1. Outline & Descriptions

Shimura varieties are generalizations of modular curves, whose cohomology is useful for relating automorphic and Galois representations in the context of Langlands program.  For each prime number p, the associated p-adic completed cohomology naturally captures lots of p-adic interpolations of such relations.  In this lecture series, I will start with a general introduction to Shimura varieties, and explain some background materials in complex and p-adic Hodge theories.  After these preparations, I will explain some vanishing results for the rational completed cohomology of Shimura varieties in my joint work in progress with Lue Pan.  I will provide concrete examples when possible.