Vanishing theorems for Shimura varieties

Abstract:  The Langlands program is a vast network of conjectures that connect number theory to other areas of mathematics, such as representation theory and harmonic analysis. The global Langlands correspondence can often be realised through the cohomology of Shimura varieties, which are certain moduli spaces equipped with many symmetries. In this talk, I will survey some recent vanishing results for the cohomology of Shimura varieties with mod p coefficients and mention several applications to the Langlands program and beyond. I will discuss some results that have an l-​adic flavour, where l is a prime different from p, that are primarily joint work with Peter Scholze. I will then mention some results that have a p-​adic flavour, that are primarily joint work with Dan Gulotta and Christian Johansson. I will highlight the different kinds of techniques that are needed in these different settings using the toy model of the modular curve.