東京北京パリ数論幾何セミナー（終わったもの） これからのもの

毎月第2水曜　 056号室 　午後５時半から６時半 (パリが夏時間のとき)　 午後６時から７時　(パリが冬時間のとき)　
インターネットで、東大数理とIHES とMorningside centerで双方向同時中継したものです。

オーガナイザー： 志甫　淳、辻　雄、斎藤　毅、 Ahmed Abbes (CNRS, IHES), Fabrice Orgogozo (CNRS, Ecole Polytechnique), 田　一超(Tian Yichao, Morningside center)、 鄭 維哲(Zheng Weizhe, Morningside center)
2020年5月27日(水)　 Zoomによるオンライン・セミナーなので 事前登録 が必要です。

坂内健一 （慶応大学） Kenichi Bannai (Keio University/RIKEN)
Shintani generating class and the p-adic polylogarithm for totally real fields.

In this talk, we will give a new interpretation of Shintani's work concerning the generating function of nonpositive values of Hecke $L$-functions for totally real fields. In particular, we will construct a canonical class, which we call the Shintani generating class, in the cohomology of a certain quotient stack of an infinite direct sum of algebraic tori associated with a fixed totally real field. Using our observation that cohomology classes, not functions, play an important role in the higher dimensional case, we proceed to newly define the p-adic polylogarithm function in this case, and investigate its relation to the special value of p-adic Hecke $L$-functions. Some observations concerning the quotient stack will also be discussed. This is a joint work with Kei Hagihara, Kazuki Yamada, and Shuji Yamamoto.
2020年5月13日(水)　 Zoomによるオンライン・セミナーなので 事前登録 が必要です。

Yifeng Liu (Yale University)
On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives

In this talk, we will explain the final outcome on the Beilinson-Bloch-Kato conjecture for motives coming from certain automorphic representations of GL(n) x GL(n+1), of our recent project with Yichao Tian, Liang Xiao, Wei Zhang, and Xinwen Zhu. In particular, we show that the nonvanishing of the central L-value of the motive implies the vanishing of the corresponding Bloch-Kato Selmer group. We will also explain the main ideas and ingredients of the proof.
2020年4月22日(水)　17時30分から

Arthur-César Le Bras (CNRS & Université Paris 13)
Prismatic Dieudonné theory

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.
2019年11月20日(水)　18時00分から

Vasudevan Srinivas (Tata Institute of Fundamental Research)
Algebraic versus topological entropy for surfaces over finite fields

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 H'el`ene Esnault, and some subsequent work of K. Shuddhodan.
2019年10月16日(水)　17時30分から　117

Liang Xiao 肖梁 (BICMR, Peking University)
On slopes of modular forms

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.
2019年6月5日(水)　17時30分から

服部　新　(東京都市大学)
Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms

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4月24日(水)　17時30分から

Joseph Ayoub (University of Zurich):
P^1-localisation and a possible definition of arithmetic Kodaira-Spencer classes

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 \UTF{00E0} 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4月10日(水)　17時30分から

Zongbin Chen (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

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.
2019年1月16日(水)　18時00分から

Lei Fu 扶磊　(Yau Mathematical Sciences Center, Tsinghua University)
p-adic Gelfand-Kapranov-Zelevinsky systems

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.
2018年12月12日(水)　18時00分から

G. Chenevier (CNRS, Universit\UTF{00E9} Paris-Sud)
A higher weight (and automorphic) generalization of the Hermite-Minkowski theorem

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.
2018年11月14日(水)　18時00分から

斎藤　秀司（東京大学）
A motivic construction of ramification filtrations

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 Ru\CID{1}lling.
2018年10月10日(水)　18時00分から

Yichao Tian 田　一超(Universite de Strasbourg)
Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives

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.
2018年6月6日(水)　17時30分から

Nicolas Templier (Cornell University)
On the Ramanujan conjecture for automorphic forms over function fields

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.
2018年5月9日(水)　17時30分から

Sug Woo Shin (University of California, Berkeley)
Endoscopy and cohomology of U(n-1,1)

