Titres et résumés

Programme et vidéos

A. Beilinson (University of Chicago)
Titre : The singular support of a constructible sheaf
Résumé : According to Kashiwara and Schapira for every constructible sheaf F on a complex manifold X there is a conical closed subset SS(F) of the cotangent bundle whose dimension equals dim X and such that for any function f on X with df disjoint from SS(F) the vanishing cycles complex \phi_f (F) is acyclic. I will explain a similar result for étale sheaves on algebraic varieties over base field of arbitrary characteristic.

G. Boxer (Harvard)
Titre : Construction of torsion Galois representations
Résumé : Recently, Scholze has constructed Galois representations associated to torsion classes in the cohomology of certain arithmetic locally symmetric spaces. I will explain a different construction of some of these Galois representations. The ingredients are a construction of boundary cohomology inspired by the earlier work of Harris-Lan-Taylor-Thorne and a construction of congruences using the theory of "generalized Hasse invariants".

F. Calegari (Northwestern University)
Titre : Non-minimal modularity lifting theorems for imaginary quadratic fields
Résumé : I will explain how previous (conditional) minimal modularity lifting results (in the presence of torsion) may be adapted to the non-minimal case in the context of imaginary quadratic fields. This is joint work with David Geraghty.

G. Faltings (MPIM, Bonn)
Titre : The category MF in the semistable case
Résumé : For smooth schemes the category MF (defined by Fontaine for DVR's) realises the "mysterious functor", and provides natural systems of coeffients for crystalline cohomology. We generalise it to schemes with semistable singularities. The new technical features consist mainly of different methods in commutative algebra.

T. Gee (Imperial College London)
Titre : Moduli stacks of potentially Barsotti-Tate Galois representations
Résumé : I will discuss joint work with Ana Caraiani, Matthew Emerton and David Savitt, in which we construct moduli stacks of two-dimensional potentially Barsotti-Tate Galois representations, and study the relationship of their geometry to the weight part of Serre's conjecture.

M. Harris (Université Paris 7)
Titre : Construction of p-adic L-functions for unitary groups
Résumé : This is a report on the construction of p-adic L-functions attached to ordinary families of holomorphic modular forms on the unitary groups of n-dimensional hermitian vector spaces over CM fields. The results have been obtained over a period of nearly 15 years in joint work with Ellen Eischen, Jian-Shu Li, and Chris Skinner. The p-adic L-functions specialize at classical points to critical values of standard L-functions of cohomological automorphic forms on unitary groups, or equivalently of cohomological automorphic forms on GL(n) that satisfy a polarization condition. When n = 1 one recovers Katz's construction of p-adic L-functions of Hecke characters.

D. Helm (Imperial College London)
Titre : Whittaker models, converse theorems, and the local Langlands correspondence for GL_n in families
Résumé : Let F be a p-adic field, and \ell a prime different from p. In this setting there is a conjectural correspondence (due to M. Emerton and the speaker) between families of \ell-adic representations of G_F and families of smooth \ell-adic representations of GL_n(F). We describe how this correspondence can be reinterpreted in the context of the Bernstein center, and discuss recent joint work with Gil Moss which allows us to reformulate this conjecture as a purely Galois-theoretic statement.

F. Herzig (University of Toronto)
Titre : On de Rham lifts of local Galois representations
Résumé : It is an open problem to show that a given n-dimensional mod p local Galois representation \rho has a (regular) de Rham lift. We discuss several results concerning the existence of de Rham lifts of \rho of prescribed weights and types, assuming that \rho admits a nice lift to start with (for example, a Fontaine-Laffaille lift). Our arguments combine local and global methods. This is joint work with T. Gee, T. Liu, and D. Savitt.

N. Imai (University of Tokyo)
Titre : Affinoids in the Lubin-Tate perfectoid space and simple epipelagic representations
Résumé : We construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the middle cohomology of the reductions of the formal models realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for representations of exponential Swan conductor one, which we call the simple epipelagic representations. We discuss also a relation between these affinoids and CM points. This is a joint work with Takahiro Tsushima.

M. Kisin (Harvard)
Titre : Honda-Tate theory for Shimura varieties
Résumé : Honda-Tate theory asserts that an abelian variety over a finite field is isogenous to one which has a CM lifting. We will explain a result which says that, under some mild conditions, the analogous statement holds for isogeny classes on a Shimura variety of Hodge type.
This is joint work with Keerthi Madapusi and Sug-Woo Shin.

