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過去の記録 ～04/16｜次回の予定｜今後の予定 04/17～
| 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
2020年04月22日(水)
17:30-18:30 オンライン開催
Arthur-César Le Bras 氏 (CNRS & Université Paris 13)
Prismatic Dieudonné theory (ENGLISH)
Arthur-César Le Bras 氏 (CNRS & Université Paris 13)
Prismatic Dieudonné theory (ENGLISH)
[ 講演概要 ]
I would like to explain a classification result for p-divisible groups, which unifies many of the existing results in the literature. The main tool is the theory of prisms and prismatic cohomology recently developed by Bhatt and Scholze. This is joint work with Johannes Anschütz.
本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる
東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．
今回はパリからの中継です．
I would like to explain a classification result for p-divisible groups, which unifies many of the existing results in the literature. The main tool is the theory of prisms and prismatic cohomology recently developed by Bhatt and Scholze. This is joint work with Johannes Anschütz.
本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる
東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．
今回はパリからの中継です．
2019年12月04日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
竹内 大智 氏 (東京大学数理科学研究科)
Characteristic epsilon cycles of l-adic sheaves on varieties (ENGLISH)
竹内 大智 氏 (東京大学数理科学研究科)
Characteristic epsilon cycles of l-adic sheaves on varieties (ENGLISH)
[ 講演概要 ]
For l-adic sheaves on varieties over finite fields, the constant terms of the functional equations of the L-functions, called global epsilon factors, are important arithmetic invariants. When the varieties are curves, Deligne and Laumon show that they admit product formulae in terms of local epsilon factors.
In this talk, I will explain that, attaching some coefficients to irreducible components of singular supports, we can define refinements of characteristic cycles. We will see that, after taking modulo roots of unity, they give product formulae of global epsilon factors for higher dimensional varieties.
I will also explain that these results can be generalized to arbitrary perfect fields of any characteristic.
For l-adic sheaves on varieties over finite fields, the constant terms of the functional equations of the L-functions, called global epsilon factors, are important arithmetic invariants. When the varieties are curves, Deligne and Laumon show that they admit product formulae in terms of local epsilon factors.
In this talk, I will explain that, attaching some coefficients to irreducible components of singular supports, we can define refinements of characteristic cycles. We will see that, after taking modulo roots of unity, they give product formulae of global epsilon factors for higher dimensional varieties.
I will also explain that these results can be generalized to arbitrary perfect fields of any characteristic.
2019年11月27日(水)
17:00-18:00 数理科学研究科棟(駒場) 117号室
Ahmed Abbes 氏 (CNRS & IHÉS)
The relative Hodge-Tate spectral sequence (ENGLISH)
Ahmed Abbes 氏 (CNRS & IHÉS)
The relative Hodge-Tate spectral sequence (ENGLISH)
[ 講演概要 ]
It is well known that the p-adic étale cohomology of a smooth and proper variety over a p-adic field admits a Hodge-Tate decomposition and that it is the abutment of a spectral sequence called Hodge-Tate; these two properties are incidentally equivalent. The Hodge-Tate decomposition was generalized in higher dimensions to Hodge-Tate local systems by Hyodo, and was studied by Faltings, Tsuji and others. But the generalization of the Hodge-Tate spectral sequence to a relative situation has not yet been considered (not even conjectured), with the exception of a special case of abelian schemes by Hyodo. This has now been done in a joint work with Michel Gros. The relative Hodge-Tate spectral sequence that we construct takes place in the Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work. The relative Hodge-Tate spectral sequence sheds new light on the fact that the relative p-adic étale cohomology is Hodge-Tate, but the two properties are not equivalent in general.
It is well known that the p-adic étale cohomology of a smooth and proper variety over a p-adic field admits a Hodge-Tate decomposition and that it is the abutment of a spectral sequence called Hodge-Tate; these two properties are incidentally equivalent. The Hodge-Tate decomposition was generalized in higher dimensions to Hodge-Tate local systems by Hyodo, and was studied by Faltings, Tsuji and others. But the generalization of the Hodge-Tate spectral sequence to a relative situation has not yet been considered (not even conjectured), with the exception of a special case of abelian schemes by Hyodo. This has now been done in a joint work with Michel Gros. The relative Hodge-Tate spectral sequence that we construct takes place in the Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work. The relative Hodge-Tate spectral sequence sheds new light on the fact that the relative p-adic étale cohomology is Hodge-Tate, but the two properties are not equivalent in general.
