## Number Theory Seminar

Seminar information archive ～09/22｜Next seminar｜Future seminars 09/23～

Date, time & place | Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | Naoki Imai, Shane Kelly |

**Seminar information archive**

### 2023/07/05

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Duality for motivic cohomology over local fields and applications to class field theory. (English)

**Thomas Geisser**(Rikkyo University)Duality for motivic cohomology over local fields and applications to class field theory. (English)

[ Abstract ]

We give an outline a (conjectural) construction of cohomology groups for smooth and proper varieties over local fields with values in the heart of the derived category of locally compact groups.

This theory should satisfy a Pontrjagin duality theorem, and for certain weights, we give an ad hoc construction which satisfies such a duality unconditionally.

As an application we discuss class field theory for smooth and proper varieties over local fields.

We give an outline a (conjectural) construction of cohomology groups for smooth and proper varieties over local fields with values in the heart of the derived category of locally compact groups.

This theory should satisfy a Pontrjagin duality theorem, and for certain weights, we give an ad hoc construction which satisfies such a duality unconditionally.

As an application we discuss class field theory for smooth and proper varieties over local fields.

### 2023/06/28

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

The integral models of the RSZ Shimura varieties (日本語)

**Yuta Nakayama**(University of Tokyo)The integral models of the RSZ Shimura varieties (日本語)

[ Abstract ]

We prove that the integral models of Shimura varieties by Rapoport, Smithling and Zhang proposed to describe variants of the arithmetic Gan–Gross–Prasad conjecture are isomorphic to the models by Pappas and Rapoport. This extends our previous work that compares the former models and the Kisin–Pappas models. We rely on the construction of the models of Pappas and Rapoport, not on their characterization.

We prove that the integral models of Shimura varieties by Rapoport, Smithling and Zhang proposed to describe variants of the arithmetic Gan–Gross–Prasad conjecture are isomorphic to the models by Pappas and Rapoport. This extends our previous work that compares the former models and the Kisin–Pappas models. We rely on the construction of the models of Pappas and Rapoport, not on their characterization.

### 2023/06/21

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

On moduli of principal bundles under non-connected reductive groups (英語)

**Stefan Reppen**(Stockholm University)On moduli of principal bundles under non-connected reductive groups (英語)

[ Abstract ]

Let $C$ be a smooth, connected projective curve over an algebraically closed field $k$ of characteristic 0, and let $G$ be a non-connected reductive group over $k$. I will explain how to decompose the stack of $G$-bundles $\text{Bun}_G$ into open and closed substacks $X_i$ which admits finite torsors $\text{Bun}_{\mathcal{G}_i} \to X_i$, for some connected reductive group schemes $\mathcal{G}_i$ over $C$. I explain how to use this to obtain a projective good moduli space of semistable $G$-bundles over $C$, for a suitable notion of semistability. Finally, after stating a result concerning finite subgroups of connected reductive groups over $k$, I explain how to see that essentially finite $H$-bundles are not dense in the moduli space of semistable degree 0 $H$-bundles, for any connected reductive group $H$ not equal to a torus.

Let $C$ be a smooth, connected projective curve over an algebraically closed field $k$ of characteristic 0, and let $G$ be a non-connected reductive group over $k$. I will explain how to decompose the stack of $G$-bundles $\text{Bun}_G$ into open and closed substacks $X_i$ which admits finite torsors $\text{Bun}_{\mathcal{G}_i} \to X_i$, for some connected reductive group schemes $\mathcal{G}_i$ over $C$. I explain how to use this to obtain a projective good moduli space of semistable $G$-bundles over $C$, for a suitable notion of semistability. Finally, after stating a result concerning finite subgroups of connected reductive groups over $k$, I explain how to see that essentially finite $H$-bundles are not dense in the moduli space of semistable degree 0 $H$-bundles, for any connected reductive group $H$ not equal to a torus.

