Arithmetic Day: Bonn - Essen - Münster
Date: 15.01.2025
Venue: the 5th Floor, Thea-Leymann-Str. 9, 45127 Essen
Directions: Enter the building through the entrance from Thea-Leymann-Str. (there should be a park behind you). Take the elevator to the 5th floor. You can get to the 5th floor only from this staircase. If you enter the building through a different entrance (Altendorfer Str., next to a bike shop) then go to the 3rd floor, turn right and keep going until the next staircase, then go up to the 5th floor.
Program
11:30 – 12:30 Sally Gilles (Essen), The v-Picard group of Stein spaces.
Lunch
14:00 – 15:00 Lucas Mann (Münster), A 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve
15:30 – 16:30 Ioannis Zachos (Münster), On regular integral models for some Shimura varieties.
16:45 – 17:45 Arnaud Eteve (Bonn), Spectral action on isocrystals.
18:15 Dinner
Abstracts
Arnaud Eteve: Spectral action on isocrystals. This is joint work in progress with D. Gaitsgory, A. Genestier and V. Lafforgue. Let G be a reductive group on a local function field F = Fq((t)) of characteristic p. In this situation, Fargues and Scholze have proposed a geometrization of the local Langlands correspondence by constructing an action of the category of perfect complexes on the stack of local Langlands parameters on the category of étale sheaves on Bun_G(X_FF), the stack of G-torsors on the Fargues-Fontaine curve. The goal of this talk is to report on work in progress for the construction of an action of the same category of perfect complexes on the category of sheaves on the stack of isocrystals, following proposals of Gaitsgory and Zhu. With this new method, we expect strong forms of local-global compatibility and compatibility with constructions coming from geometric representation theory.
Sally Gilles: The v-Picard group of Stein spaces. In this talk, I will present a computation of the image of the Hodge-Tate logarithm map (defined by Heuer) in the case of smooth Stein varieties. When the variety is the affine space, Heuer has proved that this image is equal to the group of closed differential forms. In general, we will see that the image always contains such forms but the quotient can be non-trivial: it contains a Zp-module that maps, via the Bloch-Kato exponential map, to integral classes in the pro-étale cohomology. This is based on a joint work with V. Ertl and W. Niziol.
Ioannis Zachos: On regular integral models for some Shimura varieties. Local models of Shimura varieties are projective flat schemes over the spectrum of a discrete valuation ring. The importance of local models lies in the fact that under some assumptions they model the singularities that arise in the reduction modulo p of Shimura varieties. In this talk, we will first introduce the notion of local models for some unitary and orthogonal Shimura varieties. Building on this, we will resolve the singularities of these models, leading to regular integral models for the corresponding Shimura varieties.
Lucas Mann: A 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve. Given a rigid-analytic variety X over a p-adic non-archimedian field, we study the pro-étale cohomology of X with coefficients in Z_p and Q_p. As was observed by many people (including Hansen and Colmez-Gilles-Niziol), this pro-étale cohomology has some unexpected behavior. For example, one can construct pro-étale Q_p-local systems on the projective line whose cohomology is infinite dimensional, defying any naive version of Poincaré duality. Based on ideas coming from prismatic cohomology, we provide a fix for this situation: Instead of working with pro-étale sheaves, we propose to work with quasi-coherent sheaves on the relative Fargues-Fontaine curve. This perspective allows a full 6-functor formalism and provides explanations for the pathologies occurring in more naive approaches to pro-étale cohomology. It is joint work with Johannes Anschütz and Arthur-César Le Bras and it is related to work of Colmez-Gilles-Niziol.