Oort stratum is an iterated P^1-fibration over another quaternionic Shimura variety in characteristic p. Now suppose that [F:Q] is even and that p is inert in F. An iteration of this construction gives rise to many algebraic cycles of middle codimension on the characteristic p fibre of Hilbert modular varieties of prime-to-p level. We show that the cohomological classes of these cycles generate a large subspace of the Tate cycles, which, in some special cases, coincides with the prediction of the Tate conjecture for the Hilbert modular variety over finite fields. This is a joint work with Liang Xiao.

Let B be a quaternionic algebra over a totally real field F, and p be a prime at least 3 unramified in F. We consider a Shimura variety X associated to B^* of level prime to p. A generalization of Deligne-Carayol's "modèle étrange" allows us to define an integral model for X. We will then define a Goren-Oort stratification on the characteristic p fiber of X, and show that each closed Goren-Oort stratum is an iterated P^1-fibration over another quaternionic Shimura variety in characteristic p. Now suppose that [F:Q] is even and that p is inert in F. An iteration of this construction gives rise to many algebraic cycles of middle codimension on the characteristic p fibre of Hilbert modular varieties of prime-to-p level. We show that the cohomological classes of these cycles generate a large subspace of the Tate cycles, which, in some special cases, coincides with the prediction of the Tate conjecture for the Hilbert modular variety over finite fields. This is a joint work with Liang Xiao.

### 2013/10/30

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

Explicit integral cocycles on GLn and special values of p-adic partial zeta functions (ENGLISH)

**Pierre Charollois**(Université Paris 6)Explicit integral cocycles on GLn and special values of p-adic partial zeta functions (ENGLISH)

[ Abstract ]

Building on earlier work by Sczech, we contruct an explicit integral valued cocycle on GLn(Z).

It allows for the detailed analysis of the order of vanishing and of the special value at s=0 of the p-adic partial zeta functions introduced by Pi. Cassou-Noguès and Deligne-Ribet. In particular we recover a result of Wiles (1990) on Gross conjecture.

Another construction, now based on Shintani's method, is shown to lead to a cohomologous cocycle. This is joint work with S. Dasgupta and M. Greenberg.

Building on earlier work by Sczech, we contruct an explicit integral valued cocycle on GLn(Z).

It allows for the detailed analysis of the order of vanishing and of the special value at s=0 of the p-adic partial zeta functions introduced by Pi. Cassou-Noguès and Deligne-Ribet. In particular we recover a result of Wiles (1990) on Gross conjecture.

Another construction, now based on Shintani's method, is shown to lead to a cohomologous cocycle. This is joint work with S. Dasgupta and M. Greenberg.

### 2013/10/16

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

Shimura varieties with infinite level, and torsion in the cohomology of locally symmetric spaces (ENGLISH)

**Peter Scholze**(Universität Bonn)Shimura varieties with infinite level, and torsion in the cohomology of locally symmetric spaces (ENGLISH)

[ Abstract ]

We will discuss the p-adic geometry of Shimura varieties with infinite level at p: They are perfectoid spaces, and there is a new period map defined at infinite level. As an application, we will discuss some results on torsion in the cohomology of locally symmetric spaces, and in particular the existence of Galois representations in this setup.

We will discuss the p-adic geometry of Shimura varieties with infinite level at p: They are perfectoid spaces, and there is a new period map defined at infinite level. As an application, we will discuss some results on torsion in the cohomology of locally symmetric spaces, and in particular the existence of Galois representations in this setup.

### 2013/07/24

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

The determinant of a double covering of the projective space of even dimension and the discriminant of the branch locus (JAPANESE)

**Yasuhiro Terakado**(University of Tokyo)The determinant of a double covering of the projective space of even dimension and the discriminant of the branch locus (JAPANESE)

### 2013/07/10

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

On ramification filtration of local fields of equal characteristic (JAPANESE)

**Yuri Yatagawa**(University of Tokyo)On ramification filtration of local fields of equal characteristic (JAPANESE)

### 2013/07/03

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

A general formula for the discriminant of polynomials over $¥mathbb{F}_2$ determining the parity of the number of prime factors

(JAPANESE)

**Takehito Yoshiki**(University of Tokyo)A general formula for the discriminant of polynomials over $¥mathbb{F}_2$ determining the parity of the number of prime factors

(JAPANESE)

[ Abstract ]

In order to find irreducible polynomials over $\\mathbb{F}_2$ efficiently, the method using Swan's theorem is known. Swan's theorem determines the parity of the numberof irreducible factors of a polynomial $f$ over $\\mathbb{F}_2$ with no repeated root, by using the discriminant ${\\rm D}(\\tilde{f})\\pmod 8$, where $\\tilde{f}$ is a monic polynomial over $\\mathbb{Z}_2$ such that $\\tilde{f}=f\\pmod 2$. In the lecture, we will give the formula for the discriminant ${\\rm D}(\\tilde{f}) \\pmod 8$ for a polynomial $f$ over $\\mathbb{F}_2$ with no repeated root. By applying this formula to various types of polynomials, we shall get the parity of the number of irreducible factors of them.

In order to find irreducible polynomials over $\\mathbb{F}_2$ efficiently, the method using Swan's theorem is known. Swan's theorem determines the parity of the numberof irreducible factors of a polynomial $f$ over $\\mathbb{F}_2$ with no repeated root, by using the discriminant ${\\rm D}(\\tilde{f})\\pmod 8$, where $\\tilde{f}$ is a monic polynomial over $\\mathbb{Z}_2$ such that $\\tilde{f}=f\\pmod 2$. In the lecture, we will give the formula for the discriminant ${\\rm D}(\\tilde{f}) \\pmod 8$ for a polynomial $f$ over $\\mathbb{F}_2$ with no repeated root. By applying this formula to various types of polynomials, we shall get the parity of the number of irreducible factors of them.