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過去の記録 ～10/06｜次回の予定｜今後の予定 10/07～
| 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
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。
2018年05月09日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Sug Woo Shin 氏 (University of California, Berkeley)
Endoscopy and cohomology of U(n-1,1) (ENGLISH)
Sug Woo Shin 氏 (University of California, Berkeley)
Endoscopy and cohomology of U(n-1,1) (ENGLISH)
[ 講演概要 ]
We apply the endoscopic classification of automorphic representations for inner forms of unitary groups to bound the growth of cohomology in congruence towers of locally symmetric spaces associated with U(n-1,1). Our bound is sharper than the bound predicted by Sarnak-Xue for general locally symmetric spaces. This is joint work with Simon Marshall.
（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．）
We apply the endoscopic classification of automorphic representations for inner forms of unitary groups to bound the growth of cohomology in congruence towers of locally symmetric spaces associated with U(n-1,1). Our bound is sharper than the bound predicted by Sarnak-Xue for general locally symmetric spaces. This is joint work with Simon Marshall.
（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．）
2018年04月18日(水)
16:00-17:00 数理科学研究科棟(駒場) 002号室
Ildar Gaisin 氏 (東京大学数理科学研究科)
Fargues' conjecture in the GL_2-case (ENGLISH)
Ildar Gaisin 氏 (東京大学数理科学研究科)
Fargues' conjecture in the GL_2-case (ENGLISH)
[ 講演概要 ]
Recently Fargues announced a conjecture which attempts to geometrize the (classical) local Langlands correspondence. Just as in the geometric Langlands story, there is a stack of G-bundles and a Hecke stack which one can define. The conjecture is based on some conjectural objects, however for a cuspidal Langlands parameter and a minuscule cocharacter, we can define every object in the conjecture, assuming only the local Langlands correspondence. We study the geometry of the non-semi-stable locus in the Hecke stack and as an application we will show the Hecke eigensheaf property of Fargues conjecture holds in the GL_2-case and a cuspidal Langlands parameter. This is joint work with Naoki Imai.
Recently Fargues announced a conjecture which attempts to geometrize the (classical) local Langlands correspondence. Just as in the geometric Langlands story, there is a stack of G-bundles and a Hecke stack which one can define. The conjecture is based on some conjectural objects, however for a cuspidal Langlands parameter and a minuscule cocharacter, we can define every object in the conjecture, assuming only the local Langlands correspondence. We study the geometry of the non-semi-stable locus in the Hecke stack and as an application we will show the Hecke eigensheaf property of Fargues conjecture holds in the GL_2-case and a cuspidal Langlands parameter. This is joint work with Naoki Imai.
2018年04月18日(水)
17:10-18:10 数理科学研究科棟(駒場) 002号室
阿部紀行 氏 (東京大学数理科学研究科)
p進代数群の法p表現とHecke環 (JAPANESE)
阿部紀行 氏 (東京大学数理科学研究科)
p進代数群の法p表現とHecke環 (JAPANESE)
[ 講演概要 ]
p進代数群の，標数pの体上における表現（法p表現）について，付随するHecke環の表現論の関わりとともにお話をします．これは，G. Henniart，F. HerzigおよびM.-F. Vignérasとの共同研究に基づきます．
p進代数群の，標数pの体上における表現（法p表現）について，付随するHecke環の表現論の関わりとともにお話をします．これは，G. Henniart，F. HerzigおよびM.-F. Vignérasとの共同研究に基づきます．
2018年04月11日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Minhyong Kim 氏 (University of Oxford)
Non-abelian cohomology and Diophantine geometry (ENGLISH)
Minhyong Kim 氏 (University of Oxford)
Non-abelian cohomology and Diophantine geometry (ENGLISH)
[ 講演概要 ]
This lecture will review the construction of moduli schemes of torsors for sheaves of pro-unipotent groups and their applications to the resolution of Diophantine problems.
（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．）
This lecture will review the construction of moduli schemes of torsors for sheaves of pro-unipotent groups and their applications to the resolution of Diophantine problems.
