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3-5 juin 2026 CAEN (France)
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ExposésOn holonomic D-cap-modules on rigid analytic spaces Andreas Bode Résumé: We describe an analogue of Caro's notion of overholonomicity for Ardakov-Wadsley's D-cap-modules on rigid analytic spaces and establish large parts of a corresponding six-functor formalism. We also discuss various examples from joint work with Finn Wiersig, including the extensions of Gauss-Manin connections and, more generally, extensions of log connections arising from de Rham local systems.
Motivic Hecke algebras Mattia Cavicchi Résumé: The action of Hecke operators on cuspidal modular forms of weight 2 can be realized as the action of an algebra of algebraic cycles on the square of a smooth projective curve, with composition defined in terms of intersection theory. With a view toward applications to the arithmetic of the associated Galois representations, one would like to generalize this description to the case of higher weights. The aim of this talk is to present this circle of ideas and to explain the construction of a so‑called motivic Hecke algebra, providing such a generalization. In fact, this construction works in the much broader setting of automorphic forms appearing in the cohomology of any PEL‑type Shimura variety.
The $p$-curvature conjecture for physically rigid local systems Hélène Esnault Résumé: Joint work with Michael Groechenig. We define in higher dimension the notion of Physically Rigid Local Systems due to Katz in dimension one and prove the $p$-curvature conjecture for those.
\G_m-cohomology of p-adic Stein spaces Sally Gilles Résumé: In this talk I will explain a computation we did with Damien Junger, in which we determined the étale cohomology with \G_m coefficients of some p-adic analytic spaces. One difficulty is that rigid analytic spaces do not have enough points and it is not possible to deduce the global behavior of \G_m from its stalks. Instead we consider the quotient \G_m/U by the group of principal units U. The cohomology of this quotient (in some cases) can then be computed by looking at what happens over a point, whereas the cohomology of U can be computed by passing to the pro-étale site, via p-adic methods.
Characteristic cycle for Dcap-modules in dimension one
Raoul Hallopeau Résumé: Let X be a smooth, rigid analytic space. Ardakov-Wadsley have introduced a sheaf Dcap of rapidly converging differential operators over X, together with a category of coadmissible Dcap-modules that play the role of "coherent objects". I have defined a characteristic variety and a characteristic cycle for these modules in the one-dimensional case. In particular, a notion of "sub-holonomicity" for coadmissible Dcap-modules over a smooth, rigid curve follows. I will present this construction.
Fourier transform for coadmissible D-modules Christine Huygue Résumé: I will explain how to construct a Fourier transform for coadmissible D-modules over a rigid analytic vector space and explain some properties of this transform.
Singularities in the Ekedahl--Oort stratification Lorenzo La Porta Résumé: "Special fibres of abelian type Shimura varieties at hyperspecial level are naturally endowed with a finite stratification, called the Ekedahl--Oort (EO, for short) stratification. While any given EO stratum is smooth, its closure may have singularities which encode interesting arithmetic information. I will present recent results on the study of such singularities which were obtained by working with the stacks of G-zips of Pink--Wedhorn--Ziegler. I will discuss some computational aspects and applications to the theory of generalised Hasse invariants, as well as some concrete examples. This is joint work with Jean-Stefan Koskivirta and Stefan Reppen."
Comparisons between overconvergent isocrystals and arithmetic D-modules Christopher Lazda Résumé: According to a philosophy of Grothendieck, every good cohomology theory should have a six functor formalism. Arithmetic D-modules were introduced by Berthelot to provide the theory of rigid cohomology with exactly such a formalism. However, it is not clear that cohomology groups computed via the theory of arithmetic D-modules coincide with the analogous rigid cohomology groups. In this talk I will describe an 'overconvergent Riemann-Hilbert correspondence' that can be used to settle this question.
On the $p$-adic deformation for the K-theory of semistable schemes Alberto Merici Résumé: We provide a generalization of the fiber square of Bloch--Esnault--Kerz and Beilinson for $p$-adic log schemes, relating relative continuous K-theory to logarithmic cyclic homology. We deduce that for a semistable scheme over a mixed characteristic DVR, the obstruction problem to lifting K-theory classes in continuous K-theory is reduced to the Chern classes in Hyodo--Kato cohomology.
Effective Artin-Schreier-Witt theory for curves Rubén Muños-Bertrand Résumé: Given a (projective, algebraic, normal) curve defined over a finite field of characteristic p, Witt vector theory allows us to easily construct covers of this curve having a p-cyclic Galois group. Among these, we will focus on étale covers. Artin-Schreier-Witt theory enables us to characterise them using an étale cohomology group. In theory, it is known since Serre's work how we can construct them all. In practice, starting from a curve given by an equation, it is a different story. We will see what difficulties arise when we undertake such a computation, and then we will explain how to overcome them. We will discuss applications of this project, in collaboration with Christophe Levrat (Inria Saclay), to computation of étale cohomology.
Tobias Schmidt Résumé: Let G be a p-adic Lie group. In this talk, I will first give some background on G-equivariant D-modules on rigid analytic spaces and explain why one is interested in them. I will then state a general irreducibility result for certain induced modules and discuss some applications. This is joint work with K. Ardakov.
Local model of Shimura varieties, nearby cycles, and center of the pro-p-Iwahori Hecke algebra Benoît Stroh Résumé: Au début des années 2000, deux constructions importantes ont vu le jour, très reliées entre elles malgré le fait qu'elles portent sur des domaines mathématiques sans lien apparents : d'un côté les modèles locaux de variétés de Shimura dûs à Rapoport et Zink dans le cadre du programme de Langlands, et de l'autre les cycles proches centraux de Gaitsgory et Beilinson en théorie géométrique des représentations. Ces constructions valent pour des structures de niveau iwahorique, qu'on peut également qualifier de purement unipotentes. |