### 2023/06/07

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

On the dimension of spaces of p-ordinary half-integral weight Siegel modular forms

of degree 2 (日本語)

**Hirofumi Yamamoto**(The University of Tokyo)On the dimension of spaces of p-ordinary half-integral weight Siegel modular forms

of degree 2 (日本語)

[ Abstract ]

Hecke eigenforms whose eigenvalues at $p$ are $p$-adic units are called $p$-ordinary. The dimension of the spaces spanned by $p$-ordinary Siegel modular eigenforms is known to be bounded regardless of the power of $p$ of levels and weights. Furthermore, Hida showed that the degree is bounded in the case of half-integral weight modular forms, using the Shimura correspondence. In this talk, I will explain that a similar result holds for half-integral weight $p$-ordinary Siegel modular forms of degree 2. Let $F$ be a $p$-ordinary Hecke eigen cuspform, and $\pi_F$ be the corresponding cuspidal representation of $Mp_4(\mathbb{A}_\mathbb{Q})$. Then the weight of $F$ determines the Hecke eigenvalue of $\pi_F$ at $p$. Therefore, we can relate $F$ to $p$-ordinary integral weight Siegel modular forms or elliptic modular forms by using the local theta correspondence and the method of Ishimoto(Ibukiyama conjecture).

Hecke eigenforms whose eigenvalues at $p$ are $p$-adic units are called $p$-ordinary. The dimension of the spaces spanned by $p$-ordinary Siegel modular eigenforms is known to be bounded regardless of the power of $p$ of levels and weights. Furthermore, Hida showed that the degree is bounded in the case of half-integral weight modular forms, using the Shimura correspondence. In this talk, I will explain that a similar result holds for half-integral weight $p$-ordinary Siegel modular forms of degree 2. Let $F$ be a $p$-ordinary Hecke eigen cuspform, and $\pi_F$ be the corresponding cuspidal representation of $Mp_4(\mathbb{A}_\mathbb{Q})$. Then the weight of $F$ determines the Hecke eigenvalue of $\pi_F$ at $p$. Therefore, we can relate $F$ to $p$-ordinary integral weight Siegel modular forms or elliptic modular forms by using the local theta correspondence and the method of Ishimoto(Ibukiyama conjecture).

### 2023/05/31

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Quadratic $\ell$-adic sheaf and its Heisenberg group (日本語)

**Daichi Takeuchi**(RIKEN)Quadratic $\ell$-adic sheaf and its Heisenberg group (日本語)

[ Abstract ]

Quadratic Gauss sums are usually defined only for finite fields of odd characteristic. However, it is known that there is a reformulation in which one can uniformly treat the case of even characteristic. In this talk, I will introduce a new class of $\ell$-adic sheaf, which I call quadratic sheaf. This is a sheaf-theoretic enhancement of the reformulation of quadratic Gauss sum, in the sense of the function-sheaf dictionary. After explaining its cohomological properties and consequences, such as a version of Hasse-Davenport relation, I will show that a certain finite Heisenberg group naturally acts on a quadratic sheaf. I will also report various results that can be deduced from this action.

Quadratic Gauss sums are usually defined only for finite fields of odd characteristic. However, it is known that there is a reformulation in which one can uniformly treat the case of even characteristic. In this talk, I will introduce a new class of $\ell$-adic sheaf, which I call quadratic sheaf. This is a sheaf-theoretic enhancement of the reformulation of quadratic Gauss sum, in the sense of the function-sheaf dictionary. After explaining its cohomological properties and consequences, such as a version of Hasse-Davenport relation, I will show that a certain finite Heisenberg group naturally acts on a quadratic sheaf. I will also report various results that can be deduced from this action.

### 2023/05/17

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Indecomposable higher Chow cycles on Kummer surfaces (日本語)

**Ken Sato**(Tokyo Institute of Technology)Indecomposable higher Chow cycles on Kummer surfaces (日本語)

[ Abstract ]

The higher Chow group $\mathrm{CH}^p(X,q)$ introduced by Bloch is a generalization of the classical Chow groups. It satisfies many interesting properties, but its structure is still mysterious for almost all varieties when $p$ is greater than 1. In this talk, I will explain the explicit construction of higher Chow cycles in $\mathrm{CH}^2(X,1)$ on a family of Kummer surfaces. By computing their images under the Beilinson regulator map, in very general cases, these cycles generate at least rank 18 subgroup of $\mathrm{CH}^2(X,1)_{\mathrm{ind}}$, which is the quotient of $\mathrm{CH}^2(X,1)$ by the images of the intersection product maps. To compute the images under the regulator map, we use automorphisms of the family and the explicit description of the action of the automorphisms on the Picard-Fuchs differential equations of the family.

