Appears in collections : Conférences Paris Pékin Tokyo, Fields medallists - 2018

Shimura varieties with infinite level, and torsion in the cohomology of locally symmetric spaces. We will discuss the p-adic geometry of Shimura varieties with infinite level at p: They are perfectoid spaces, and there is a new period map defined at infinite level. As an application, we will discuss some results on torsion in the cohomology of locally symmetric spaces, and in particular the existence of Galois representations in this setup.