## Number Theory Seminar

Seminar information archive ～05/25｜Next seminar｜Future seminars 05/26～

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**

### 2024/05/22

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

On the (φ,Γ)-modules corresponding to crystalline representations and semi-stable representations

**Takumi Watanabe**(University of Tokyo)On the (φ,Γ)-modules corresponding to crystalline representations and semi-stable representations

[ Abstract ]

From the 1980s to the 1990s, J.-M. Fontaine constructed an equivalence of categories between the category of (φ, Γ)-modules and the category of p-adic Galois representations. After recalling it, I will present my result on the (φ, Γ)-modules corresponding to crystalline representations and semi-stable representations. As for the crystalline case, this can be seen, in a sense, as a generalization of Wach module in the ramified case. If time permits, I will explain my ongoing research on the (φ, Γ)-modules corresponding to de Rham representations.

From the 1980s to the 1990s, J.-M. Fontaine constructed an equivalence of categories between the category of (φ, Γ)-modules and the category of p-adic Galois representations. After recalling it, I will present my result on the (φ, Γ)-modules corresponding to crystalline representations and semi-stable representations. As for the crystalline case, this can be seen, in a sense, as a generalization of Wach module in the ramified case. If time permits, I will explain my ongoing research on the (φ, Γ)-modules corresponding to de Rham representations.

### 2024/05/15

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

Equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic (日本語)

**Yuta Takaya**(University of Tokyo)Equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic (日本語)

[ Abstract ]

Shimura varieties are of central interest in arithmetic geometry and affine Deligne-Lusztig varieties are closely related to their special fibers. These varieties are group-theoretical objects and can be defined even for non-miniscule local Shimura data. In this talk, I will explain the proof of the equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic.

The main ingredient is a local foliation of affine Deligne-Lusztig varieties in mixed characteristic. In equal characteristic, this local structure was previously introduced by Hartl and Viehmann.

Shimura varieties are of central interest in arithmetic geometry and affine Deligne-Lusztig varieties are closely related to their special fibers. These varieties are group-theoretical objects and can be defined even for non-miniscule local Shimura data. In this talk, I will explain the proof of the equidimensionality of affine Deligne-Lusztig varieties in mixed characteristic.

The main ingredient is a local foliation of affine Deligne-Lusztig varieties in mixed characteristic. In equal characteristic, this local structure was previously introduced by Hartl and Viehmann.

### 2024/05/08

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

The pro-modularity in the residually reducible case (English)

**Xinyao Zhang**(University of Tokyo)The pro-modularity in the residually reducible case (English)

[ Abstract ]

For a continuous odd two dimensional Galois representation over a finite field of characteristic p, it is conjectured that its universal deformation ring is isomorphic to some p-adic big Hecke algebra, called the big R=T theorem. Recently, Deo explored the residually reducible case and proved a big R=T theorem for Q under the assumption of the cyclicity of some cohomology group. However, his method is unavailable for totally real fields since the assumption does not hold any longer. In this talk, we follow the strategy of the work from Skinner-Wiles and Pan on the Fontaine-Mazur conjecture and give a pro-modularity result for some totally real fields, which is an analogue to the big R=T theorem.

For a continuous odd two dimensional Galois representation over a finite field of characteristic p, it is conjectured that its universal deformation ring is isomorphic to some p-adic big Hecke algebra, called the big R=T theorem. Recently, Deo explored the residually reducible case and proved a big R=T theorem for Q under the assumption of the cyclicity of some cohomology group. However, his method is unavailable for totally real fields since the assumption does not hold any longer. In this talk, we follow the strategy of the work from Skinner-Wiles and Pan on the Fontaine-Mazur conjecture and give a pro-modularity result for some totally real fields, which is an analogue to the big R=T theorem.

### 2024/05/01

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

On the potential automorphy and the local-global compatibility for the monodromy operators at p ≠ l over CM fields. (日本語)

**Kojiro Matsumoto**(University of Tokyo)On the potential automorphy and the local-global compatibility for the monodromy operators at p ≠ l over CM fields. (日本語)

[ Abstract ]

Let F be a totally real field or CM field, n be a positive integer, l be a prime, π be a cohomological cuspidal automorphic representation of GLn over F and v be a non-l-adic finite place of F. In 2014, Harris-Lan-Taylor-Thorne constructed the l-adic Galois representation corresponding to π. (Scholze also constructed this by another method.) The compatibility of this construction and the local Langlands correspondence at v was proved up to semisimplification by Ila Varma(2014), but the compatibility for the monodromy operators was known only in conjugate self-dual cases and some special 2-dimensional cases. In this talk, we will prove the local-global compatibility in some self-dual cases and sufficiently regular weight cases by using some new potential automorphy theorems. Moreover, if we have time, we will also prove the Ramanujan conjecture for the cohomological cuspidal automorphic representations of GL2 over F, which was proved in parallel weight cases by Boxer-Calegari-Gee-Newton-Thorne (2023).

