Points on Shimura varieties modulo primes

Abstract

We survey recent developments on the Langlands–Rapoport conjecture for Shimura varieties modulo primes and an analogous conjecture for Igusa varieties. We discuss resulting implications on the automorphic decomposition of the Hasse–Weil zeta functions and $\ell$-adic cohomology of Shimura varieties, along with further applications to the Langlands correspondence and related problems.