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.
Program
11:30 – 12:30 Sally Gilles (Essen), The v-Picard group of Stein spaces.
Lunch
14:00 – 15:00 Alex Ivanov (Bochum)
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
Please register until 06.01 for lectures and for dinner if you want to attend.
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.