Joint IAS/PU Arithmetic Geometry

Studying the \’etale cohomology of Shimura varieties with Hecke and Galois actions provides an avenue toward understanding the Langlands correspondence. 
While the structure of the rational cohomology groups is predicted conjectures of Kottwitz and Arthur,
the integral or mod $\ell$ cohomology groups are more intricate. Although a conjectural formula exists for computing these groups, it relies on an additional conjecture on the existence of moduli spaces of global Galois representations.
However, much more can be said if we focus on these cohomology groups with Hecke and local Galois actions. Specifically, 
I will present an unconditional formula for computing the mod $\ell$ cohomology groups of many Shimura varieties with Iwahori level at $p(\neq \ell$), expressed in terms of coherent sheaves on the moduli space of local Galois representations. This result not only provides evidence supporting the aforementioned conjectures but also has surprising arithmetic applications.