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.
2018年4月11日(水)　17時30分から

Minhyong Kim (University of Oxford)
Non-abelian cohomology and Diophantine geometry

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.
2018年1月17日(水)　18時

Ana Caraiani (Imperial College)
On the vanishing of cohomology for certain Shimura varieties

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.
2017年12月13日(水)　18時

J. Fresán (Ecole Polytechnique)
Exponential motives

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 ¥UTF{00E9}tale 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 ¥UTF{00E9}tale realisation related to exponential sums. This is a joint work with Peter Jossen (ETH).
2017年11月8日(水)　18時

X. Wan (万¥UTF{6615}) (Morningside Center for Mathematics(中国科学院晨¥UTF{5174}数学中心))
Iwasawa theory and Bloch-Kato conjecture for modular forms

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.
2017年10月11日(水)　17時半

M. Temkin (The Hebrew University of Jerusalem)
Logarithmic resolution of singularities

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9月27日(水)　17時半

加藤和也（シカゴ大学）
Height functions for motives, Hodge analogues, and Nevanlinna analogues.

Abstract. 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6月14日(水)

Hu Yongquan (胡永泉) (Morningside center of Mathematics 中国科学院晨光数学中心)
Multiplicity one for the mod p cohomology of Shimura curves

abstract
2017年5月17日(水)

Olivier Fouquet (Université Paris-Sud)
The Equivariant Tamagawa Number Conjecture for modular motives with coefficients in Hecke algebras

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4月11日(火) 夏時間　１１７ いつもと曜日と部屋が違います。

Peter Scholze (Bonn)
The geometric Satake equivalence in mixed characteristic

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1月11日(水)

扶磊（Fu Lei） (清華大学)
Deformation and rigidity of $¥ell$-adic sheaves

Let $X$ be a smooth connected algebraic curve over an algebraically closed field, let $S$ be a finite closed subset in $X$, and let $¥mathcal F_0$ be a lisse $¥ell$-torsion sheaf on $X-S$. We study the deformation of $¥mathcal F_0$. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $¥overline{¥mathbb Q}_¥ell$-sheaf $¥mathcal F$ is irreducible and physically rigid, then it is cohomologically rigid in the sense that $¥chi(X,j_¥ast¥mathcal End(¥mathcal F))=2$, where $j:X-S¥to X$ is the open immersion.
2016年12月14日(水)

Luc Illusie (Orsay)
On vanishing cycles and duality, after A. Beilinson

It was proved by Gabber in the early 1980's that R¥Psi commutes with duality, and that R¥Phi preserves perversity up to shift. It had been in the folklore since then that this last result was in fact a consequence of a finer one, namely the compatibility of R¥Phi with duality. In this talk I'll give a proof of this, using a method explained to me by A. Beilinson.
2016年11月9日(水)

Emmanuel Ullmo (IHES)
Flows on Abelian Varieties and Shimura Varieties

I will discuss several questions and some results about algebraic flows, o-minimal flows and holomorphic flows on abelian varieties and Shimura varieties.
2016年11月2日(水)

Y. Andre (CNRS, Institut de Mathematiques de Jussieu)
Direct summand conjecture and perfectoid Abhyankar lemma: an overview

According to Hochster's direct summand conjecture (1973), a regular ring R is a direct summand, as an R-module, of every finite extension ring. We shall outline our recent proof which relies on perfectoid techniques. Similar arguments also establish the existence of big Cohen-Macaulay algebras for complete local domains of mixed characteristics.
2016年10月12日(水)

Uwe Jannsen (Regensburg/東京大学)
Filtered de Rham Witt complexes and wildly ramified higher class field theory over finite fields (joint work with Shuji Saito and Yigeng Zhao)

We will consider abelian coverings of smooth projective varieties over finite fields which are wildly ramified along a divisor D with normal crossings, and will describe the corresponding abelianized fundamental group via modified logarithmic de Rham-Witt sheaves.
6月8日(水)

