本文印刷
全画面プリント
代数学コロキウム
過去の記録 ～05/01｜次回の予定｜今後の予定 05/02～
| 開催情報 | 水曜日 17:00～18:00 数理科学研究科棟(駒場) 117号室 |
|---|---|
| 担当者 | 今井 直毅，ケリー シェーン |
過去の記録
2023年06月28日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
中山裕大 氏 (東京大学大学院数理科学研究科)
The integral models of the RSZ Shimura varieties (日本語)
中山裕大 氏 (東京大学大学院数理科学研究科)
The integral models of the RSZ Shimura varieties (日本語)
[ 講演概要 ]
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年06月21日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Stefan Reppen 氏 (Stockholm University)
On moduli of principal bundles under non-connected reductive groups (英語)
Stefan Reppen 氏 (Stockholm University)
On moduli of principal bundles under non-connected reductive groups (英語)
[ 講演概要 ]
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年06月07日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
山本 寛史 氏 (東京大学)
p-通常的半整数重さ次数 2 ジーゲルモジュラー形式の空間の次元について (日本語)
山本 寛史 氏 (東京大学)
p-通常的半整数重さ次数 2 ジーゲルモジュラー形式の空間の次元について (日本語)
[ 講演概要 ]
$p$-通常的半整数重さ次数 2 ジーゲルモジュラー形式の空間の次元について $p$ での固有値が $p$ 進単数である Hecke 固有形式を $p$ 通常的固有形式という. $p$ 通常的な Siegel 固有形式や半整数重さモジュラー形式ではられる空間の次元は保型形式の重さやレベルの $p$ 冪に関わらず上から抑えられていることが知られている.本講演で,私は同様の結果が半整数重さ,次数 2 の Siegel モジュラー形式でも成り立つことを示す. $F$ を $p$ 通常的 Hecke 固有カス
プ形式とし,$\pi_F$ を対応する $Mp_4(\mathbb{A}_\mathbb{Q})$ のカスプ表現とする.このとき, $\pi_F$ の Hecke 固有値が
$F$ の重さによって決まることがわかる.このことにより,局所テータ対応や石本氏の結果 (伊
吹山予想) を用いることで, $F$ が整数重さの $p$ 通常的 Siegel モジュラー形式や楕円モジュラー形式に対応することが示せる.
$p$-通常的半整数重さ次数 2 ジーゲルモジュラー形式の空間の次元について $p$ での固有値が $p$ 進単数である Hecke 固有形式を $p$ 通常的固有形式という. $p$ 通常的な Siegel 固有形式や半整数重さモジュラー形式ではられる空間の次元は保型形式の重さやレベルの $p$ 冪に関わらず上から抑えられていることが知られている.本講演で,私は同様の結果が半整数重さ,次数 2 の Siegel モジュラー形式でも成り立つことを示す. $F$ を $p$ 通常的 Hecke 固有カス
プ形式とし,$\pi_F$ を対応する $Mp_4(\mathbb{A}_\mathbb{Q})$ のカスプ表現とする.このとき, $\pi_F$ の Hecke 固有値が
$F$ の重さによって決まることがわかる.このことにより,局所テータ対応や石本氏の結果 (伊
吹山予想) を用いることで, $F$ が整数重さの $p$ 通常的 Siegel モジュラー形式や楕円モジュラー形式に対応することが示せる.
2023年05月31日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
竹内大智 氏 (理化学研究所)
Quadratic $\ell$-adic sheaf and its Heisenberg group (日本語)
竹内大智 氏 (理化学研究所)
Quadratic $\ell$-adic sheaf and its Heisenberg group (日本語)
[ 講演概要 ]
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年05月17日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
佐藤 謙 氏 (東京工業大学)
Indecomposable higher Chow cycles on Kummer surfaces (日本語)
佐藤 謙 氏 (東京工業大学)
Indecomposable higher Chow cycles on Kummer surfaces (日本語)
[ 講演概要 ]
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年05月10日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Guy Henniart 氏 (パリ第11大学)
Swan exponent of Galois representations and fonctoriality for classical groups over p-adic fields (English)
Guy Henniart 氏 (パリ第11大学)
Swan exponent of Galois representations and fonctoriality for classical groups over p-adic fields (English)
[ 講演概要 ]
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年04月26日(水)
18:00-19:30 数理科学研究科棟(駒場) 117号室
IHESからの中継、注意：開始時間は通常より1時間遅く
Dustin Clausen 氏 (Institut des Hautes Études Scientifiques)
A Conjectural Reciprocity Law for Realizations of Motives
https://indico.math.cnrs.fr/event/9634/
IHESからの中継、注意：開始時間は通常より1時間遅く
Dustin Clausen 氏 (Institut des Hautes Études Scientifiques)
A Conjectural Reciprocity Law for Realizations of Motives
[ 講演概要 ]
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.