The higher Chow group $\mathrm{CH}^p(X,q)$ introduced by Bloch is a generalization of the classical Chow groups. It satisfies many interesting properties, but its structure is still mysterious for almost all varieties when $p$ is greater than 1. In this talk, I will explain the explicit construction of higher Chow cycles in $\mathrm{CH}^2(X,1)$ on a family of Kummer surfaces. By computing their images under the Beilinson regulator map, in very general cases, these cycles generate at least rank 18 subgroup of $\mathrm{CH}^2(X,1)_{\mathrm{ind}}$, which is the quotient of $\mathrm{CH}^2(X,1)$ by the images of the intersection product maps. To compute the images under the regulator map, we use automorphisms of the family and the explicit description of the action of the automorphisms on the Picard-Fuchs differential equations of the family.

### 2023/05/10

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Swan exponent of Galois representations and fonctoriality for classical groups over p-adic fields (English)

**Guy Henniart**(Paris-Sud University)Swan exponent of Galois representations and fonctoriality for classical groups over p-adic fields (English)

[ Abstract ]

This is joint work with Masao Oi in Kyoto. Let F be a p-adic field for some prime number p,

F^ac an algebraic closure of F, and G_F the Galois group of F^ac/F. A continuous finite dimensional

representation σ (on a complex vector space W) has a Swan exponent s(σ), a non-negative integer

which measures how "wildly ramified" σ is. Langlands functoriality makes it of interest

to compare s(σ) and s(r o σ) when r is an algebraic representation of Aut_C(W). The first cases

for r are the determinant, the adjoint representation, the symmetric square representation and

the alternating square representation. I shall give some relations (inequalities mostly, with

equality in interesting cases) between the Swan exponents of those representations r o σ. I shall

also indicate how such relations can be used to explicit the local Langlands correspondence of

Arthur for some simple cuspidal representations of split classical groups over F.

This is joint work with Masao Oi in Kyoto. Let F be a p-adic field for some prime number p,

F^ac an algebraic closure of F, and G_F the Galois group of F^ac/F. A continuous finite dimensional

representation σ (on a complex vector space W) has a Swan exponent s(σ), a non-negative integer

which measures how "wildly ramified" σ is. Langlands functoriality makes it of interest

to compare s(σ) and s(r o σ) when r is an algebraic representation of Aut_C(W). The first cases

for r are the determinant, the adjoint representation, the symmetric square representation and

the alternating square representation. I shall give some relations (inequalities mostly, with

equality in interesting cases) between the Swan exponents of those representations r o σ. I shall

also indicate how such relations can be used to explicit the local Langlands correspondence of

Arthur for some simple cuspidal representations of split classical groups over F.

### 2023/04/26

18:00-19:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Live transmission from IHES, Warning: Start time is one hour later than usual.

A Conjectural Reciprocity Law for Realizations of Motives

https://indico.math.cnrs.fr/event/9634/

Live transmission from IHES, Warning: Start time is one hour later than usual.

**Dustin Clausen**(Institut des Hautes Études Scientifiques)A Conjectural Reciprocity Law for Realizations of Motives

[ Abstract ]

A motive over a scheme S is a bit of linear algebra which is supposed to "universally" capture the cohomology of smooth proper S-schemes. Motives can be studied via various "realizations", which are objects of more concrete linear algebraic categories attached to S. It is known that over certain S, these different realizations are related to one another via comparison isomorphisms, as in Hodge theory. In this talk, I will try to explain that for completely general S, there is a much more subtle kind of relationship between these realizations, which takes a similar form to classical reciprocity laws in number theory.

[ Reference URL ]A motive over a scheme S is a bit of linear algebra which is supposed to "universally" capture the cohomology of smooth proper S-schemes. Motives can be studied via various "realizations", which are objects of more concrete linear algebraic categories attached to S. It is known that over certain S, these different realizations are related to one another via comparison isomorphisms, as in Hodge theory. In this talk, I will try to explain that for completely general S, there is a much more subtle kind of relationship between these realizations, which takes a similar form to classical reciprocity laws in number theory.

https://indico.math.cnrs.fr/event/9634/

### 2023/04/19

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

The conjugate uniformization in the semistable case (English)

https://sites.google.com/site/nclmzzr/

**Nicola Mazzari**(University of Padua)The conjugate uniformization in the semistable case (English)

[ Abstract ]

We will review some recent results by Iovita-Morrow-Zaharescu about p-adic uniformization of abelian varieties with good reduction. Most of it relies on the theory developed by Fontaine especially about almost Cp-representations. These results were recently generalised by Howe-Morrow-Wear, via p-divisible groups.

We will explain how to treat the semistable case with focus on some really basic example, like the Tate elliptic curve.

