Perfectoid Shimura varieties
P. Scholze
Abstract: 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.