Xu Shen(申旭) (Morningside Center of Mathematics)
Local and global geometric structures of perfectoid Shimura varieties


In this talk, we will investigate some geometric structural properties of perfectoid Shimura varieties of abelian type. In the global part, we will construct some minimal and toroidal type compactifications for these spaces, and we will describe explicitly the degeneration of Hodge-Tate period map at the boundaries. In the local part, we will show that each Newton stratum of these perfectoid Shimura varieties can be described by the related (generalized) Rapoport-Zink space and Igusa variety. As a consequence of our local and global constructions, we can compute the stalks of the relative cohomology under the Hodge-Tate period map of the intersection complex (on the minimal compactification), in terms of cohomology of Igusa varieties at the boundary with truncated coefficients.
5月11日(水)

W. Niziol (CNRS & ENS de Lyon)
Syntomic complexes and p-adic nearby cycles

I will present a proof of a comparison isomorphism, up to some universal constants, between truncated sheaves of p-adic nearby cycles and syntomic cohomology sheaves on semistable schemes over a mixed characteristic local rings. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. This is a joint work with Pierre Colmez.
2016年4月13日(水)

玉川　安騎男（京大数理研）
Semisimplicity of geometric monodromy on etale cohomology (joint work with Anna Cadoret and Chun Yin Hui)

Let K be a function field over an algebraically closed field of characteritic p ¥geq 0, X a proper smooth K-scheme, and l a prime distinct from p. Deligne proved that the Q_l-coefficient etale cohomology groups of the geometric fiber of X --> K are always semisimple as G_K-modules. In this talk, we consider a similar problem for the F_l-coefficient etale cohomology groups. Among other things, we show that if p=0 (resp. in general), they are semisimple for all but finitely many l's ( resp. for all l's in a set of density 1).
3月29日(火)

M. Morrow (Universitaet Bonn)
Motivic cohomology of formal schemes in characteristic p

The logarithmic Hodge-Witt sheaves of Illusie, Milne, Kato, et al. of a smooth variety in characteristic p provide a concrete realisation of its p-adic motivic cohomology, thanks to results of Geisser-Levine and Bloch-Kato-Gabber which link them to algebraic K-theory. I will explain an analogous theory for formal schemes, as well as applications to algebraic cycles, such as a weak Lefschetz theorem for formal Chow groups.
12月9日（水）6時00分--7時00分 056

Ted Chinburg (University of Pennsylvania & IHES):
Chern classes in Iwasawa theory

Many of the main conjectures in Iwasawa theory can be phrased as saying that the first Chern class of an Iwasawa module is generated by a p-adic L-series. In this talk I will describe how higher Chern classes pertain to the higher codimension behavior of Iwasawa modules. I'll then describe a template for conjectures which would link such higher Chern classes to elements in the K-theory of Iwasawa algebras which are constructed from tuples of Katz p-adic L-series. I will finally describe an instance in which a result of this kind, for the second Chern class of an unramified Iwasawa module, can be proved over an imaginary quadratic field. This is joint work with F. Bleher, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi and M. J. Taylor.
11月17日（火）6時00分--7時00分 117

Dennis Gaitsgory (Harvard & IHES):
The Tamagawa number formula over function fields.

Let G be a semi-simple and simply connected group and X an algebraic curve. We consider Bun_G(X), the moduli space of G-bundles on X. In their celebrated paper, Atiyah and Bott gave a formula for the cohomology of Bun_G, namely H^*(Bun_G)=Sym(H_*(X)¥otimes V), where V is the space of generators for H^*_G(pt). When we take our ground field to be a finite field, the Atiyah-Bott formula implies the Tamagawa number conjecture for the function field of X. The caveat here is that the A-B proof uses the interpretation of Bun_G as the space of connection forms modulo gauge transformations, and thus only works over complex numbers (but can be extend to any field of characteristic zero). In the talk we will outline an algebro-geometric proof that works over any ground field. As its main geometric ingredient, it uses the fact that {¥it the space of rational maps from X to G is homologically contractible.} Because of the nature of the latter statement, the proof necessarily uses tools from higher category theory. So, it can be regarded as an example how the latter can be used to prove something concrete: a construction at the level of 2-categories leads to an equality of numbers.
10月27日（火）6時00分--7時00分 002

