Japan. J. Math. 11, 15--32 (2016)

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.