[ 参考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年04月19日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Nicola Mazzari 氏 (パドヴァ大学)
The conjugate uniformization in the semistable case (English)
https://sites.google.com/site/nclmzzr/
Nicola Mazzari 氏 (パドヴァ大学)
The conjugate uniformization in the semistable case (English)
[ 講演概要 ]
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.
[ 参考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年01月18日(水)
17:00-18:00 ハイブリッド開催
Kestutis Cesnavicius 氏 (Paris-Saclay University)
The affine Grassmannian as a presheaf quotient (English)
Kestutis Cesnavicius 氏 (Paris-Saclay University)
The affine Grassmannian as a presheaf quotient (English)
[ 講演概要 ]
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年01月04日(水)
17:00-18:00 ハイブリッド開催
伊藤和広 氏 (東京大学カブリ数物連携宇宙研究機構)
G-displays over prisms and deformation theory (Japanese)
伊藤和広 氏 (東京大学カブリ数物連携宇宙研究機構)
G-displays over prisms and deformation theory (Japanese)
[ 講演概要 ]
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年11月30日(水)
17:00-18:00 ハイブリッド開催
Xinyao Zhang 氏 (東京大学大学院数理科学研究科)
The modularity of elliptic curves over some number fields (English)
Xinyao Zhang 氏 (東京大学大学院数理科学研究科)
The modularity of elliptic curves over some number fields (English)
[ 講演概要 ]
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年11月16日(水)
17:00-18:00 ハイブリッド開催
Zijian Yao 氏 (University of Chicago)
The eigencurve over the boundary of the weight space (English)
Zijian Yao 氏 (University of Chicago)
The eigencurve over the boundary of the weight space (English)
[ 講演概要 ]
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年11月02日(水)
17:00-18:00 ハイブリッド開催
Laurent Fargues 氏 (Mathematics Institute of Jussieu–Paris Rive Gauche・東京大学大学院数理科学研究科)
Some compact generators of D_{lis} (Bun_G,\Lambda) (English)
Laurent Fargues 氏 (Mathematics Institute of Jussieu–Paris Rive Gauche・東京大学大学院数理科学研究科)
Some compact generators of D_{lis} (Bun_G,\Lambda) (English)
[ 講演概要 ]
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年10月19日(水)
17:00-18:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
Shane Kelly 氏 (東京大学大学院数理科学研究科)
A nilpotent variant cdh-topology (English)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
Shane Kelly 氏 (東京大学大学院数理科学研究科)
A nilpotent variant cdh-topology (English)
[ 講演概要 ]
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年10月12日(水)
17:00-18:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
Abhinandan 氏 (東京大学大学院数理科学研究科)
Syntomic complex with coefficients (English)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
Abhinandan 氏 (東京大学大学院数理科学研究科)
Syntomic complex with coefficients (English)
[ 講演概要 ]
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年09月28日(水)
17:00-18:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
Jens Eberhardt 氏 (University of Wuppertal)
A K-theoretic approach to geometric representation theory (ENGLISH)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
Jens Eberhardt 氏 (University of Wuppertal)
A K-theoretic approach to geometric representation theory (ENGLISH)
[ 講演概要 ]
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年07月20日(水)
15:30-18:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
清水 康司 氏 (UC Berkeley) 15:30-16:30
Completed prismatic F-crystals and crystalline local systems (ENGLISH)
Twisted differential operators in several variables (ENGLISH)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
清水 康司 氏 (UC Berkeley) 15:30-16:30
Completed prismatic F-crystals and crystalline local systems (ENGLISH)
[ 講演概要 ]
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 氏 (Université de Caen) 17:00-18:00Bhatt 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.