（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01月17日(水)
18:00-19:00 数理科学研究科棟(駒場) 002号室
Ana Caraiani 氏 (Imperial College)
On the vanishing of cohomology for certain Shimura varieties (ENGLISH)
Ana Caraiani 氏 (Imperial College)
On the vanishing of cohomology for certain Shimura varieties (ENGLISH)
[ 講演概要 ]
I will prove that the compactly supported cohomology of certain unitary or symplectic Shimura varieties at level Gamma_1(p^\infty) vanishes above the middle degree. The key ingredients come from p-adic Hodge theory and studying the Bruhat decomposition on the Hodge-Tate flag variety. I will describe the steps in the proof using modular curves as a toy model. I will also mention an application to Galois representations for torsion classes in the cohomology of locally symmetric spaces for GL_n. This talk is based on joint work in preparation with D. Gulotta, C.Y. Hsu, C. Johansson, L. Mocz, E. Reineke, and S.C. Shih.
（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 prove that the compactly supported cohomology of certain unitary or symplectic Shimura varieties at level Gamma_1(p^\infty) vanishes above the middle degree. The key ingredients come from p-adic Hodge theory and studying the Bruhat decomposition on the Hodge-Tate flag variety. I will describe the steps in the proof using modular curves as a toy model. I will also mention an application to Galois representations for torsion classes in the cohomology of locally symmetric spaces for GL_n. This talk is based on joint work in preparation with D. Gulotta, C.Y. Hsu, C. Johansson, L. Mocz, E. Reineke, and S.C. Shih.
（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．）
2017年12月13日(水)
18:00-19:00 数理科学研究科棟(駒場) 056号室
Javier Fresán 氏 (École polytechnique)
Exponential motives (ENGLISH)
Javier Fresán 氏 (École polytechnique)
Exponential motives (ENGLISH)
[ 講演概要 ]
What motives are to algebraic varieties, exponential motives are to pairs (X, f) consisting of an algebraic variety over some field k and a regular function f on X. In characteristic zero, one is naturally led to define the de Rham and rapid decay cohomology of such pairs when dealing with numbers like the special values of the gamma function or the Euler constant gamma which are not expected to be periods in the usual sense. Over finite fields, the étale and rigid cohomology groups of (X, f) play a pivotal role in the study of exponential sums.
Following ideas of Katz, Kontsevich, and Nori, we construct a Tannakian category of exponential motives when k is a subfield of the complex numbers. This allows one to attach to exponential periods a Galois group that conjecturally governs all algebraic relations among them. The category is equipped with a Hodge realisation functor with values in mixed Hodge modules over the affine line and, if k is a number field, with an étale realisation related to exponential sums. This is a joint work with Peter Jossen (ETH).
（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．）
What motives are to algebraic varieties, exponential motives are to pairs (X, f) consisting of an algebraic variety over some field k and a regular function f on X. In characteristic zero, one is naturally led to define the de Rham and rapid decay cohomology of such pairs when dealing with numbers like the special values of the gamma function or the Euler constant gamma which are not expected to be periods in the usual sense. Over finite fields, the étale and rigid cohomology groups of (X, f) play a pivotal role in the study of exponential sums.
Following ideas of Katz, Kontsevich, and Nori, we construct a Tannakian category of exponential motives when k is a subfield of the complex numbers. This allows one to attach to exponential periods a Galois group that conjecturally governs all algebraic relations among them. The category is equipped with a Hodge realisation functor with values in mixed Hodge modules over the affine line and, if k is a number field, with an étale realisation related to exponential sums. This is a joint work with Peter Jossen (ETH).
（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．）
2017年11月08日(水)
18:00-19:00 数理科学研究科棟(駒場) 056号室
Xin Wan 氏 (Morningside Center for Mathematics)
Iwasawa theory and Bloch-Kato conjecture for modular forms (ENGLISH)
Xin Wan 氏 (Morningside Center for Mathematics)
Iwasawa theory and Bloch-Kato conjecture for modular forms (ENGLISH)
[ 講演概要 ]
Bloch and Kato formulated conjectures relating sizes of p-adic Selmer groups with special values of L-functions. Iwasawa theory is a useful tool for studying these conjectures and BSD conjecture for elliptic curves. For example the Iwasawa main conjecture for modular forms formulated by Kato implies the Tamagawa number formula for modular forms of analytic rank 0.
In this talk I'll first briefly review the above theory. Then we will focus on a different Iwasawa theory approach for this problem. The starting point is a recent joint work with Jetchev and Skinner proving the BSD formula for elliptic curves of analytic rank 1. We will discuss how such results are generalized to modular forms. If time allowed we may also explain the possibility to use it to deduce Bloch-Kato conjectures in both analytic rank 0 and 1 cases. In certain aspects such approach should be more powerful than classical Iwasawa theory, and has some potential to attack cases with bad ramification at p.
