On the cohomology of Hilbert modular varieties with torsion coefficients

Abstract: Shimura varieties are certain moduli spaces equipped with many symmetries, that play an important role in the Langlands programme. For example, Hilbert modular varieties are quotients of the product of several copies of the upper-half plane by certain arithmetic groups. I will discuss a general conjecture on the cohomology of Shimura varieties with torsion coefficients, which states that the non-degenerate part of their cohomology is concentrated in the middle degree. I will give an overview of an approach to this conjecture introduced in joint work with Peter Scholze. This approach relies on the geometry of the Hodge-Tate period morphism, which I will describe, and on certain technical computations. I will then specialise to the case of Hilbert modular varieties and explain a modified version of this approach that relies on an instance of geometric Jacquet-Langlands functoriality for the fibers of the Hodge-Tate period morphism. This is joint work with Matteo Tamiozzo.

algebraic geometry

Audience: researchers in the topic

ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

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