[ Reference URL ]We will review some recent results by Iovita-Morrow-Zaharescu about p-adic uniformization of abelian varieties with good reduction. Most of it relies on the theory developed by Fontaine especially about almost Cp-representations. These results were recently generalised by Howe-Morrow-Wear, via p-divisible groups.

We will explain how to treat the semistable case with focus on some really basic example, like the Tate elliptic curve.

https://sites.google.com/site/nclmzzr/

### 2023/01/18

17:00-18:00 Hybrid

The affine Grassmannian as a presheaf quotient (English)

**Kestutis Cesnavicius**(Paris-Saclay University)The affine Grassmannian as a presheaf quotient (English)

[ Abstract ]

The affine Grassmannian of a reductive group G is usually defined as the étale sheafification of the quotient of the loop group LG by the positive loop subgroup. I will discuss various triviality results for G-torsors which imply that this sheafification is often not necessary.

The affine Grassmannian of a reductive group G is usually defined as the étale sheafification of the quotient of the loop group LG by the positive loop subgroup. I will discuss various triviality results for G-torsors which imply that this sheafification is often not necessary.

### 2023/01/04

17:00-18:00 Hybrid

G-displays over prisms and deformation theory (Japanese)

**Kazuhiro Ito**(The University of Tokyo, Kavli IPMU)G-displays over prisms and deformation theory (Japanese)

[ Abstract ]

The notion of display, which was introduced by Zink, has been successfully applied to the deformation theory of p-divisible groups. Recently, for a reductive group G over the ring of p-adic integers, Lau introduced the notion of G-display. In this talk, following the approach of Lau, we study displays and G-displays over the prismatic site of Bhatt-Scholze, and explain the deformation theory for them. As an application, we give an alternative proof of the classification of p-divisible groups over a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field, using our deformation theory.

The notion of display, which was introduced by Zink, has been successfully applied to the deformation theory of p-divisible groups. Recently, for a reductive group G over the ring of p-adic integers, Lau introduced the notion of G-display. In this talk, following the approach of Lau, we study displays and G-displays over the prismatic site of Bhatt-Scholze, and explain the deformation theory for them. As an application, we give an alternative proof of the classification of p-divisible groups over a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field, using our deformation theory.

### 2022/11/30

17:00-18:00 Hybrid

The modularity of elliptic curves over some number fields (English)

**Xinyao Zhang**(University of Tokyo)The modularity of elliptic curves over some number fields (English)

[ Abstract ]

As a non-trivial case of the Langlands reciprocity conjecture, the modularity of elliptic curves always intrigues number theorists, and a famous result was proved for semistable elliptic curves over \mathbb{Q} by Andrew Wiles, implying Fermat's Last Theorem. In recent years, many new results have been proved using sufficiently powerful modularity lifting theorems. For instance, Thorne proved that elliptic curves over the cyclotomic \mathbb{Z}_p-extension of \mathbb{Q} are modular. In this talk, I will sketch some of these results and try to give a new one that elliptic curves over the cyclotomic \mathbb{Z}_p-extension of a real quadratic field are modular under some technical assumptions.

As a non-trivial case of the Langlands reciprocity conjecture, the modularity of elliptic curves always intrigues number theorists, and a famous result was proved for semistable elliptic curves over \mathbb{Q} by Andrew Wiles, implying Fermat's Last Theorem. In recent years, many new results have been proved using sufficiently powerful modularity lifting theorems. For instance, Thorne proved that elliptic curves over the cyclotomic \mathbb{Z}_p-extension of \mathbb{Q} are modular. In this talk, I will sketch some of these results and try to give a new one that elliptic curves over the cyclotomic \mathbb{Z}_p-extension of a real quadratic field are modular under some technical assumptions.

### 2022/11/16

17:00-18:00 Hybrid

The eigencurve over the boundary of the weight space (English)

**Zijian Yao**(University of Chicago)The eigencurve over the boundary of the weight space (English)

[ Abstract ]

The eigencurve is a geometric object that p-adically interpolates eigenforms of finite slope. The global geometry of the eigencurve is somewhat mysterious, except that over the boundary, it is predicted to behave rather nicely (by the so-called Halo conjecture). This conjecture has been studied by Liu--Wan--Xiao for definite quaternion algebras. In this talk, we will report on some work in progress on this conjecture in the case of GL2. If time permits, we will discuss some generalizations towards groups beyond GL2. This is partially joint with H. Diao.

