代数学コロキウム

過去の記録 ~04/13次回の予定今後の予定 04/14~

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

2024年02月21日(水)

17:00-18:00   数理科学研究科棟(駒場) 117号室
Jens Niklas Eberhardt 氏 (University of Bonn)
K-motives and Local Langlands (English)
[ 講演概要 ]
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年01月24日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Yong Suk Moon 氏 (BIMSA)
Purity for p-adic Galois representations (English)
[ 講演概要 ]
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年01月10日(水)

17:00-18:00   数理科学研究科棟(駒場) 117号室
谷田川友里 氏 (東京工業大学)
階数1の層の特性サイクルと部分的に対数的な特性サイクル (Japanese)
[ 講演概要 ]
完全体上なめらかな代数多様体上の階数1の層に対して定義されるいくつかの特性サイクルの関係について考える。この特性サイクルのうちの一つはBeilinson-斎藤により構成可能層に対して消失輪体を用いて定義されるものである。
講演では、まず、加藤による対数的な特性サイクルの類似として、階数1の層に対し、対数的および非対数的な分岐理論を用いて部分的に対数的な特性サイクルを構成する。その後、部分的に対数的な特性サイクルを導入する利点や性質、特性サイクルとの関係についてわかったことを紹介する。

2023年12月20日(水)

17:00-18:00   数理科学研究科棟(駒場) 117号室
Jinhyun Park 氏 (KAIST)
Accessing the big de Rham-Witt complex via algebraic cycles with a vanishing condition (English)
[ 講演概要 ]
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年11月22日(水)

17:00-18:00   数理科学研究科棟(駒場) 117号室
山崎隆雄 氏 (中央大学)
曲面のねじれ双有理モチーフ不変量と不分岐コホモロジー (Japanese)
[ 講演概要 ]
対角線の分解を許容する曲面の捻じれ双有理モチーフについて,不分岐コホモロジーが普遍的な不変量を与えるという結果について講演する.整係数Hodge予想への新たな反例についても触れる.また,正標数では単純な類似が成立し得ないことを論じる.(佐藤周友氏との共同研究)

2023年11月01日(水)

17:00-18:00   数理科学研究科棟(駒場) 117号室
Alex Youcis 氏 (東京大学大学院数理科学研究科)
Prismatic realization functor for Shimura varieties of abelian type (English)
[ 講演概要 ]
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年10月25日(水)

17:00-18:00   数理科学研究科棟(駒場) 117号室
Linus Hamann 氏 (Stanford University)
Geometric Eisenstein Series over the Fargues-Fontaine curve (English)
[ 講演概要 ]
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年10月18日(水)

17:00-18:00   数理科学研究科棟(駒場) 117号室
Wansu Kim 氏 (KAIST/東京大学大学院数理科学研究科)
On Igusa varieties (English)
[ 講演概要 ]
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年07月05日(水)

17:00-18:00   数理科学研究科棟(駒場) 117号室
Thomas Geisser 氏 (立教大学)
Duality for motivic cohomology over local fields and applications to class field theory. (English)
[ 講演概要 ]
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)
[ 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.

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)
[ 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.

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)
[ 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.

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)
[ 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.

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)
[ 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.

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)
[ 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.

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
[ 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 ]
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)
[ 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 ]
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)
[ 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.

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)
[ 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.

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)
[ 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.

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)
[ 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.

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)
[ 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.

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)
[ 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.

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)
[ 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).

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)
[ 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.

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