朝倉政典氏（北海道大学）
On the period conjecture of Gross-Deligne for fibrations

The period conjecture of Gross-Deligne asserts that the periods of algebraic varieties with complex multiplication are products of values of the gamma function at rational numbers. This is proved for CM elliptic curves by Lerch-Chowla-Selberg, and for abelian varieties by Shimura-Deligne-Anderson. However the question in the general case is still open. In this talk, we verify an alternating variant of the period conjecture for the cohomology of fibrations with relative multiplication.The proof relies on the Saito-Terasoma product formula for epsilon factors of integrable regular singular connections and the Riemann-Roch-Hirzebruch theorem. This is a joint work with Javier Fresan.
5月20日（水）5時30分--6時30分

張寿武氏(Shouwu Zhang) (Princeton大学)
Colmez' conjecture in average

This is a report on a joint work with Xinyi Yuan on a conjectured formula of Colmez about the Faltings heights of CM abelian varieties. I will sketch a deduction of this formula in average of CM types from our early work on Gross8212Zagier formula. When combined with a recent work of Tsimerman, this result implies the Andre-Oort conjecture for the moduli of abelian varieties.
Our method is different than a recently announced proof of a weaker form of the average formula by Andreatta, Howard, Goren, and Madapusi Pera: we use neither high dimensional Shimura varieties nor Borcherds’ liftings.
4月8日（水）5時30分--6時30分

安田　正大氏 (大阪大学)
Integrality of p-adic multiple zeta values and application to finite multiple zeta values.

I will give a proof of an integrality of p-adic multiple zeta values. I would also like to explain how it can be applied to give an upper bound of the dimension of finite multiple zeta values.
1月21日（水）6時00分--6時10分

Luc Illusie (Universite de Paris-Sud)
Remembering the SGA's

6時10分--7時10分

Ofer Gabber (CNRS & IHES)
Spreading-out of rigid-analytic families and observations on p-adic Hodge theory

(Joint work with Brian Conrad.) Let K be a complete rank 1 valued field with ring of integers OK , A an adic noetherian ring and f : A → OK an adic morphism. If g : X → Y is a proper flat morphism between rigid analytic spaces over K then locally on Y a flat formal model of g spreads out to a proper flat morphism between formal schemes topologically of finite type over A. As an application one can prove that for proper smooth g and K of characteristic 0, the Hodge to de Rham spectral sequence for g degenerates and the Rq g Ω p are locally free.
12月17日（水）6時00分--7時00分

Konstantin Ardakov (University of Oxford)
Equivariant D-modules on rigid analytic spaces

Locally analytic representations of p-adic Lie groups are of interest in several branches of arithmetic algebraic geometry, notably the p-adic local Langlands program. I will discuss some work in progress towards a Beilinson-Bernstein style localisation theorem for admissible locally analytic representations of semisimple compact p-adic Lie groups using equivariant formal models of rigid analytic flag varieties.
11月12日（水）6時00分--7時00分

Ruochuan Liu (BICMR)
Relative (φ, Γ)-modules

In this talk, we will introduce the theory of (φ, Γ)-modules for general adic spaces. This is a joint work with Kedlaya.
10月14日（火）5時30分--6時30分 （いつもと曜日が違います）

Fabrizio Andreatta (Universitá Statale di Milano)
A p-adic criterion for good reduction of curves

Given a curve over a dvr of mixed characteristic 0-p with smooth generic fiber and with semistable reduction, I will present a criterion for good reduction in terms of the (unipotent) p-adic étale fundamental group of its generic fiber.
6月17日（火）5時30分--6時30分 （いつもと曜日が違います）

Ngo Bao Chau (University of Chicago, VIASM)
Vingberg's monoid and automorphis L-functions

We will explain a generalisation of the construction of the local factors of Godement-Jacquet's L-functions, based on Vingberg's method.

