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Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo

Ahmed Abbes (CNRS, IHÉS), Fabrice Orgogozo (CNRS, École polytechnique)

(Takeshi Saito)志甫　淳(Atsushi Shiho) 辻　雄(Takeshi Tsuji)　(東大数理)

田野(Ye Tian)、田一超(Yichao Tian)郑维喆(Weizhe Zheng)　(晨兴)

2017年12月13日（周三）16:00—17:00

Javier Fresán (école 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 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).

2017年11月8日（周三）17:00—18:00

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9月27日（周三）16:30—17:30

Height functions for motives, Hodge analogues, and Nevanlinna analogues
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（周三）16:30—17:30

Yongquan Hu（Chinese Academy of Sciences, Morningside Center of Mathematics）
Multiplicity one for the mod p cohomology of Shimura curves
At present, the mod $p$ (and $p$-adic) local Langlands correspondence is only well understood for the group $\mathrm{GL}_2(\mathbb{Q}_p)$. One of the main difficulties is that little is known about supersingular representations besides this case, and we do know that there is no simple one-to-one correspondence between representations of $\mathrm{GL}_2(K)$ with two-dimensional representations of $\mathrm{Gal}(\overline{K}/K)$, at least when $K/\mathbb{\mathbb{Q}}_p$ is (non-trivial) finite unramified.
However, the Buzzard-Diamond-Jarvis conjecture and the mod $p$ local-global compatibility for $\mathrm{GL}_2/\mathbb{Q}$ suggest that this hypothetical correspondence may be realized in the cohomology of Shimura curves with characteristic $p$ coefficients (cut out by some modular residual global representation $\bar{r}$). Moreover, the work of Gee, Breuil and Emerton-Gee-Savitt show that, to get information about the $\mathrm{GL}_2(K)$-action on the cohomology, one could instead study the geometry of certain Galois deformation rings of the $p$-component of $\bar{r}$.
In a work in progress with Haoran Wang, we push forward their analysis of the structure of potentially Barsotti-Tate deformation rings and, as an application, we prove a multiplicity one result of the cohomology at full congruence level when $\bar{r}$ is reducible generic \emph{non-split} at $p$. (The semi-simple case was previously proved by Le-Morra-Schraen and by ourselves.)

2017年10月11日（周三）16:30—17:30

Michael Temkin(耶路撒冷希伯莱大学)
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5月17日（周三）16:30—17:30
Olivier Fouquet（南巴黎大学）
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（周二）16:30—17:30

Peter Scholze（波恩大学）

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（周三）17:00至18:00

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（周三）17:00至18:00

Luc Illusie (Université Paris-Sud)

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2日（周三）17:00至18:00

Yves André (CNRS, Institut de Mathématiques 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（周三）16:30至17:30

U. Jannsen (Universities of Regensburg & Tokyo)

Filtered de Rham Witt complexes and wildly ramified higher class field theory over finite fields
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.

2016年6月8日（周三）16:30至17:30

Xu Shen (AMSS)

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.

2016年5月11日（周三）16:30至17:30

Wiesława Nizioł (CNRS, ÉNS 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（周三）16:30至17:30

Akio Tamagawa (RIMS, Kyoto University)

Semisimplicity of geometric monodromy on etale cohomology.
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).

2016年3月29日（周二）16:30至17:30

Matthew Morrow(University of 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.

2015年12月9日（周三）17:00至18:00

Ted Chinburg (University of Pennsylvania & IHÉS)

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.

2015年11月17日（周二）17:00至18:00
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 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.

2015年10月27日（周二）17:00至18:00

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.

2015年520日（周三）16:3017:30

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 Gross—Zagier 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.

201548日（周三）16:3017:30

Seidai Yasuda (Osaka University)

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.

2015121日（周三）17:0018:10

17:0017:10, Luc Illusie (Université Paris-Sud)

Remembering the SGA's

17:1018:10, Ofer Gabber (CNRS & IHÉS)

(Joint work with Brian Conrad.) Let $K$ be a complete rank 1 valued field with ring of integers $O_K$, A an adic noetherian ring and $f : A → O_K$ 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 $R^q g* Ω^p_{X/Y}$ are locally free.

20141217日（周三）17:0018:00

Konstantin Ardakov(University of Oxford)

Equivariant $\mathcal{D}$ modules on rigid analytic spaces

Locally analytic representations ofp-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.

20141112日（周三）17:0018:00

Relative (φ, Γ)-modules

In this talk, we will introduce the theory of (φ, Γ)-modules for general adic spaces. This is joint work with Kedlaya.

20141014日（周二）16:3017:30

Fabrizio Andreatta (Università 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.

2014617日（周二）16:3017:30

Ngô Bảo Châu (The University of Chicago, VIASM)

Vinberg's monoid and automorphic L-functions

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

201452116:3017:30

Parity of Betti numbers in étale cohomology

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 étale 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 étale cohomology. Some related results will be discussed. This is joint work with Weizhe Zheng.

201441616:3017:30

Olivier Wittenberg (ÉNS, 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 Hélène Esnault.

201412217:0018:00

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.

2013121817:0018:00

Heights of motives

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 varieties 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.

2013111317:0018:00

Goren-Oort stratification and Tate cycles on Hilbert modular varieties

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 “modèle étrange” 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 P1-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 fiber 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.

2013101616:3017:30

Peter Scholze (Universität Bonn)

Shimura varieties with infinite level, and torsion in the cohomology of locally symmetric spaces

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.

201361216:3017:30

Hodge index theorem for adelic line bundles

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.

201342416:3017:30

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 GLh, which are unramified in some sense. In this talk, we introduce other affinoids in the Lubin-Tate perfectoid 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.

201341016:3017:30

Deepam Patel (Universiteit van 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 Pn minus n+2 hyperplanes in general position.

201311617:0018:00

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 NdR(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 NdR, which are due to Adriano Marmora in the perfect residue field case.

2012121217:0018:00

François Charles (CNRS & Université 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.

2012111417:0018:00

Pierre Berthelot (Université de Rennes 1)

De Rham-Witt complexes with coefficients and rigid cohomology

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.

2017年4月11日（周二）16:30—17:30
Peter Scholze（波恩大学）
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.

A book of Yau - A History in Sum