K. Nakamura (Hokkaido University)
Titre : Local epsilon isomorphisms for rank two p-adic representations of Gal(\bar{Q}_p/Q_p) and a functional equation of Kato's Euler systems
Résumé : Local epsilon isomorphisms are conjectural bases of the determinants of the Galois cohomologies of families of p-adic representations of Gal(\bar{Q}_p/Q_p), which interpolate the de Rham epsilon isomorphisms which are explicitly defined by using local (L-and epsilon) constants and Bloch-Kato exponential for de Rham representations. In our talk, we propose a conjectural definition of the local epsilon isomorphisms using (a multivariable version of) Colmez' convolution pairing. Moreover, we prove (almost parts of) our conjecture for the rank two case using Colmez' theory of p-adic local Langlands correspondence for GL_2(Q_p). As an application, we prove a functional equation of Kato's Euler systems associated to modular forms, which is of the same form as predicted by Kato's global epsilon conjecture.

T. Saito (University of Tokyo)
Titre : The characteristic cycle and the singular support of an étale sheaf
Résumé : We define the characteristic cycle of an étale sheaf on a smooth variety of arbitrary dimension in positive characteristic using the singular support, constructed by Beilinson very recently. The characteristic cycle satisfies a Milnor formula for vanishing cycles and an index formula for the Euler-Poincaré characteristic.

P. Scholze (Universität Bonn)
Titre : The Witt vector affine Grassmannian
Résumé : (joint with Bhargav Bhatt) We prove that the space of W(k)-lattices in W(k)[1/p]^n, for a perfect field k of characteristic p, has a natural structure as an ind-(perfect scheme). This improves on recent results of Zhu by constructing a natural ample line bundle on the space of such lattices.

B. Schraen (CNRS, Université de Versailles Saint-Quentin)
Titre : Classicality on eigenvarieties
Résumé : I'll explain how one can use the patching method to obtain finer classicality results on eigenvarieties for definite unitary groups. We obtain new cases of classicality for finite slope p-adic automorphic forms whose associated Galois representation are crystalline at p. Patching methods translate the problem to a local analysis of the space of trianguline representations. This is joint work with Christophe Breuil and Eugen Hellmann.

S. W. Shin (UC Berkeley)
Titre : Galois representations in the cohomology of Shimura varieties
Résumé : I will report on work in progress to describe the cohomology of Shimura varieties of abelian type (with Mark Kisin and Yihang Zhu) with an application in the case of general symplectic groups (with Arno Kret).

Y. Tian (Morningside Center for Mathematics, Beijing)
Titre : Generic Tate cycles on certain unitary Shimura varieties over finite fields
Résumé : Let F be a quadratic real field, and E/F be a CM extension. Let p be a prime number inert in F and split in E. Consider a Shimura variety of PEL type associated to a unitary group for E/F of type G(U(1,n-1)*U(n-1,1)) , and denote by X its fiber in characteristic p. We will construct n series of algebraic cycles in X such that each of them is parametrized by another unitary Shimura variety of type G(U(0,n)*U(n,0)). We prove that, under certain genericity conditions, these algebraic cycles give rise to almost all obvious Tate cycles on X. This is a work joint with Liang Xiao and David Helm.

T. Tsuji (University of Tokyo)
Titre : On p-adic étale cohomology of perverse sheaves
Résumé : I will talk about a generalization of the comparison theorem of the cohomologies of crystalline p-adic étale sheaves and filtered F-isocrystals by G. Faltings, to p-adic perverse sheaves for the stratification associated to a simple normal crossing divisor. A comparison map is constructed via descriptions of both cohomologies in terms of "nearby cycles" along each stratum; we use certain fibered topos whose fibers are topos associated to strata with some log structures.


A. Abbes (CNRS, IHÉS), C. Breuil (CNRS, Université Paris-Sud),
G. Chenevier (CNRS, École Polytechnique) et T. Saito (University of Tokyo)

Organisée en partenariat avec


Conférence finale du programme ANR Théorie de Hodge p-adique et Développements (ThéHopaD)
Cinquième conference franco-japonaise en géométrie arithmétique
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