2019年11月20日(水)
18:00-19:00 数理科学研究科棟(駒場) 056号室
Vasudevan Srinivas 氏 (Tata Institute of Fundamental Research)
Algebraic versus topological entropy for surfaces over finite fields (ENGLISH)
Vasudevan Srinivas 氏 (Tata Institute of Fundamental Research)
Algebraic versus topological entropy for surfaces over finite fields (ENGLISH)
[ 講演概要 ]
For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は東京からの中継です．）
For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は東京からの中継です．）
2019年10月16日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Liang Xiao 氏 (BICMR, Peking University)
On slopes of modular forms (ENGLISH)
Liang Xiao 氏 (BICMR, Peking University)
On slopes of modular forms (ENGLISH)
[ 講演概要 ]
In this talk, I will survey some recent progress towards understanding the slopes of modular forms, with or without level structures. This has direct application to the conjecture of Breuil-Buzzard-Emerton on the slopes of Kisin's crystabelline deformation spaces. In particular, we obtain certain refined version of the spectral halo conjecture, where we may identify explicitly the slopes at the boundary when given a reducible non-split generic residual local Galois representation. This is a joint work in progress with Ruochuan Liu, Nha Truong, and Bin Zhao.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は北京からの中継です．）
In this talk, I will survey some recent progress towards understanding the slopes of modular forms, with or without level structures. This has direct application to the conjecture of Breuil-Buzzard-Emerton on the slopes of Kisin's crystabelline deformation spaces. In particular, we obtain certain refined version of the spectral halo conjecture, where we may identify explicitly the slopes at the boundary when given a reducible non-split generic residual local Galois representation. This is a joint work in progress with Ruochuan Liu, Nha Truong, and Bin Zhao.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は北京からの中継です．）
2019年10月09日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
Yuanqing Cai 氏 (京都大学)
Twisted doubling integrals for classical groups (ENGLISH)
Yuanqing Cai 氏 (京都大学)
Twisted doubling integrals for classical groups (ENGLISH)
[ 講演概要 ]
In the 1980s, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula.
In this talk, we present a family of Rankin-Selberg integrals (the twisted doubling method, in joint work with Friedberg, Ginzburg, and Kaplan) for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. This can be viewed as a generalization of the doubling integrals of Piatetski-Shapiro and Rallis. Time permitting, we will discuss the twisted doubling integrals for Brylinski-Deligne covers of classical groups.
In the 1980s, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula.
In this talk, we present a family of Rankin-Selberg integrals (the twisted doubling method, in joint work with Friedberg, Ginzburg, and Kaplan) for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. This can be viewed as a generalization of the doubling integrals of Piatetski-Shapiro and Rallis. Time permitting, we will discuss the twisted doubling integrals for Brylinski-Deligne covers of classical groups.