Let F be a totally real field or CM field, n be a positive integer, l be a prime, π be a cohomological cuspidal automorphic representation of GLn over F and v be a non-l-adic finite place of F. In 2014, Harris-Lan-Taylor-Thorne constructed the l-adic Galois representation corresponding to π. (Scholze also constructed this by another method.) The compatibility of this construction and the local Langlands correspondence at v was proved up to semisimplification by Ila Varma(2014), but the compatibility for the monodromy operators was known only in conjugate self-dual cases and some special 2-dimensional cases. In this talk, we will prove the local-global compatibility in some self-dual cases and sufficiently regular weight cases by using some new potential automorphy theorems. Moreover, if we have time, we will also prove the Ramanujan conjecture for the cohomological cuspidal automorphic representations of GL2 over F, which was proved in parallel weight cases by Boxer-Calegari-Gee-Newton-Thorne (2023).

### 2024/04/17

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

Functoriality of the p-adic Simpson correspondence by proper push forward (English)

**Ahmed Abbes**(Institut des Hautes Études Scientifiques, University of Tokyo)Functoriality of the p-adic Simpson correspondence by proper push forward (English)

[ Abstract ]

Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. After recalling the one initiated by myself with Michel Gros, I will present our initial result on the functoriality of the p-adic Simpson correspondence by proper push forward, leading to a generalization of the relative Hodge-Tate spectral sequence. If time permits, I will give a brief overview of an ongoing project with Michel Gros and Takeshi Tsuji, aimed at establishing a more robust framework for achieving broader functoriality results of the p-adic Simpson correspondence, by both proper push forward and pullback.

Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. After recalling the one initiated by myself with Michel Gros, I will present our initial result on the functoriality of the p-adic Simpson correspondence by proper push forward, leading to a generalization of the relative Hodge-Tate spectral sequence. If time permits, I will give a brief overview of an ongoing project with Michel Gros and Takeshi Tsuji, aimed at establishing a more robust framework for achieving broader functoriality results of the p-adic Simpson correspondence, by both proper push forward and pullback.

### 2024/02/21

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

K-motives and Local Langlands (English)

**Jens Niklas Eberhardt**(University of Bonn)K-motives and Local Langlands (English)

[ Abstract ]

In this talk, we construct a geometric realisation of the category of representations of the affine Hecke algebra. For this, we introduce a formalism of K-theoretic sheaves (called K-motives) on stacks. The affine Hecke algebra arises from the K-theory of the Steinberg stack, and we explain how to “category” using K-motives.

Lastly, we briefly discuss the relation to the local Langlands program.

In this talk, we construct a geometric realisation of the category of representations of the affine Hecke algebra. For this, we introduce a formalism of K-theoretic sheaves (called K-motives) on stacks. The affine Hecke algebra arises from the K-theory of the Steinberg stack, and we explain how to “category” using K-motives.

Lastly, we briefly discuss the relation to the local Langlands program.

### 2024/01/24

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

Purity for p-adic Galois representations (English)

**Yong Suk Moon**(BIMSA)Purity for p-adic Galois representations (English)

[ Abstract ]

Given a smooth p-adic formal scheme, Tsuji proved a purity result for crystalline local systems on its generic fiber. In this talk, we will discuss a generalization for log-crystalline local systems on the generic fiber of a semistable p-adic formal scheme. This is based on a joint work with Du, Liu, and Shimizu.

Given a smooth p-adic formal scheme, Tsuji proved a purity result for crystalline local systems on its generic fiber. In this talk, we will discuss a generalization for log-crystalline local systems on the generic fiber of a semistable p-adic formal scheme. This is based on a joint work with Du, Liu, and Shimizu.

### 2024/01/10

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

Characteristic cycle and partially logarithmic characteristic cycle of a rank 1 sheaf (Japanese)

**Yuri Yatagawa**(Tokyo Institute of Technology)Characteristic cycle and partially logarithmic characteristic cycle of a rank 1 sheaf (Japanese)

### 2023/12/20

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

Accessing the big de Rham-Witt complex via algebraic cycles with a vanishing condition (English)

**Jinhyun Park**(KAIST)Accessing the big de Rham-Witt complex via algebraic cycles with a vanishing condition (English)

[ Abstract ]

The big de Rham-Witt complexes of certain good rings over a field are known to admit certain motivic descriptions, namely via cycles with a modulus condition, e.g. additive higher Chow groups. This allowed us to define the trace maps on the de Rham-Witt forms in geometric terms, for instance.