5月21日（水）5時30分--6時30分

Shenghao Sun (Mathematical Sciences Center of Tsinghua University)
Parity of Betti numbers in ¥UTF{00E9}tale cohomology

Abstract: By Hodge symmetry, the Betti numbers of a complex projective smooth variety in odd degrees are even. When the base field has characteristic $p,$ Deligne proved the hard Lefschetz theorem in ¥UTF{00E9}tale cohomology, and the parity result follows from this. Suh has generalized this to proper smooth varieties in characteristic $p$, using crystalline cohomology. The purity of intersection cohomology group of proper varieties suggests that the same parity property should hold for these groups in characteristic $p$. We proved this by investigating the symmetry in the categorical level. In particular, we reproved Suh's result, using merely ¥'etale cohomology. Some related results will be discussed. This is joint work with Weizhe Zheng.

4月16日（水）5時30分--6時30分

Olivier Wittenberg (ENS et CNRS)
On the cycle class map for zero-cycles over local fields

The Chow group of zero-cycles of a smooth and projective variety defined over a field k is an invariant of an arithmetic and geometric nature which is well understood only when k is a finite field (by higher-dimensional class field theory). In this talk, we will discuss the case of local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus over p-adic fields. The same statement holds for semistable K3 surfaces over C((t)), but does not hold in general for surfaces over C((t)) or over the maximal unramified extension of a p-adic field. This is a joint work with Helene Esnault.

2014年1月22日（水）18:00-19:00
柏原正樹 (京都大学　数理研)
Riemann-Hilbert correspondence for irregular holonomic D-modules

The classical Riemann-Hilbert correspondence establishes an equivalence between the derived category of regular holonomic D-modules and the derived category of constructible sheaves. Recently, I, with Andrea D'Agnolo, proved a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular (arXiv:1311.2374). In this correspondence, we have to replace the derived category of constructible sheaves with a full subcategory of ind-sheaves on the product of the base space and the real projective line. The construction is therefore based on the theory of ind-sheaves by Kashiwara-Schapira, and also it is influenced by Tamarkin's work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Takuro Mochizuki and Kiran Kedlaya.

2013年12月18日（水）18:00-19:00
加藤　和也 (シカゴ大学)
Heights of motives.

Abstract. The height of a rational number a/b (a, b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion height is so naive, height has played fundamental roles in number theory. There are important variants of this notion. In 1983, when Faltings proved Mordell conjecture, Faltings first proved Tate conjecture for abelian variaties by defining heights of abelian varieties, and then he deduced Mordell conjecture from the latter conjecture.I explain that his height of an abelian variety is generalized to the height of a motive. This generalization of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded heights, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.

2013年11月13日（水）18:00-19:00
Yichao Tian (Morningside Center for Mathematics)
Goren-Oort stratification and Tate cycles on Hilbert modular varieties

Abstract: Let B be a quaternionic algebra over a totally real field F, and p be a prime at least 3 unramified in F. We consider a Shimura variety X associated to B^* of level prime to p. A generalization of Deligne-Carayol's "modele etrange" allows us to define an integral model for X. We will then define a Goren-Oort stratification on the characteristic p fiber of X, and show that each closed Goren-Oort stratum is an iterated P^1-fibration over another quaternionic Shimura variety in characteristic p. Now suppose that [F:Q] is even and that p is inert in F. An iteration of this construction gives rise to many algebraic cycles of middle codimension on the characteristic p fibre of Hilbert modular varieties of prime-to-p level. We show that the cohomological classes of these cycles generate a large subspace of the Tate cycles, which, in some special cases, coincides with the prediction of the Tate conjecture for the Hilbert modular variety over finite fields. This is a joint work with Liang Xiao.