（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．）
Bloch and Kato formulated conjectures relating sizes of p-adic Selmer groups with special values of L-functions. Iwasawa theory is a useful tool for studying these conjectures and BSD conjecture for elliptic curves. For example the Iwasawa main conjecture for modular forms formulated by Kato implies the Tamagawa number formula for modular forms of analytic rank 0.
In this talk I'll first briefly review the above theory. Then we will focus on a different Iwasawa theory approach for this problem. The starting point is a recent joint work with Jetchev and Skinner proving the BSD formula for elliptic curves of analytic rank 1. We will discuss how such results are generalized to modular forms. If time allowed we may also explain the possibility to use it to deduce Bloch-Kato conjectures in both analytic rank 0 and 1 cases. In certain aspects such approach should be more powerful than classical Iwasawa theory, and has some potential to attack cases with bad ramification at p.
（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．）
2017年10月11日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Michael Temkin 氏 (The Hebrew University of Jerusalem)
Logarithmic resolution of singularities (ENGLISH)
Michael Temkin 氏 (The Hebrew University of Jerusalem)
Logarithmic resolution of singularities (ENGLISH)
[ 講演概要 ]
The famous Hironaka's theorem asserts that any integral algebraic variety X of characteristic zero can be modified to a smooth variety X_res by a sequence of blowings up. Later it was shown that one can make this compatible with smooth morphisms Y --> X in the sense that Y_res --> Y is the pullback of X_res --> X. In a joint project with D. Abramovich and J. Wlodarczyk, we construct a new algorithm which is compatible with all log smooth morphisms (e.g. covers ramified along exceptional divisors). We expect that this algorithm will naturally extend to an algorithm of resolution of morphisms to log smooth ones. In particular, this should lead to functorial semistable reduction theorems. In my talk I will tell about main ideas of the classical algorithm and will then discuss logarithmic and stack-theoretic modifications we had to make in the new algorithm.
The famous Hironaka's theorem asserts that any integral algebraic variety X of characteristic zero can be modified to a smooth variety X_res by a sequence of blowings up. Later it was shown that one can make this compatible with smooth morphisms Y --> X in the sense that Y_res --> Y is the pullback of X_res --> X. In a joint project with D. Abramovich and J. Wlodarczyk, we construct a new algorithm which is compatible with all log smooth morphisms (e.g. covers ramified along exceptional divisors). We expect that this algorithm will naturally extend to an algorithm of resolution of morphisms to log smooth ones. In particular, this should lead to functorial semistable reduction theorems. In my talk I will tell about main ideas of the classical algorithm and will then discuss logarithmic and stack-theoretic modifications we had to make in the new algorithm.
2017年09月27日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
加藤和也 氏 (University of Chicago)
Height functions for motives, Hodge analogues, and Nevanlinna analogues (ENGLISH)
加藤和也 氏 (University of Chicago)
Height functions for motives, Hodge analogues, and Nevanlinna analogues (ENGLISH)
[ 講演概要 ]
We compare height functions for (1) points of an algebraic variety over a number field, (2) motives over a number field, (3) variations of Hodge structure with log degeneration on a projective smooth curve over the complex number field, (4) horizontal maps from the complex plane C to a toroidal partial compactification of the period domain. Usual Nevanlinna theory uses height functions for (5) holomorphic maps f from C to a compactification of an agebraic variety V and considers how often the values of f lie outside V. Vojta compares (1) and (5). In (4), V is replaced by a period domain. The comparisons of (1)--(4) provide many new questions to study.
We compare height functions for (1) points of an algebraic variety over a number field, (2) motives over a number field, (3) variations of Hodge structure with log degeneration on a projective smooth curve over the complex number field, (4) horizontal maps from the complex plane C to a toroidal partial compactification of the period domain. Usual Nevanlinna theory uses height functions for (5) holomorphic maps f from C to a compactification of an agebraic variety V and considers how often the values of f lie outside V. Vojta compares (1) and (5). In (4), V is replaced by a period domain. The comparisons of (1)--(4) provide many new questions to study.