The eigencurve is a geometric object that p-adically interpolates eigenforms of finite slope. The global geometry of the eigencurve is somewhat mysterious, except that over the boundary, it is predicted to behave rather nicely (by the so-called Halo conjecture). This conjecture has been studied by Liu--Wan--Xiao for definite quaternion algebras. In this talk, we will report on some work in progress on this conjecture in the case of GL2. If time permits, we will discuss some generalizations towards groups beyond GL2. This is partially joint with H. Diao.

### 2022/11/02

17:00-18:00 Hybrid

Some compact generators of D_{lis} (Bun_G,\Lambda) (English)

**Laurent Fargues**(Mathematics Institute of Jussieu–Paris Rive Gauche, University of Tokyo)Some compact generators of D_{lis} (Bun_G,\Lambda) (English)

[ Abstract ]

I will speak about some aspect of my joint work with Scholze on the geomerization of the local Langlands correspondence. More precisely, I will explain how to construct explicitly some compact generators of the derived category of étale sheaves on Bun_G, the Artin v-stack of G-bundles on the curve. Those compact generators generalize the classical compactly induced representations in the classical local Langlands program. For this we construct some particular charts on Bun_G and this will be the occasion to review some geometric constructions in our joint work.

I will speak about some aspect of my joint work with Scholze on the geomerization of the local Langlands correspondence. More precisely, I will explain how to construct explicitly some compact generators of the derived category of étale sheaves on Bun_G, the Artin v-stack of G-bundles on the curve. Those compact generators generalize the classical compactly induced representations in the classical local Langlands program. For this we construct some particular charts on Bun_G and this will be the occasion to review some geometric constructions in our joint work.

### 2022/10/19

17:00-18:00 Hybrid

A nilpotent variant cdh-topology (English)

**Shane Kelly**(University of Tokyo)A nilpotent variant cdh-topology (English)

[ Abstract ]

I will speak about a version of the cdh-topology which can see nilpotents, and applications to algebraic K-theory. This is joint work in progress with Shuji Saito.

I will speak about a version of the cdh-topology which can see nilpotents, and applications to algebraic K-theory. This is joint work in progress with Shuji Saito.

### 2022/10/12

17:00-18:00 Hybrid

Syntomic complex with coefficients (English)

**Abhinandan**(University of Tokyo)Syntomic complex with coefficients (English)

[ Abstract ]

In the proof of $p$-adic crystalline comparison theorem, one of the most important steps in the approach of Fontaine and Messing is to establish a comparison between syntomic cohomology and p-adic étale cohomology via (Fontaine-Messing) period map. This approach was successfully generalized to the semistable case by Kato and a complete proof of crystalline and semistable comparison theorems for schemes was given by Tsuji. Few years ago, Colmez and Nizioł gave a new interpretation of the (local) Fontaine-Messing period map in terms of complexes of $(\varphi,\Gamma)$-modules and used it to prove semistable comparison theorem for $p$-adic formal schemes. We will present a generalisation (of crystalline version of this interpretation by Colmez and Nizioł) to coefficients arising from relative Fontaine-Laffaille modules of Faltings (on syntomic side) and relative Wach modules introduced by the speaker (on $(\varphi,\Gamma)$-module side).

In the proof of $p$-adic crystalline comparison theorem, one of the most important steps in the approach of Fontaine and Messing is to establish a comparison between syntomic cohomology and p-adic étale cohomology via (Fontaine-Messing) period map. This approach was successfully generalized to the semistable case by Kato and a complete proof of crystalline and semistable comparison theorems for schemes was given by Tsuji. Few years ago, Colmez and Nizioł gave a new interpretation of the (local) Fontaine-Messing period map in terms of complexes of $(\varphi,\Gamma)$-modules and used it to prove semistable comparison theorem for $p$-adic formal schemes. We will present a generalisation (of crystalline version of this interpretation by Colmez and Nizioł) to coefficients arising from relative Fontaine-Laffaille modules of Faltings (on syntomic side) and relative Wach modules introduced by the speaker (on $(\varphi,\Gamma)$-module side).

### 2022/09/28

17:00-18:00 Hybrid

A K-theoretic approach to geometric representation theory (ENGLISH)

**Jens Eberhardt**(University of Wuppertal)A K-theoretic approach to geometric representation theory (ENGLISH)

[ Abstract ]

Perverse sheaves and intersection cohomology are central objects in geometric representation theory. This talk is about their long-lost K-theoretic cousins, called K-motives. We will discuss definitions and basic properties of K-motives and explore potential applications to geometric representation theory. For example, K-motives shed a new light on Beilinson--Ginzburg--Soergel's Koszul duality -- a remarkable symmetry in the representation theory and geometry of two Langlands dual reductive groups. We will see that this new form of Koszul duality does not involve any gradings or mixed geometry which are as essential as mysterious in the classical approaches.