2019年07月03日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
佐藤謙 氏 (東京大学数理科学研究科)
Explicit calculation of values of the regulator maps on a certain type of Kummer surfaces (Japanese)
佐藤謙 氏 (東京大学数理科学研究科)
Explicit calculation of values of the regulator maps on a certain type of Kummer surfaces (Japanese)
[ 講演概要 ]
複素数に埋め込まれた体K上定義された射影代数多様体Xに対して、レギュレーター写像というモチヴィックコホモロジーからDeligneコホモロジーへの写像がBeilinsonにより定義された。特にKが有理数体の時、Beilinsonによりレギュレーター写像の値はモチーフのL関数の特殊値の無理数部分と結びつくと予想されているが、予想が成り立つことが知られている例は少ない。しかしながら、レギュレーター写像の値を超幾何関数のような特殊関数を用いて表す研究は朝倉政典氏や大坪紀之氏の研究に見られるように近年盛んである。本講演では、楕円曲線の直積に付随するようなKummer曲面に対し、高次Chow群との同型を用いてモチヴィックコホモロジーの中に具体的に元を構成し、その元のレギュレーター写像による値を考察する。またその応用として、上記の曲面のモチヴィックコホモロジーのindecomposal partが複素数体上十分一般の場合に消えていないことを示す。
複素数に埋め込まれた体K上定義された射影代数多様体Xに対して、レギュレーター写像というモチヴィックコホモロジーからDeligneコホモロジーへの写像がBeilinsonにより定義された。特にKが有理数体の時、Beilinsonによりレギュレーター写像の値はモチーフのL関数の特殊値の無理数部分と結びつくと予想されているが、予想が成り立つことが知られている例は少ない。しかしながら、レギュレーター写像の値を超幾何関数のような特殊関数を用いて表す研究は朝倉政典氏や大坪紀之氏の研究に見られるように近年盛んである。本講演では、楕円曲線の直積に付随するようなKummer曲面に対し、高次Chow群との同型を用いてモチヴィックコホモロジーの中に具体的に元を構成し、その元のレギュレーター写像による値を考察する。またその応用として、上記の曲面のモチヴィックコホモロジーのindecomposal partが複素数体上十分一般の場合に消えていないことを示す。
2019年06月12日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
笠浦一海 氏 (東京大学数理科学研究科)
On extension of overconvergent log isocrystals on log smooth varieties (Japanese)
笠浦一海 氏 (東京大学数理科学研究科)
On extension of overconvergent log isocrystals on log smooth varieties (Japanese)
[ 講演概要 ]
Kを混標数の完備な非アルキメデス付値体とし，kをその剰余体とする．
Kedlayaおよび志甫の研究により，k上の滑らかな代数多様体Xとその上の単純正規交叉因子Zについて，X ¥setminus Z上の過収束アイソクリスタルのうちZの周りである種のモノドロミーを持つものは，XにZから定まる対数的構造を入れた対数的代数多様体上の収束対数的アイソクリスタルに延長できることが知られている．
本講演では，この結果の，適当な条件を満たす一般の対数的に滑らかな代数多様体と，その対数的構造から定められる部分スキーム上の過収束対数的アイソクリスタルへの拡張について議論する．
Kを混標数の完備な非アルキメデス付値体とし，kをその剰余体とする．
Kedlayaおよび志甫の研究により，k上の滑らかな代数多様体Xとその上の単純正規交叉因子Zについて，X ¥setminus Z上の過収束アイソクリスタルのうちZの周りである種のモノドロミーを持つものは，XにZから定まる対数的構造を入れた対数的代数多様体上の収束対数的アイソクリスタルに延長できることが知られている．
本講演では，この結果の，適当な条件を満たす一般の対数的に滑らかな代数多様体と，その対数的構造から定められる部分スキーム上の過収束対数的アイソクリスタルへの拡張について議論する．
2019年06月05日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
服部新 氏 (東京都市大学)
Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms (English)
服部新 氏 (東京都市大学)
Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms (English)
[ 講演概要 ]
Let p be a rational prime, q>1 a p-power and P a non-constant irreducible polynomial in F_q[t]. The notion of Drinfeld modular form is an analogue over F_q(t) of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite v-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.
Let p be a rational prime, q>1 a p-power and P a non-constant irreducible polynomial in F_q[t]. The notion of Drinfeld modular form is an analogue over F_q(t) of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite v-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.
2019年05月29日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
沖泰裕 氏 (東京大学数理科学研究科)
On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)
沖泰裕 氏 (東京大学数理科学研究科)
On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)
[ 講演概要 ]
PEL型志村多様体のp進整数環上の整モデルは, Abel多様体と付加構造のモジュライ空間として定義される. その幾何的特殊ファイバーのうち, 超特異Abel多様体に対応する点からなる閉部分スキームを超特異部分という. 超特異部分の構造の明示的な記述は, arithmetic intersectionをはじめとする整数論への応用をもつことが知られている.
本講演では, 2次四元数ユニタリ群に対する志村多様体の超特異部分の明示的記述に関して, 講演者が得た結果を紹介する. また, 関連するRapoport-Zink空間の底空間に対する同様の結果についても言及する.
