Journée de géométrie arithmétique en l'honneur de Michel Gros

jeudi 17 oct. 2024, 09:00 → 18:30 Europe/Paris

Centre de conférences Marilyn et James Simons (Le Bois-Marie)

Michel Gros a officiellement pris sa retraite le 1er janvier de cette année. À cette occasion, ses collègues et amis souhaitent lui rendre hommage et célébrer une carrière remarquable, principalement axée sur la géométrie arithmétique, domaine dans lequel il a obtenu des contributions importantes sur des questions variées allant des cohomologies p-adiques et modulo p à la théorie des représentations, en passant par les régulateurs syntoniques. 

Michel a obtenu un doctorat en 1983 à Orsay, à l'issue d'une thèse intitulée « Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique » dirigée par Luc Illusie. Après un séjour postdoctoral à l'Université de Tokyo auprès de Kazuya Kato, il a été recruté par le CNRS et affecté depuis les années 90 à l'Institut de Recherche Mathématique de Rennes où il participe à la vie scientifique du groupe initialement créé par Pierre Berthelot.

Orateurs invités :
- Cédric Pépin, Univ. Sorbonne Paris-Nord
- Emanuel Reinecke, IHES
- Simon Riche, Univ. Clermont Auvergne
- Takeshi Tsuji, The University of Tokyo

Organisateurs : Ahmed Abbes (CNRS & IHES), Fabrice Orgogozo (CNRS & IMJ-PRG) et Julien Sebag (Univ. de Rennes).

09:30 → 10:25 Café d'accueil

10:25 → 10:30 Introduction par Ahmed Abbes

10:30 → 11:30 q-Higgs Modules and the Nygaard Filtration

Similarly to the theory of crystalline cohomology, prismatic crystals and their cohomology over a bounded prism (R,I) are described in terms of q-Higgs modules when (R,I) is defined over the q-crystalline prism, or more generally when R/I contains a primitive pth root of unity. We see how this allows us to understand the isogeny property of Frobenius, the de Rham specialization, and the Nygaard filtration for the prismatic cohomology of a prismatic F-crystal.

Orateur: Takeshi Tsuji (The University of Tokyo)

11:45 → 12:45 Relative Poincaré Duality in Non Archimedean Geometry

In my talk, I will explain a relative version of mod-p Poincare duality for any proper morphism of rigid-analytic varieties over a p-adic field. A key ingredient in the proof will be a novel construction of trace maps for smooth morphisms and for proper morphisms of rigid-analytic varieties. Along the way, I will sketch a new and essentially diagrammatic proof of mod-p Poincare duality for smooth and proper morphisms. Joint work with Shizhang Li and Bogdan Zavyalov.

Orateur: Emanuel Reinecke (IHES)

12:45 → 14:00 Déjeuner

14:00 → 15:00 On the Left Adjoint of mod p Smooth Parabolic Induction

Let $F/\mathbb{Q}_p$ be a finite field extension with ring of integers $\mathcal{O}_F$ and residue field $k_F$. Let $\mathbf{G}$ be a split connected reductive group over $\mathcal{O}_F$, and $D(G)$ be the derived category of smooth representations of $G:=\mathbf{G}(F)$ over a fixed field extension of $k_F$. For a Borel $\mathbf{B}=\mathbf{T}\mathbf{U}\subset\mathbf{G}$, the $t$-exact parabolic induction functor ${\rm Ind}_B^G:D(T)\rightarrow D(G)$ admits a left adjoint $L(U,-)$, as proved by Heyer. We study the functor $L(U,-)$ on algebraic weights $L(\lambda)$ which are $p$-small. When $F$ is unramified, we show that $L(U,ind_{G(\mathcal{O}_F)}^G(L(\lambda)))\in D(T)$ splits completely (under some mild modular assumptions). This is joint work with Karol Koziol.

Orateur: Cédric Pépin (Université Sorbonne Paris Nord)

15:15 → 16:15 Localization Theory for Harish-Chandra Bimodules in Positive Characteristic

Given a connected reductive algebraic group G over an algebraically closed field of positive characteristic, a Harish-Chandra bimodule is a G-equivariant bimodule for the enveloping algebra of the Lie algebra of G such that the differential of the G-action coincides with the diagonal action of the Lie algebra. Such objects are interesting in particular because they realize translation functors for representations of G. We will explain how Harish-Chandra bimodules with a regular central character can be related to equivariant coherent sheaves on the Steinberg variety of G, building on the localization theory of Bezrukavnikov-Mirkovic-Rumynin. This relation is an essential step in the construction of an equivalence relating such coherent sheaves to perverse sheaves on the affine flag variety of the Langlands dual group, and opens the way to a study of certain categories of representations of the Lie algebra of G using constructible sheaves.

Orateur: Simon Riche (Université Clermont Auvergne)

16:30 → 18:00 Cocktail