2017年06月14日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Yongquan Hu 氏 (Chinese Academy of Sciences, Morningside Center of Mathematics)
Multiplicity one for the mod p cohomology of Shimura curves (ENGLISH)
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~t-saito/title_Hu.pdf
Yongquan Hu 氏 (Chinese Academy of Sciences, Morningside Center of Mathematics)
Multiplicity one for the mod p cohomology of Shimura curves (ENGLISH)
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~t-saito/title_Hu.pdf
2017年05月31日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
坂本龍太郎 氏 (東京大学数理科学研究科)
Stark Systems over Gorenstein Rings (JAPANESE)
坂本龍太郎 氏 (東京大学数理科学研究科)
Stark Systems over Gorenstein Rings (JAPANESE)
[ 講演概要 ]
Gorenstein環上の代数体のGalois表現とSelmer構造に対するStark系の定義について紹介する．
これは佐野昂迪氏とBarry Mazur氏，Karl Rubin氏によって独立に定義された単項イデアル環上のStark系の一般化になっている．
さらに，Stark系の成す加群が階数1の自由加群である事，stark系を用いてSelmer群のFittingイデアル全てを記述できる事を示す．
Gorenstein環上の代数体のGalois表現とSelmer構造に対するStark系の定義について紹介する．
これは佐野昂迪氏とBarry Mazur氏，Karl Rubin氏によって独立に定義された単項イデアル環上のStark系の一般化になっている．
さらに，Stark系の成す加群が階数1の自由加群である事，stark系を用いてSelmer群のFittingイデアル全てを記述できる事を示す．
2017年05月17日(水)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Olivier Fouquet 氏 (Université Paris-Sud)
The Equivariant Tamagawa Number Conjecture for modular motives with coefficients in Hecke algebras (ENGLISH)
Olivier Fouquet 氏 (Université Paris-Sud)
The Equivariant Tamagawa Number Conjecture for modular motives with coefficients in Hecke algebras (ENGLISH)
[ 講演概要 ]
The Equivariant Tamagawa Number Conjecture (ETNC) of Kato is an awe-inspiring web of conjectures predicting the special values of L-functions of motives as well as their behaviors under the action of algebras acting on motives. In this talk, I will explain the statement of the ETNC with coefficients in Hecke algebras for motives attached to modular forms, show some consequences in Iwasawa theory and outline a proof (under mild hypotheses on the residual representation) using a combination of the methods of Euler and Taylor-Wiles systems.
The Equivariant Tamagawa Number Conjecture (ETNC) of Kato is an awe-inspiring web of conjectures predicting the special values of L-functions of motives as well as their behaviors under the action of algebras acting on motives. In this talk, I will explain the statement of the ETNC with coefficients in Hecke algebras for motives attached to modular forms, show some consequences in Iwasawa theory and outline a proof (under mild hypotheses on the residual representation) using a combination of the methods of Euler and Taylor-Wiles systems.
2017年05月10日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
加藤大輝 氏 (東京大学数理科学研究科)
Wild ramification and restrictions to curves (JAPANESE)
加藤大輝 氏 (東京大学数理科学研究科)
Wild ramification and restrictions to curves (JAPANESE)
[ 講演概要 ]
スキーム上のエタール層の暴分岐がすべての曲線への制限のArtin導手で決まるかどうかを調べ、特異点解消を仮定するとそれが正しいこと、特に、スキームが2次元の場合には正しいことを示した。
またその帰結として、(次元に関する仮定なしに)体上の多様体のエタール層のEuler－Poincare標数や、局所体上の多様体のエタール層から定まるGalois表現のSwan導手の交代和もすべての"曲線"への制限のArtin導手で決まるという結果を得た。
スキーム上のエタール層の暴分岐がすべての曲線への制限のArtin導手で決まるかどうかを調べ、特異点解消を仮定するとそれが正しいこと、特に、スキームが2次元の場合には正しいことを示した。
またその帰結として、(次元に関する仮定なしに)体上の多様体のエタール層のEuler－Poincare標数や、局所体上の多様体のエタール層から定まるGalois表現のSwan導手の交代和もすべての"曲線"への制限のArtin導手で決まるという結果を得た。
2017年04月12日(水)
17:00-18:00 数理科学研究科棟(駒場) 056号室
跡部発 氏 (東京大学数理科学研究科)
A conjecture of Gross-Prasad and Rallis for metaplectic groups (JAPANESE)
跡部発 氏 (東京大学数理科学研究科)
A conjecture of Gross-Prasad and Rallis for metaplectic groups (JAPANESE)
[ 講演概要 ]
p-進簡約代数群の既約スムース表現が generic であるとは、それが Whittaker 模型を持つ時に言う。
Whittaker 模型の一意性により、generic 表現は表現論及び数論の両分野で多くの応用を持つ。
一方で、局所 Langlands 予想 (LLC) は既約スムース表現を L-パラメーターで分類する。
Gross-Prasad は Rallis に触発されて、generic 表現に対応する L-パラメーターの判定法を予想した。
これを Gross-Prasad と Rallis の予想 (GPR) という。