PEL型志村多様体のp進整数環上の整モデルは, Abel多様体と付加構造のモジュライ空間として定義される. その幾何的特殊ファイバーのうち, 超特異Abel多様体に対応する点からなる閉部分スキームを超特異部分という. 超特異部分の構造の明示的な記述は, arithmetic intersectionをはじめとする整数論への応用をもつことが知られている.
本講演では, 2次四元数ユニタリ群に対する志村多様体の超特異部分の明示的記述に関して, 講演者が得た結果を紹介する. また, 関連するRapoport-Zink空間の底空間に対する同様の結果についても言及する.
2019年05月08日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
山本祐輝 氏 (東京大学数理科学研究科)
On the types for supercuspidal representations of inner forms of GL_n (Japanese)
山本祐輝 氏 (東京大学数理科学研究科)
On the types for supercuspidal representations of inner forms of GL_n (Japanese)
[ 講演概要 ]
Aを非アルキメデス的局所体F上の中心的単純環とし，Gをその乗法群とする．
Gのsmooth表現を考察する際に有用な理論の一つとしてtypeの理論が存在する．
type (J, ¥lambda) とはGのコンパクト部分群Jと J の既約部分表現 ¥lambda の組であって，Gの既約表現をある意味で分類することのできるものである．
S¥'echerre-Stevenにより，Gのtypeの族としてsimple typeという概念が構成されている．
本講演ではGのtypeについて説明した後，JをGの極大コンパクト部分群Kとして固定した場合にtypeがどれだけ存在するかについて議論する．
特に，Gのsupercuspidal表現 ¥pi に対し，¥pi がsimple typeとしてある種の不分岐的な条件を満たすようなものを含むときに，Kの表現で ¥pi に対応するtypeがGでの共役を除き一意であることを示す．
Aを非アルキメデス的局所体F上の中心的単純環とし，Gをその乗法群とする．
Gのsmooth表現を考察する際に有用な理論の一つとしてtypeの理論が存在する．
type (J, ¥lambda) とはGのコンパクト部分群Jと J の既約部分表現 ¥lambda の組であって，Gの既約表現をある意味で分類することのできるものである．
S¥'echerre-Stevenにより，Gのtypeの族としてsimple typeという概念が構成されている．
本講演ではGのtypeについて説明した後，JをGの極大コンパクト部分群Kとして固定した場合にtypeがどれだけ存在するかについて議論する．
特に，Gのsupercuspidal表現 ¥pi に対し，¥pi がsimple typeとしてある種の不分岐的な条件を満たすようなものを含むときに，Kの表現で ¥pi に対応するtypeがGでの共役を除き一意であることを示す．
2019年04月30日(火)
17:00-18:00 数理科学研究科棟(駒場) 122号室
Jean-Francois Dat 氏 (Sorbonne University)
Moduli space of l-adic Langlands parameters and the stable Bernstein center (English)
Jean-Francois Dat 氏 (Sorbonne University)
Moduli space of l-adic Langlands parameters and the stable Bernstein center (English)
[ 講演概要 ]
Motivated by the description of the integral l-adic cohomology of certain Shimura varieties in middle degree, Emerton and Helm have conjectured the existence of a certain local Langlands correspondence for l-adic families of n-dimensional Galois representations. The proof of this conjecture by Helm and Moss relies on a beautiful isomorphism between the ring of functions of the moduli space of l-adic representations and the integral Bernstein center of GL_n(F). We will present a work in progress with Helm, Korinczuk and Moss towards a generalization of this result for arbitrary (tamely ramified) reductive groups.
Motivated by the description of the integral l-adic cohomology of certain Shimura varieties in middle degree, Emerton and Helm have conjectured the existence of a certain local Langlands correspondence for l-adic families of n-dimensional Galois representations. The proof of this conjecture by Helm and Moss relies on a beautiful isomorphism between the ring of functions of the moduli space of l-adic representations and the integral Bernstein center of GL_n(F). We will present a work in progress with Helm, Korinczuk and Moss towards a generalization of this result for arbitrary (tamely ramified) reductive groups.