2013年10月16日（水）17:30-18:30
Peter Scholze (Universite de Bonn)
Shimura varieties with infinite level, and torsion in the cohomology of locally symmetric spaces

Abstract: We will discuss the p-adic geometry of Shimura varieties with infinite level at p: They are perfectoid spaces, and there is a new period map defined at infinite level. As an application, we will discuss some results on torsion in the cohomology of locally symmetric spaces, and in particular the existence of Galois representations in this setup.

2013年6月12日（水）17:30-18:30
Xinyi Yuan
Hodge index theorem for adelic line bundles

Abstract: The Hodge index theorem of Faltings and Hriljac asserts that the Neron--Tate height pairing on a projective curve over a number field is equal to certain intersection pairing in the setting of Arakelov geometry. In the talk, I will present an extension of the result to adelic line bundles on higher dimensional varieties over finitely generated fields. Then we will talk about its relation to the non-archimedean Calabi--Yau theorem and the its application to algebraic dynamics. This is a joint work with Shou-Wu Zhang.

2013年5月22日（水）17:30-18:30 とりやめになりました。
Peter Scholze（Uni Bonn）
The Siegel moduli space with infinite level at p

abstract : We prove that (the minimal compactification of) the Siegel moduli space becomes perfectoid by passing to infinite level at p, and that a new kind of period map exists on it, called the Hodge-Tate period map. Among other things, we give a new and simple proof of existence of the canonical subgroup, directly from the results of Illusie's thesis.

2013年4月24日（水）17:30-18:30
今井　直毅さん（東大数理）
Good reduction of ramified affinoids in the Lubin-Tate perfectoid space

Recently, Weinstein finds some affinoids in the Lubin-Tate perfectoid space and computes their reduction in equal characteristic case. The cohomology of the reduction realizes the local Langlands correspondence for some representations of GL_h, which are unramified in some sense. In this talk, we introduce other affinoids in the Lubin-Tate perfectiod space in equal characteristic case, whose reduction realizes "ramified" representations of conductor exponent h+1. We call them ramified affinoids. We study the cohomology of the reduction and its relation with the local Langlands correspondence. This is a joint work with Takahiro Tsushima.

2013年4月10日（水）17:30-18:30
Deepam Patel（Amsterdam）
Motivic structure on higher homotopy of non-nilpotent spaces

In his fundamental paper on the projective line minus three points, Deligne constructed certain extensions of mixed Tate motives arising from the fundamental group of the projective line minus three points. Since then, motivic structures on homotopy groups have been studied by many authors. In this talk, we will construct a motivic structure on the (nilpotent completion of) n-th homotopy group of P^{n} minus n+2 hyperplanes in general position.

2013年1月16日（水）18:00-19:00
大久保　俊さん（東大数理）
On differential modules associated to de Rham representations in the imperfect residue field case

Let K be a CDVF of mixed characteristic (0,p) and G the absolute Galois group of K. When the residue field of K is perfect, Laurent Berger constructed a p-adic differential equation N_dR(V) for any de Rham representation V of G. In this talk, we will generalize his construction when the residue field of K is not perfect. We also explain some ramification properties of our N_dR, which are due to Adriano Marmora in the perfect residue field case.

2012年12月12日（水）18:00-19:00
Francois Charles (CNRS & Universite de Rennes 1)
The Tate conjecture for K3 surfaces and holomorphic symplectic varieties over finite fields

We prove the Tate conjecture for divisors on reductions of holomorphic symplectic varieties over finite fields -- with some restrictions on the characteristic of the base field. We will be concerned mostly with the supersingular case. As a special case, we prove the Tate conjecture, also known as Artin's conjecture in our case, for K3 surfaces over finite fields of characteristic at least 5 and for codimension 2 cycles on cubic fourfolds.

2012年11月14日(水) 18:00-19:00
Pierre Berthelot (Universite de Rennes 1)
De Rham-Witt complexes with coefficients and rigid cohomology

Abstract: For a smooth scheme over a perfect field of characteristic p, we will explain a generalization of the classical comparison theorem between crystalline cohomology and de Rham-Witt cohomology to the case of cohomologies with coefficients in a p-torsion free crystal. This provides in particular a description of the rigid cohomology of a proper singular scheme in terms of a de Rham-Witt complex built from a closed immersion into a smooth scheme.

