The cohomology of Shimura varieties with torsion coefficients

Abstract

In this article, we survey recent work on some vanishing conjectures for the cohomology of Shimura varieties with torsion coefficients, under both local and global conditions. We discuss the $p$-adic geometry of Shimura varieties and of the associated Hodge–Tate period morphism, and explain how this can be used to make progress on these conjectures. Finally, we describe some applications of these results, in particular to the proof of the Sato–Tate conjecture for elliptic curves over CM fields.