Perverse sheaves and intersection cohomology are central objects in geometric representation theory. This talk is about their long-lost K-theoretic cousins, called K-motives. We will discuss definitions and basic properties of K-motives and explore potential applications to geometric representation theory. For example, K-motives shed a new light on Beilinson--Ginzburg--Soergel's Koszul duality -- a remarkable symmetry in the representation theory and geometry of two Langlands dual reductive groups. We will see that this new form of Koszul duality does not involve any gradings or mixed geometry which are as essential as mysterious in the classical approaches.

### 2022/07/20

15:30-18:00 Hybrid

Completed prismatic F-crystals and crystalline local systems (ENGLISH)

Twisted differential operators in several variables (ENGLISH)

**Koji Shimizu**(UC Berkeley) 15:30-16:30Completed prismatic F-crystals and crystalline local systems (ENGLISH)

[ Abstract ]

Bhatt and Scholze introduced the absolute prismatic site of a p-adic ring and proved the equivalence of categories between prismatic F-crystals and lattices in crystalline representations in the CDVR case with perfect residue field. We will define a wider category of completed prismatic F-crystals in the relative case and explain its relation to the category of crystalline local systems. This is joint work with Heng Du, Tong Liu, and Yong Suk Moon.

Bhatt and Scholze introduced the absolute prismatic site of a p-adic ring and proved the equivalence of categories between prismatic F-crystals and lattices in crystalline representations in the CDVR case with perfect residue field. We will define a wider category of completed prismatic F-crystals in the relative case and explain its relation to the category of crystalline local systems. This is joint work with Heng Du, Tong Liu, and Yong Suk Moon.

**Pierre Houedry**(Université de Caen) 17:00-18:00Twisted differential operators in several variables (ENGLISH)

[ Abstract ]

The aim of my presentation is to give an overview of the results I obtained during the first year of my PhD. The theory of $q$-differences equations appeared a long time ago with the Birkhoff's work. It is well understood in the complex setting. In 2004, Lucia Di Vizio and Yves André, in the paper $q$-differences and p-adic local monodromy, gave an equivalence between certain type of $q$-differences equations and a certain type of classical differential equations in the p-adic setting. Recently, Adolfo Quiros, Bernard Le Stum and Michel Gros have been working on a generalization of this result not looking only for $q$-differences equations but also twisted equations in general. The framework that they develop is working for equations in one variable. The goal of my thesis is to generalize those results in several variables.

The aim of my presentation is to give an overview of the results I obtained during the first year of my PhD. The theory of $q$-differences equations appeared a long time ago with the Birkhoff's work. It is well understood in the complex setting. In 2004, Lucia Di Vizio and Yves André, in the paper $q$-differences and p-adic local monodromy, gave an equivalence between certain type of $q$-differences equations and a certain type of classical differential equations in the p-adic setting. Recently, Adolfo Quiros, Bernard Le Stum and Michel Gros have been working on a generalization of this result not looking only for $q$-differences equations but also twisted equations in general. The framework that they develop is working for equations in one variable. The goal of my thesis is to generalize those results in several variables.

### 2022/07/06

17:00-18:00 Hybrid

The characteristic cycles of non-confluent $\ell$-adic GKZ hypergeometric sheaves (ENGLISH)

**Peijiang Liu**(University of Tokyo)The characteristic cycles of non-confluent $\ell$-adic GKZ hypergeometric sheaves (ENGLISH)

[ Abstract ]

$\ell$-adic GKZ hypergeometric sheaves are defined to be étale analogues of GKZ hypergeometric $\mathcal{D}$-modules. We introduce an algorithm of computing the characteristic cycles of certain type of $\ell$-adic GKZ hypergeometric sheaves. We compute the irreducible components by a push-forward formula for characteristic cycles of étale sheaves, and compute the multiplicities by considering a comparison theorem between the characteristic cycles of non-confluent $\ell$-adic GKZ hypergeometric sheaves and those of non-confluent GKZ hypergeometric $\mathcal{D}$-modules. We also explain the limitation of our algorithm by an example.

