Locally symmetric spaces: $p$-adic aspects

Locally symmetric spaces: pp-adic aspects - Laurent Fargues

Laurent Fargues
Institut de Mathématiques de Jussieu
November 30, 2017
$p$-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic $p$-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. The proof consists in a thorough study of modifications of G-bundles on the curve. As an application we can compute the p-adic period space of K3 surfaces with supersingular reduction and other period spaces related to the basic locus in some Shimura varieties. This is joint work with Miaofen Chen and Xu Shen.