近年、古典群に関して (GPR) は Gan-Ichino により証明された。
本講演では、シンプレクティック群の二重被覆であるメタプレクティック群に関して (GPR) を議論する。
p-進簡約代数群の既約スムース表現が generic であるとは、それが Whittaker 模型を持つ時に言う。
Whittaker 模型の一意性により、generic 表現は表現論及び数論の両分野で多くの応用を持つ。
一方で、局所 Langlands 予想 (LLC) は既約スムース表現を L-パラメーターで分類する。
Gross-Prasad は Rallis に触発されて、generic 表現に対応する L-パラメーターの判定法を予想した。
これを Gross-Prasad と Rallis の予想 (GPR) という。
近年、古典群に関して (GPR) は Gan-Ichino により証明された。
本講演では、シンプレクティック群の二重被覆であるメタプレクティック群に関して (GPR) を議論する。
2017年04月11日(火)
17:30-18:30 数理科学研究科棟(駒場) 117号室
Peter Scholze 氏 (University of Bonn)
The geometric Satake equivalence in mixed characteristic (ENGLISH)
Peter Scholze 氏 (University of Bonn)
The geometric Satake equivalence in mixed characteristic (ENGLISH)
[ 講演概要 ]
In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa Q_p. These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec Z. The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting.
In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa Q_p. These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec Z. The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting.
2017年03月30日(木)
16:40-17:40 数理科学研究科棟(駒場) 056号室
いつもと曜日が異なりますのでご注意ください．
Haoyu Hu 氏 (東京大学数理科学研究科)
Logarithmic ramifications via pull-back to curves (English)
いつもと曜日が異なりますのでご注意ください．
Haoyu Hu 氏 (東京大学数理科学研究科)
Logarithmic ramifications via pull-back to curves (English)
[ 講演概要 ]
Let X be a smooth variety over a perfect field of characteristic p>0, D a strict normal crossing divisor of X, U the complement of D in X, j:U—>X the canonical injection, and F a locally constant and constructible sheaf of F_l-modules on U (l is a prime number different from p). Using Abbes and Saito’s logarithmic ramification theory, we define a Swan divisor SW(j_!F), which supported on D. Let i:C-->X be a quasi-finite morphism from a smooth curve C to X. Following T. Saito’s idea, we compare the pull-back of SW(j_!F) to C with the Swan divisor of the pull-back of j_!F to C. It answers an expectation of Esnault and Kerz and generalizes the same result of Barrientos for rank 1 sheaves. As an application, we obtain a lower semi-continuity property for Swan divisors of an l-adic sheaf on a smooth fibration, which gives a generalization of Deligne and Laumon’s lower semi-continuity property of Swan conductors of l-adic sheaves on relative curves to higher relative dimensions. This application is a supplement of the semi-continuity of total dimension of vanishing cycles due to T. Saito and the lower semi-continuity of total dimension divisors due to myself and E. Yang.
Let X be a smooth variety over a perfect field of characteristic p>0, D a strict normal crossing divisor of X, U the complement of D in X, j:U—>X the canonical injection, and F a locally constant and constructible sheaf of F_l-modules on U (l is a prime number different from p). Using Abbes and Saito’s logarithmic ramification theory, we define a Swan divisor SW(j_!F), which supported on D. Let i:C-->X be a quasi-finite morphism from a smooth curve C to X. Following T. Saito’s idea, we compare the pull-back of SW(j_!F) to C with the Swan divisor of the pull-back of j_!F to C. It answers an expectation of Esnault and Kerz and generalizes the same result of Barrientos for rank 1 sheaves. As an application, we obtain a lower semi-continuity property for Swan divisors of an l-adic sheaf on a smooth fibration, which gives a generalization of Deligne and Laumon’s lower semi-continuity property of Swan conductors of l-adic sheaves on relative curves to higher relative dimensions. This application is a supplement of the semi-continuity of total dimension of vanishing cycles due to T. Saito and the lower semi-continuity of total dimension divisors due to myself and E. Yang.