Twisted differential operators in several variables (ENGLISH)
[ 講演概要 ]
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年07月06日(水)
17:00-18:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
劉 沛江 氏 (東京大学大学院数理科学研究科)
The characteristic cycles of non-confluent $\ell$-adic GKZ hypergeometric sheaves (ENGLISH)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
劉 沛江 氏 (東京大学大学院数理科学研究科)
The characteristic cycles of non-confluent $\ell$-adic GKZ hypergeometric sheaves (ENGLISH)
[ 講演概要 ]
$\ell$-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年06月22日(水)
17:00-18:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
高梨 悠吾 氏 (東京大学大学院数理科学研究科)
Parity of conjugate self-dual representations of inner forms of $\mathrm{GL}_n$ over $p$-adic fields (JAPANESE)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
高梨 悠吾 氏 (東京大学大学院数理科学研究科)
Parity of conjugate self-dual representations of inner forms of $\mathrm{GL}_n$ over $p$-adic fields (JAPANESE)
[ 講演概要 ]
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年06月20日(月)
15:00-16:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
加藤大輝 氏 (パリ・サクレー大学)
完全交差に対する$p$進ウェイトモノドロミー予想 (Japanese)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
加藤大輝 氏 (パリ・サクレー大学)
完全交差に対する$p$進ウェイトモノドロミー予想 (Japanese)
[ 講演概要 ]
ウェイトモノドロミー予想は$\ell$進コホモロジーに関する予想であるが、$p$進コホモロジーに対しても同様の主張を考えることができる。通常の$\ell$進の場合に、完全交差に対するウェイトモノドロミー予想がScholzeによって証明されたことは非常に有名である。彼はパーフェクトイド空間の理論を用いて等標数の場合に帰着することでそれを証明した。$p$進の場合にも等標数類似が(Crew、Lazda--Palにより)すでに証明されていることを踏まえるとScholzeと同様の議論を$p$進の場合にも行えないかと考えるのは自然なことである。私はFederico BindaとAlberto Vezzaniと共に、リジット空間のモチーフの理論を用いてそれを実行し、(スキーム論的)完全交差に対して$p$進ウェイトモノドロミー予想を証明したのでそれについて話す。
ウェイトモノドロミー予想は$\ell$進コホモロジーに関する予想であるが、$p$進コホモロジーに対しても同様の主張を考えることができる。通常の$\ell$進の場合に、完全交差に対するウェイトモノドロミー予想がScholzeによって証明されたことは非常に有名である。彼はパーフェクトイド空間の理論を用いて等標数の場合に帰着することでそれを証明した。$p$進の場合にも等標数類似が(Crew、Lazda--Palにより)すでに証明されていることを踏まえるとScholzeと同様の議論を$p$進の場合にも行えないかと考えるのは自然なことである。私はFederico BindaとAlberto Vezzaniと共に、リジット空間のモチーフの理論を用いてそれを実行し、(スキーム論的)完全交差に対して$p$進ウェイトモノドロミー予想を証明したのでそれについて話す。
2022年06月15日(水)
17:00-18:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
小泉 淳之介 氏 (東京大学大学院数理科学研究科)
Steinberg symbols and reciprocity sheaves (JAPANESE)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
小泉 淳之介 氏 (東京大学大学院数理科学研究科)
Steinberg symbols and reciprocity sheaves (JAPANESE)
[ 講演概要 ]
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年05月25日(水)
17:00-18:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
松田 光智 氏 (東京大学大学院数理科学研究科)
Torsion points of elliptic curves over cyclotomic fields (JAPANESE)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
松田 光智 氏 (東京大学大学院数理科学研究科)
Torsion points of elliptic curves over cyclotomic fields (JAPANESE)
[ 講演概要 ]
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年05月18日(水)
17:00-18:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
石本 宙 氏 (東京大学大学院数理科学研究科)
Local Langlands correspondence for non-quasi-split odd special orthogonal groups (JAPANESE)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
石本 宙 氏 (東京大学大学院数理科学研究科)
Local Langlands correspondence for non-quasi-split odd special orthogonal groups (JAPANESE)
[ 講演概要 ]
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年05月11日(水)
17:00-18:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
Joseph Muller 氏 (東京大学大学院数理科学研究科)
Cohomology of the unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ (ENGLISH)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
Joseph Muller 氏 (東京大学大学院数理科学研究科)
Cohomology of the unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ (ENGLISH)
[ 講演概要 ]
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.
2022年04月27日(水)
17:00-18:00 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
山本 修司 氏 (東京大学大学院数理科学研究科)
The Kaneko-Zagier conjecture on finite and symmetric multiple zeta values for general integer indices (JAPANESE)
数理科学研究科所属以外の方は、オンラインでのご参加をお願いいたします。
山本 修司 氏 (東京大学大学院数理科学研究科)
The Kaneko-Zagier conjecture on finite and symmetric multiple zeta values for general integer indices (JAPANESE)
[ 講演概要 ]
Kaneko and Zagier introduced two variants of multiple zeta values, which we call A-MZVs and S-MZVs, and conjectured that the algebraic structures of them are isomorphic. While these values were originally defined for positive integer (multi-)indices, recently, Komori extended the definition of S-MZVs to general integer indices. Since A-MZVs can also be defined for general integers, Komori's work suggests a generalization of the Kaneko-Zagier conjecture, from positive to general integers. In this talk, we will show how this generalization is reduced to the original conjecture. This is a joint work with Masataka Ono.
Kaneko and Zagier introduced two variants of multiple zeta values, which we call A-MZVs and S-MZVs, and conjectured that the algebraic structures of them are isomorphic. While these values were originally defined for positive integer (multi-)indices, recently, Komori extended the definition of S-MZVs to general integer indices. Since A-MZVs can also be defined for general integers, Komori's work suggests a generalization of the Kaneko-Zagier conjecture, from positive to general integers. In this talk, we will show how this generalization is reduced to the original conjecture. This is a joint work with Masataka Ono.