$\ell$-adic GKZ hypergeometric sheaves are defined to be étale analogues of GKZ hypergeometric $\mathcal{D}$-modules. We introduce an algorithm of computing the characteristic cycles of certain type of $\ell$-adic GKZ hypergeometric sheaves. We compute the irreducible components by a push-forward formula for characteristic cycles of étale sheaves, and compute the multiplicities by considering a comparison theorem between the characteristic cycles of non-confluent $\ell$-adic GKZ hypergeometric sheaves and those of non-confluent GKZ hypergeometric $\mathcal{D}$-modules. We also explain the limitation of our algorithm by an example.

### 2022/06/22

17:00-18:00 Hybrid

Parity of conjugate self-dual representations of inner forms of $\mathrm{GL}_n$ over $p$-adic fields (JAPANESE)

**Yugo Takanashi**(University of Tokyo)Parity of conjugate self-dual representations of inner forms of $\mathrm{GL}_n$ over $p$-adic fields (JAPANESE)

[ Abstract ]

There are two parametrizations of discrete series representations of $\mathrm{GL}_n$ over $p$-adic fields. One is the local Langlands correspondence, and the other is the local Jacquet-Langlands correspondence. The composite of these two maps the discrete series representations of an inner form of $\mathrm{GL}_n$ to Galois representations called discrete L-parameters. On the other hand, we can define the parity for each self-dual representation depending on whether the representation is orthogonal or symplectic. The composite preserves the notion of self-duality, and it transforms the parity in a nontrivial manner. Prasad and Ramakrishnan computed the transformation law, and Mieda proved its conjugate self-dual analog under some conditions on groups and representations. We will talk about the proof of the general case of this analog. We use the globalization method, as in the proof of Prasad and Ramakrishnan.

There are two parametrizations of discrete series representations of $\mathrm{GL}_n$ over $p$-adic fields. One is the local Langlands correspondence, and the other is the local Jacquet-Langlands correspondence. The composite of these two maps the discrete series representations of an inner form of $\mathrm{GL}_n$ to Galois representations called discrete L-parameters. On the other hand, we can define the parity for each self-dual representation depending on whether the representation is orthogonal or symplectic. The composite preserves the notion of self-duality, and it transforms the parity in a nontrivial manner. Prasad and Ramakrishnan computed the transformation law, and Mieda proved its conjugate self-dual analog under some conditions on groups and representations. We will talk about the proof of the general case of this analog. We use the globalization method, as in the proof of Prasad and Ramakrishnan.

### 2022/06/20

15:00-16:00 Hybrid

p-adic weight-monodromy conjecture for complete intersections (Japanese)

**Hiroki Kato**(Paris-Saclay University)p-adic weight-monodromy conjecture for complete intersections (Japanese)

### 2022/06/15

17:00-18:00 Hybrid

Steinberg symbols and reciprocity sheaves (JAPANESE)

**Junnosuke Koizumi**(University of Tokyo)Steinberg symbols and reciprocity sheaves (JAPANESE)

[ Abstract ]

The norm residue symbol and the differential symbol are known to satisfy the common relation $(a,1-a)=0$ which is called the Steinberg relation. Hu-Kriz showed that the Steinberg relation can be understood as a relation between certain morphisms in the stable motivic homotopy category. On the other hand, there is also an “additive variant” of the Steinberg relation, namely $(a,a)+(1-a,1-a)=0$, for which the classical motivic theory is no longer applicable. In this talk we will explain how the theory of reciprocity sheaves due to Kahn-Saito-Yamazaki can be utilized to generalize the theory of Hu-Kriz to include the additive Steinberg relation.

The norm residue symbol and the differential symbol are known to satisfy the common relation $(a,1-a)=0$ which is called the Steinberg relation. Hu-Kriz showed that the Steinberg relation can be understood as a relation between certain morphisms in the stable motivic homotopy category. On the other hand, there is also an “additive variant” of the Steinberg relation, namely $(a,a)+(1-a,1-a)=0$, for which the classical motivic theory is no longer applicable. In this talk we will explain how the theory of reciprocity sheaves due to Kahn-Saito-Yamazaki can be utilized to generalize the theory of Hu-Kriz to include the additive Steinberg relation.