2012年4月11日(水) 17:30-18:30
D. Rossler (CNRS, Universite de Toulouse)
Around the Mordell-Lang conjecture in positive characteristic

Abstract: Let V be a subvariety of an abelian variety A over C and let G¥subseteq A(C) be a subgroup. The classical Mordell-Lang conjecture predicts that if V is of general type and G¥otimesQ is finite dimensional, then V¥cap G is not Zariski dense in V. This statement contains the Mordell conjecture as well as the Manin-Mumford conjecture (for curves). The positive characteristic analog of the Mordell-Lang conjecture makes sense, when A is supposed to have no subquotient, which is defined over a finite field. This positive characteristic analog was proven in 1996 by E. Hrushovski using model-theoretic methods. We shall discuss the prehistory and context of this proof. We shall also discuss the proof (due to the speaker) of the fact that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture (whereas this seems far from true in characteristic 0).

2012年2月22日(水) 18:00-19:00
望月拓郎 (京大数理研)
Twistor $D$-module and harmonic bundle

Abstract: We shall overview the theory of twistor $D$-modules and harmonic bundles. I am planning to survey the following topics, motivated by the Hard Lefschetz Theorem for semisimple holonomic $D$-modules:
  1. What is a twistor $D$-module?
  2. Local structure of meromorphic flat bundles
  3. Wild harmonic bundles from local and global viewpoints

2011年10月19日
A. Suslin (Northwestern University)
K_2 of the biquaternion algebra

2011年11月9日 18:00-19:00
志甫 淳 (東大数理)
On extension and restriction of overconvergent isocrystals

Resume: First we explain two theorems concerning (log) extension of overconvergent isocrystals. One is a p-adic analogue of the theorem of logarithmic extension of regular integrable connections, and the other is a p-adic analogue of Zariski-Nagata purity. Next we explain a theorem which says that we can check certain property of overconvergent isocrystals by restricting them to curves.

2011年12月8日(木) 18:30-19:30
G. Faltings (Max-Planck Institute fuer Mathematiks Bonn)
Nonabelian p-adic Hodge theory and Frobenius

resume: Some time ago, I constructed a relation between Higgs-bundles and p-adic etale sheaves, on curves over a p-adic field. This corresponds (say in the abelian case) to a Hodge-Tate picture. In the lecture I try to explain one way to introduce Frobenius into the theory. We do not get a complete theory but at least can treat p-adic sheves close to trivial.

2012年1月12日(木) 18:15-19:15
T. Gee (Imperial College London)
New perspectives on the Breuil-Mezard conjecture (joint with M. Emerton)

resume: I will discuss joint work with Matthew Emerton on geometric approaches to the Breuil-Mezard conjecture, generalising a geometric approach of Breuil and Mezard. I will discuss a proof of the geometric version of the original conjecture, as well as work in progress on a geometric version of the conjecture which does not make use of a fixed residual representation.

2011年5月11日
M. Raynaud (Universite Paris-Sud)
Permanence following Temkin

When one proceeds to a specialization, the good properties of algebraic equations may be destroyed. Starting with a bad specialization, one can try to improve it by performing modifications under control. If, at the end of the process, the initial good properties are preserved, one speaks of permanence. I shall give old and new examples of permanence. The new one concerns the relative semi-stable reduction of curves recently proved by Temkin.

2011年6月15日
阿部知行(東大IPMU)
Product formula for $p$-adic epsilon factors

Abstract: I would like to talk about my recent work jointly with A. Marmora on a product formula for $p$-adic epsilon factors. In 80's Deligne conjectured that a constant appearing in the functional equation of $L$-function of $¥ell$-adic lisse sheaf can be written by means of local contributions, and proved some particular cases. This conjecture was proven later by Laumon, and was used in the Lafforgue's proof of the Langlands' program for functional filed case. In my talk, I would like to prove a $p$-adic analog of this product formula.