2019年04月24日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Joseph Ayoub 氏 (University of Zurich)
P^1-localisation and a possible definition of arithmetic Kodaira-Spencer classes (English)
Joseph Ayoub 氏 (University of Zurich)
P^1-localisation and a possible definition of arithmetic Kodaira-Spencer classes (English)
[ 講演概要 ]
A^1-localisation is a universal construction which produces "cohomology theories" for which the affine line A^1 is contractible. It plays a central role in the theory of motives à la Morel-Voevodsky. In this talk, I'll discuss the analogous construction where the affine line is replaced by the projective line P^1. This is the P^1-localisation which is arguably an unnatural construction since it produces "cohomology theories" for which the projective line P^1 is contractible. Nevertheless, I'll explain a few positive results and some computations around this construction which naturally lead to a definition of Kodaira-Spencer classes of arithmetic nature. (Unfortunately, it is yet unclear if these classes are really interesting and nontrivial.)
A^1-localisation is a universal construction which produces "cohomology theories" for which the affine line A^1 is contractible. It plays a central role in the theory of motives à la Morel-Voevodsky. In this talk, I'll discuss the analogous construction where the affine line is replaced by the projective line P^1. This is the P^1-localisation which is arguably an unnatural construction since it produces "cohomology theories" for which the projective line P^1 is contractible. Nevertheless, I'll explain a few positive results and some computations around this construction which naturally lead to a definition of Kodaira-Spencer classes of arithmetic nature. (Unfortunately, it is yet unclear if these classes are really interesting and nontrivial.)
2019年04月17日(水)
17:00-18:00 数理科学研究科棟(駒場) 122号室
高松哲平 氏 (東京大学数理科学研究科)
On the Shafarevich conjecture for minimal surfaces of Kodaira dimension 0 (Japanese)
高松哲平 氏 (東京大学数理科学研究科)
On the Shafarevich conjecture for minimal surfaces of Kodaira dimension 0 (Japanese)
[ 講演概要 ]
Fを代数体、Sを有限素点の有限集合とする。
Faltings-Zarhinは、固定した正整数gに対して、
Sの外で良還元を持つようなF上のg次元Abel多様体のF上の同型類の有限性を証明した。
(Abel多様体のShafarevich予想)
本講演では、K3曲面、Enriques曲面、超楕円曲面に対するこの定理の類似を議論する。
また、超楕円曲面の還元に関連する話題として、
超楕円曲面のNeronモデルについても紹介したい。
Fを代数体、Sを有限素点の有限集合とする。
Faltings-Zarhinは、固定した正整数gに対して、
Sの外で良還元を持つようなF上のg次元Abel多様体のF上の同型類の有限性を証明した。
(Abel多様体のShafarevich予想)
本講演では、K3曲面、Enriques曲面、超楕円曲面に対するこの定理の類似を議論する。
また、超楕円曲面の還元に関連する話題として、
超楕円曲面のNeronモデルについても紹介したい。
2019年04月10日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Zongbin Chen 氏 (丘成桐数学科学中心, 清華大学)
The geometry of the affine Springer fibers and Arthur's weighted orbital integrals (English)
Zongbin Chen 氏 (丘成桐数学科学中心, 清華大学)
The geometry of the affine Springer fibers and Arthur's weighted orbital integrals (English)
[ 講演概要 ]
The affine Springer fibers are geometric objects conceived for the study of orbital integrals. They have complicated geometric structures. We will explain our work on the geometry of affine Springer fibers, with emphasize on the construction of a fundamental domain, and show how the study of the affine Springer fibers can be reduced to that of its fundamental domain. As an application, we will explain how to calculate Arthur's weighted orbital integrals via counting points on the fundamental domain.
（本講演は「東京北京パリ数論幾何セミナー」として, インターネットによる東大数理, Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は東京からの中継です．）
The affine Springer fibers are geometric objects conceived for the study of orbital integrals. They have complicated geometric structures. We will explain our work on the geometry of affine Springer fibers, with emphasize on the construction of a fundamental domain, and show how the study of the affine Springer fibers can be reduced to that of its fundamental domain. As an application, we will explain how to calculate Arthur's weighted orbital integrals via counting points on the fundamental domain.