### 2022/05/25

17:00-18:00 Hybrid

Torsion points of elliptic curves over cyclotomic fields (JAPANESE)

**Koji Matsuda**(University of Tokyo)Torsion points of elliptic curves over cyclotomic fields (JAPANESE)

[ Abstract ]

By Mordell--Weil theorem, the Mordell--Weil groups of elliptic curves over number fields are finitely generated, and in particular their torsion subgroups are finite. For a fixed elliptic curve, it is easy to compute its torsion subgroups. Conversely using modular curves, we can study the possible torsion subgroups of elliptic curves. More precisely, the existence of an elliptic curve with certain torsion points is essentially equivalent to the existence of certain rational points of a modular curve. In this talk, in order to study the rational points of modular curves over cyclotomic fields, we compute the Mordell--Weil ranks of their Jacobian varieties over cyclotomic fields.

By Mordell--Weil theorem, the Mordell--Weil groups of elliptic curves over number fields are finitely generated, and in particular their torsion subgroups are finite. For a fixed elliptic curve, it is easy to compute its torsion subgroups. Conversely using modular curves, we can study the possible torsion subgroups of elliptic curves. More precisely, the existence of an elliptic curve with certain torsion points is essentially equivalent to the existence of certain rational points of a modular curve. In this talk, in order to study the rational points of modular curves over cyclotomic fields, we compute the Mordell--Weil ranks of their Jacobian varieties over cyclotomic fields.

### 2022/05/18

17:00-18:00 Hybrid

Local Langlands correspondence for non-quasi-split odd special orthogonal groups (JAPANESE)

**Hiroshi Ishimoto**(University of Tokyo)Local Langlands correspondence for non-quasi-split odd special orthogonal groups (JAPANESE)

[ Abstract ]

In 2013, Arthur established the endoscopic classification of representations of quasi-split symplectic and orthogonal groups, and Mok analogously proved the similar classification for quasi-split unitary groups. In 2014, Kaletha-Minguez-Shin-White established the classification for non-quasi-spilt unitary groups assuming Mok's results. Similarly, we can prove that for non-quasi-split odd orthogonal groups assuming Arthur's results. In this talk, I will explain the local Langlands correspondence for non-quasi-split odd special orthogonal groups, which is a part of the classification of representations.

In 2013, Arthur established the endoscopic classification of representations of quasi-split symplectic and orthogonal groups, and Mok analogously proved the similar classification for quasi-split unitary groups. In 2014, Kaletha-Minguez-Shin-White established the classification for non-quasi-spilt unitary groups assuming Mok's results. Similarly, we can prove that for non-quasi-split odd orthogonal groups assuming Arthur's results. In this talk, I will explain the local Langlands correspondence for non-quasi-split odd special orthogonal groups, which is a part of the classification of representations.

### 2022/05/11

17:00-18:00 Hybrid

Cohomology of the unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ (ENGLISH)

**Joseph Muller**(University of Tokyo)Cohomology of the unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ (ENGLISH)

[ Abstract ]

Rapoport-Zink (RZ) spaces are moduli spaces which classify the deformations of a $p$-divisible group with additional structures. It is equipped with compatible actions of $p$-adic and Galois groups, and their cohomology is believed to play a role in the local Langlands program. So far, the cohomology of RZ spaces is entirely known only in the cases of the Lubin-Tate tower and of the Drinfeld space ; in particular both of them are RZ spaces of EL type. In this talk, we consider the unramified PEL unitary RZ space with signature $(1,n-1)$. In 2011, Vollaard and Wedhorn proved that it is stratified by generalized Deligne-Lusztig varieties, whose incidence relations mimic the combinatorics of the Bruhat-Tits building of a unitary group. We compute the cohomology of these strata and we draw some consequences on the cohomology of the RZ space. When $n = 3, 4$ we deduce

an automorphic description of the cohomology of the basic stratum in the corresponding Shimura variety via p-adic uniformization.

Rapoport-Zink (RZ) spaces are moduli spaces which classify the deformations of a $p$-divisible group with additional structures. It is equipped with compatible actions of $p$-adic and Galois groups, and their cohomology is believed to play a role in the local Langlands program. So far, the cohomology of RZ spaces is entirely known only in the cases of the Lubin-Tate tower and of the Drinfeld space ; in particular both of them are RZ spaces of EL type. In this talk, we consider the unramified PEL unitary RZ space with signature $(1,n-1)$. In 2011, Vollaard and Wedhorn proved that it is stratified by generalized Deligne-Lusztig varieties, whose incidence relations mimic the combinatorics of the Bruhat-Tits building of a unitary group. We compute the cohomology of these strata and we draw some consequences on the cohomology of the RZ space. When $n = 3, 4$ we deduce

an automorphic description of the cohomology of the basic stratum in the corresponding Shimura variety via p-adic uniformization.