Perfectoid Shimura varieties

This note explains some of the author's work on understanding the torsion appearing in the cohomology of locally symmetric spaces such as arithmetic hyperbolic 3-manifolds. The key technical tool was a theory of Shimura varieties with infinite level at p: As p-adic analytic spaces, they are perfectoid, and admit a new kind of period map, called the Hodge-Tate period map, towards the flag variety. Moreover, the (semisimple) automorphic vector bundles come via pullback along the Hodge-Tate period map from the flag variety. In the case of the Siegel moduli space, the situation is fully analyzed in [12]. We explain the conjectural picture for a general Shimura variety.