（本講演は「東京北京パリ数論幾何セミナー」として, インターネットによる東大数理, Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は東京からの中継です．）
2019年01月16日(水)
18:00-19:00 数理科学研究科棟(駒場) 056号室
Lei Fu 氏 (Yau Mathematical Sciences Center, Tsinghua University)
p-adic Gelfand-Kapranov-Zelevinsky systems (ENGLISH)
Lei Fu 氏 (Yau Mathematical Sciences Center, Tsinghua University)
p-adic Gelfand-Kapranov-Zelevinsky systems (ENGLISH)
[ 講演概要 ]
Using Dwork's trace formula, we express the exponential sum associated to a Laurent polynomial as the trace of a chain map on a twisted de Rham complex for the torus over the p-adic field. Treating the coefficients of the polynomial as parameters, we obtain the p-adic Gelfand-Kapranov-Zelevinsky (GKZ) system, which is a complex of D^\dagger-modules with Frobenius structure.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は北京からの中継です．）
Using Dwork's trace formula, we express the exponential sum associated to a Laurent polynomial as the trace of a chain map on a twisted de Rham complex for the torus over the p-adic field. Treating the coefficients of the polynomial as parameters, we obtain the p-adic Gelfand-Kapranov-Zelevinsky (GKZ) system, which is a complex of D^\dagger-modules with Frobenius structure.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は北京からの中継です．）
2019年01月09日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
Laurent Berger 氏 (ENS de Lyon)
Formal groups and p-adic dynamical systems (ENGLISH)
Laurent Berger 氏 (ENS de Lyon)
Formal groups and p-adic dynamical systems (ENGLISH)
[ 講演概要 ]
A formal group gives rise to a p-adic dynamical system. I will discuss some results about formal groups that can be proved using this point of view. I will also discuss the theory of p-adic dynamical systems and some open questions.
A formal group gives rise to a p-adic dynamical system. I will discuss some results about formal groups that can be proved using this point of view. I will also discuss the theory of p-adic dynamical systems and some open questions.
2018年12月19日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Jean-Stefan Koskivirta 氏 (東京大学数理科学研究科)
Cohomology vanishing for automorphic vector bundles (ENGLISH)
Jean-Stefan Koskivirta 氏 (東京大学数理科学研究科)
Cohomology vanishing for automorphic vector bundles (ENGLISH)
[ 講演概要 ]
A Shimura variety carries naturally a family of vector bundles parametrized by the characters of a maximal torus in the attached group. We want to determine which of these vector bundles are ample, and also show cohomology vanishing results. For this we use generalized Hasse invariants on the stack of G-zips of Moonen-Pink-Wedhorn-Ziegler. It is a group-theoretical counterpart of the Shimura variety and carries a similar family of vector bundles. This is joint work with Y.Brunebarbe, W.Goldring and B.Stroh.
A Shimura variety carries naturally a family of vector bundles parametrized by the characters of a maximal torus in the attached group. We want to determine which of these vector bundles are ample, and also show cohomology vanishing results. For this we use generalized Hasse invariants on the stack of G-zips of Moonen-Pink-Wedhorn-Ziegler. It is a group-theoretical counterpart of the Shimura variety and carries a similar family of vector bundles. This is joint work with Y.Brunebarbe, W.Goldring and B.Stroh.