2010年6月9日

Fabrice Orgogozo (CNRS et Ecole polytechnique)

エタールコホモロジーの高次順像の一様構成可能性について
Constructibilité uniforme des images directes supérieures en cohomologie étale

Résumé
Motivé par une remarque de N. Katz sur le lien entre la torsion de la Z_?-cohomologie étale et les ultraproduits de groupes de F_?-cohomologie, nous démontrons un théorème d'uniformité en ? pour la constructibilité des images directes supérieures entre schémas de type fini sur un trait excellent. (Un tel théorème avait été considéré par O. Gabber il y a plusieurs années déjà.) La méthode est maintenant classique : on utilise des théorèmes de A. J. de Jong et un peu de log- géométrie.
Z_?エタールコホモロジーの捻れとF_?コホモロジーの超積の関係を巡り N. Katz氏の指摘に基づいて、高次順像に於ける?に対する 一様構成可能性についての定理を証明する。 （この様な定理は以前よりガバー氏の構想に有った。） ここでは月並みな方法で有るが、A. J. de Jong氏の定理と 少量の対数的幾何学を使う。

2010年5月26日

技術的な問題のため IHESへの中継はとりやめ になりました。 講演そのものは予定通りです。

都築 暢夫　(東北大)

Generalized hypergeometric functions and arithmetic families of Calabi-Yau varieties

abstract:

We construct an arithmetic family of Calabi-Yau varieties over the projective line minus $0, 1, ¥infty$, which is related to generalized hypergeometric functions ${}_{n+1}F_n(1/2, ¥cdots, 1/2;1, ¥cdots, 1;¥lambda)$, and calculate its cohomology with respect to various realizations.

2010年5月12日

松本　眞　(東大数理)

Differences between Galois representations in outer-automorphisms of $¥pi_1$ and those in automorphisms, implied by topology of moduli spaces

Abstract

Fix a prime $l$. Let $C$ be a proper smooth geometrically connected curve over a number field $K$, and $x$ be its closed point. Let $¥Pi$ denote the pro-$l$ completion of the geometric fundamental group of $C$ with geometric base point over $x$. We have two non-abelian Galois representations: $$ ¥rho_A : ¥mathrm{Gal}_{k(x)} ¥to ¥mathrm{Aut}(¥Pi), ¥ ¥ ¥ ¥ ¥rho_O : ¥mathrm{Gal}_K ¥to ¥mathrm{Out}(¥Pi). $$ Our question is: in the natural inclusion $¥mathrm{Ker}(¥rho_A) ¥subset ¥mathrm{Ker}(¥rho_O) ¥cap ¥mathrm{Gal}_{k(x)}$, whether the equality holds or not. ¥noindent {¥bf Theorem}: Assume that $g ¥geq 3$, $l$ divides $2g-2$. Then, there are infinitely many pairs $(C,K)$ with the following property. If $l$ does not divide the extension degree $[k(x):K]$, then $¥mathrm{Ker}(¥rho_A) ¥neq (¥mathrm{Ker}(¥rho_O) ¥cap ¥mathrm{Gal}_{k(x)})$ holds. This is in contrast to the case of the projective line minus three points and its canonical tangential base points, where the equality holds (a result of Deligne and Ihara). There are two ingredients in the proof: (1) Galois representations often contain the image of the geometric monodromy (namely, the mapping class group) [M-Tamagawa 2000] (2) A topological result [S.~Morita 98] [Hain-Reed 2000] on the cohomological obstruction of lifting the outer action of the mapping class group to automorphisms.

2010年4月14日

Gerard Laumon (CNRS and Universite Paris-Sud)

The cohomological weighted fundamental lemma

abstract: Using the Hitchin fibration, Ngo Bao Chau has proved the Langlands-Shelstad fundamental lemma. In a joint work with Pierre-Henri Chaudouard, we have extended Ngo's proof to obtain the weighted fundamental lemma which had been conjectured by Arthur. In the talk, I would like to present our main cohomological result.