Perfectoid Jacquet-Langlands and the cohomology of Hilbert modular varieties

Abstract: Deuring and Serre showed that the supersingular locus in a special fibre of a modular curve can be identified with a Shimura set attached to a definite quaternion algebra. I will discuss a perfectoid version of this result over totally real fields, comparing the cohomology of fibres of the Hodge-Tate period maps attached to different quaternionic Shimura varieties. I will then explain how this can be used to prove vanishing theorems for the cohomology with torsion coefficients of Hilbert modular varieties. This is joint work with Ana Caraiani.

number theory

Audience: researchers in the topic

London number theory seminar

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