2018年12月12日(水)
18:00-19:00 数理科学研究科棟(駒場) 056号室
Gaëtan Chenevier 氏 (CNRS, Université Paris-Sud)
A higher weight (and automorphic) generalization of the Hermite-Minkowski theorem (ENGLISH)
Gaëtan Chenevier 氏 (CNRS, Université Paris-Sud)
A higher weight (and automorphic) generalization of the Hermite-Minkowski theorem (ENGLISH)
[ 講演概要 ]
I will show that for any integer N, there are only finitely many cuspidal algebraic automorphic representations of GL_m over Q whose Artin conductor is N and whose "weights" are in the interval {0,...,23} (with m varying). Via the conjectural yoga between geometric Galois representations (or motives) and algebraic automorphic forms, this statement may be viewed as a generalization of the classical Hermite-Minkowski theorem in algebraic number theory. I will also discuss variants of these results when the base field Q is replaced by an arbitrary number field.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回はパリからの中継です．）
I will show that for any integer N, there are only finitely many cuspidal algebraic automorphic representations of GL_m over Q whose Artin conductor is N and whose "weights" are in the interval {0,...,23} (with m varying). Via the conjectural yoga between geometric Galois representations (or motives) and algebraic automorphic forms, this statement may be viewed as a generalization of the classical Hermite-Minkowski theorem in algebraic number theory. I will also discuss variants of these results when the base field Q is replaced by an arbitrary number field.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回はパリからの中継です．）
2018年11月21日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
Yves André 氏 (Université Pierre et Marie Curie)
Poncelet games, confinement of algebraic integers, and hyperbolic Ax-Schanuel (ENGLISH)
Yves André 氏 (Université Pierre et Marie Curie)
Poncelet games, confinement of algebraic integers, and hyperbolic Ax-Schanuel (ENGLISH)
[ 講演概要 ]
We shall theorize and exemplify the problem of torsion values of sections of abelian schemes. This « unlikely intersection problem », which arises in various diophantine and algebro-geometric contexts, can be reformulated in a non-trivial way in terms of Kodaira-Spencer maps. A key tool toward its general solution is then provided by recent theorems of Ax-Schanuel type (joint work with P. Corvaja, U. Zannier, and partly Z. Gao).
We shall theorize and exemplify the problem of torsion values of sections of abelian schemes. This « unlikely intersection problem », which arises in various diophantine and algebro-geometric contexts, can be reformulated in a non-trivial way in terms of Kodaira-Spencer maps. A key tool toward its general solution is then provided by recent theorems of Ax-Schanuel type (joint work with P. Corvaja, U. Zannier, and partly Z. Gao).
2018年11月14日(水)
18:00-19:00 数理科学研究科棟(駒場) 056号室
斎藤秀司 氏 (東京大学数理科学研究科)
A motivic construction of ramification filtrations (ENGLISH)
斎藤秀司 氏 (東京大学数理科学研究科)
A motivic construction of ramification filtrations (ENGLISH)
[ 講演概要 ]
We give a new interpretation of Artin conductors of characters in the framework of theory of motives with modulus. It gives a unified way to understand Artin conductors of characters and irregularities of line bundle with integrable connections as well as overconvergent F-isocrystals of rank 1. It also gives rise to new conductors, for example, for G-torsors with G a finite flat group scheme, which specializes to the classical Artin conductor in case G = Z/nZ. We also give a motivic proof of a theorem of Kato and Matsuda on the determination of Artin conductors along divisors on smooth schemes by its restrictions to curves. Its proof is based on a motivic version of a theorem of Gabber-Katz. This is a joint work with Kay Rülling.
We give a new interpretation of Artin conductors of characters in the framework of theory of motives with modulus. It gives a unified way to understand Artin conductors of characters and irregularities of line bundle with integrable connections as well as overconvergent F-isocrystals of rank 1. It also gives rise to new conductors, for example, for G-torsors with G a finite flat group scheme, which specializes to the classical Artin conductor in case G = Z/nZ. We also give a motivic proof of a theorem of Kato and Matsuda on the determination of Artin conductors along divisors on smooth schemes by its restrictions to curves. Its proof is based on a motivic version of a theorem of Gabber-Katz. This is a joint work with Kay Rülling.
2018年10月10日(水)
18:00-19:00 数理科学研究科棟(駒場) 056号室
Yichao Tian 氏 (Université de Strasbourg)
Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives (ENGLISH)
Yichao Tian 氏 (Université de Strasbourg)
Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives (ENGLISH)
[ 講演概要 ]
In my talk, I will report on my ongoing collaborating project together with Yifeng Liu, Liang Xiao, Wei Zhang, and Xinwen Zhu, which concerns the rank 0 case of the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin--Selberg motives for GL(n) x GL(n+1). I will state the main results with some examples coming from elliptic curves, sketch the strategy of the proof, and then focus on the key geometric ingredients, namely the semi-stable reduction of unitary Shimura varieties of type U(1,n) at non-quasi-split places.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回はパリからの中継です．）
In my talk, I will report on my ongoing collaborating project together with Yifeng Liu, Liang Xiao, Wei Zhang, and Xinwen Zhu, which concerns the rank 0 case of the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin--Selberg motives for GL(n) x GL(n+1). I will state the main results with some examples coming from elliptic curves, sketch the strategy of the proof, and then focus on the key geometric ingredients, namely the semi-stable reduction of unitary Shimura varieties of type U(1,n) at non-quasi-split places.