Inspired by a lemma of Kato-Saito on the class field theory and Milnor K-groups, in this talk I would introduce a recent attempt in progress, where a version of “vanishing algebraic cycles” is defined over the formal power series k[[t]]. Using these cycles, I would sketch an alternative cycle-theoretic description of the big de Rham-Witt forms.

The big de Rham-Witt complexes of certain good rings over a field are known to admit certain motivic descriptions, namely via cycles with a modulus condition, e.g. additive higher Chow groups. This allowed us to define the trace maps on the de Rham-Witt forms in geometric terms, for instance.

Inspired by a lemma of Kato-Saito on the class field theory and Milnor K-groups, in this talk I would introduce a recent attempt in progress, where a version of “vanishing algebraic cycles” is defined over the formal power series k[[t]]. Using these cycles, I would sketch an alternative cycle-theoretic description of the big de Rham-Witt forms.

### 2023/11/22

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

Torsion birational motives of surfaces and unramified cohomology (Japanese)

**Takao Yamazaki**(Chuo University)Torsion birational motives of surfaces and unramified cohomology (Japanese)

### 2023/11/01

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

Prismatic realization functor for Shimura varieties of abelian type (English)

**Alex Youcis**(University of Tokyo)Prismatic realization functor for Shimura varieties of abelian type (English)

[ Abstract ]

Shimura varieties are certain classes of schemes which play an important role in various studies of number theory. The Langlands program is one of such examples. While far from known in general, it is expected that Shimura varieties are moduli spaces of certain motives with extra structure. In this talk I discuss joint work with Naoki Imai and Hiroki Kato, which constructs prismatic objects on the integral canonical models of Shimura varieties of abelian type at hyperspecial level. These may be thought of as the prismatic realization of such a hypothetical universal motive. I will also discuss how one can use this object to characterize these integral models, even at finite level.

Shimura varieties are certain classes of schemes which play an important role in various studies of number theory. The Langlands program is one of such examples. While far from known in general, it is expected that Shimura varieties are moduli spaces of certain motives with extra structure. In this talk I discuss joint work with Naoki Imai and Hiroki Kato, which constructs prismatic objects on the integral canonical models of Shimura varieties of abelian type at hyperspecial level. These may be thought of as the prismatic realization of such a hypothetical universal motive. I will also discuss how one can use this object to characterize these integral models, even at finite level.

### 2023/10/25

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

Geometric Eisenstein Series over the Fargues-Fontaine curve (English)

**Linus Hamann**(Stanford University)Geometric Eisenstein Series over the Fargues-Fontaine curve (English)

[ Abstract ]

Geometric Eisenstein series were first studied extensively by Braverman-Gaitsgory, Laumon, and Drinfeld, in the context of function field geometric Langlands. For a Levi subgroup M inside a connected reductive group G, they are functors which send Hecke eigensheaves on the moduli stack of M-bundles to Hecke eigensheaves on the moduli stack of G-bundles via certain relative compactifications of the moduli stack of P-bundles. We will discuss what this theory has to offer in the context of the recent Fargues-Scholze geometric Langlands program. Namely, motivated by the results in the function field setting, we will explicate what the analogous results tell us in this setting of the Fargues-Scholze program, as well as discuss various consequences for the cohmology of local and global Shimura varieties, via the relation between local Shimura varieties and the p-adic shtukas appearing in the Fargues-Scholze program.

Geometric Eisenstein series were first studied extensively by Braverman-Gaitsgory, Laumon, and Drinfeld, in the context of function field geometric Langlands. For a Levi subgroup M inside a connected reductive group G, they are functors which send Hecke eigensheaves on the moduli stack of M-bundles to Hecke eigensheaves on the moduli stack of G-bundles via certain relative compactifications of the moduli stack of P-bundles. We will discuss what this theory has to offer in the context of the recent Fargues-Scholze geometric Langlands program. Namely, motivated by the results in the function field setting, we will explicate what the analogous results tell us in this setting of the Fargues-Scholze program, as well as discuss various consequences for the cohmology of local and global Shimura varieties, via the relation between local Shimura varieties and the p-adic shtukas appearing in the Fargues-Scholze program.

### 2023/10/18

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

On Igusa varieties (English)

**Wansu Kim**(KAIST/University of Tokyo)On Igusa varieties (English)

[ Abstract ]

In this talk, we construct Igusa varieties and study some basic properties in the setting of abelian-type Shimura varieties, as well as in the analogous setting for function fields (over shtuka spaces). The is joint work with Paul Hamacher.

In this talk, we construct Igusa varieties and study some basic properties in the setting of abelian-type Shimura varieties, as well as in the analogous setting for function fields (over shtuka spaces). The is joint work with Paul Hamacher.

### 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.