（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回はパリからの中継です．）
2018年06月20日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
長町一平 氏 (東京大学数理科学研究科)
Criteria for good reduction of hyperbolic polycurves (JAPANESE)
長町一平 氏 (東京大学数理科学研究科)
Criteria for good reduction of hyperbolic polycurves (JAPANESE)
[ 講演概要 ]
We give good reduction criteria for hyperbolic polycurves, i.e., successive extensions of families of curves, under mild assumption. These criteria are higher dimensional versions of the good reduction criterion for hyperbolic curves given by Oda and Tamagawa. In this talk, we construct homotopy exact sequences by using intermediate quotient groups of geometric etale fundamental groups of hyperbolic polycurves.
We give good reduction criteria for hyperbolic polycurves, i.e., successive extensions of families of curves, under mild assumption. These criteria are higher dimensional versions of the good reduction criterion for hyperbolic curves given by Oda and Tamagawa. In this talk, we construct homotopy exact sequences by using intermediate quotient groups of geometric etale fundamental groups of hyperbolic polycurves.
2018年06月06日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Nicolas Templier 氏 (Cornell University)
On the Ramanujan conjecture for automorphic forms over function fields
Nicolas Templier 氏 (Cornell University)
On the Ramanujan conjecture for automorphic forms over function fields
[ 講演概要 ]
Let G be a reductive group over a function field of large enough characteristic. We prove the temperedness at unramified places of automorphic representations of G, subject to a local assumption at one place, stronger than supercuspidality. Such an assumption is necessary, as was first shown by Saito-Kurokawa and Howe-Piatetskii-Shapiro in the 70's. Our method relies on the l-adic geometry of Bun_G, and on trace formulas. Work with Will Sawin.
（本講演は「東京北京パリ数論幾何セミナー」として, インターネットによる東大数理, Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回はパリからの中継です．）
Let G be a reductive group over a function field of large enough characteristic. We prove the temperedness at unramified places of automorphic representations of G, subject to a local assumption at one place, stronger than supercuspidality. Such an assumption is necessary, as was first shown by Saito-Kurokawa and Howe-Piatetskii-Shapiro in the 70's. Our method relies on the l-adic geometry of Bun_G, and on trace formulas. Work with Will Sawin.
（本講演は「東京北京パリ数論幾何セミナー」として, インターネットによる東大数理, Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回はパリからの中継です．）
2018年05月30日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
竹内大智 氏 (東京大学数理科学研究科)
Blow-ups and the class field theory for curves (JAPANESE)
竹内大智 氏 (東京大学数理科学研究科)
Blow-ups and the class field theory for curves (JAPANESE)
[ 講演概要 ]
幾何学的類体論とは、有限体上一変数代数関数体に対する類体論の、係数が一般の完全体の場合への拡張であり、M. Rosenlichtにより証明された定理である。一方1980年代、P. Deligneにより、順分岐の場合の別証明が見いだされた。それは曲線の対称積が、そのJacobi多様体上の射影（或いはアフィン）空間束になることを用いるものである。本講演では対称積のブローアップを考えることで、一般の分岐の場合でも類似の方法で証明できることを説明する。
幾何学的類体論とは、有限体上一変数代数関数体に対する類体論の、係数が一般の完全体の場合への拡張であり、M. Rosenlichtにより証明された定理である。一方1980年代、P. Deligneにより、順分岐の場合の別証明が見いだされた。それは曲線の対称積が、そのJacobi多様体上の射影（或いはアフィン）空間束になることを用いるものである。本講演では対称積のブローアップを考えることで、一般の分岐の場合でも類似の方法で証明できることを説明する。
