On the cohomology of compact unitary group Shimura varieties at ramified split places

Abstract

In this article, we prove results about the cohomology of compact unitary group Shimura varieties at split places. In nonendoscopic cases, we are able to give a full description of the cohomology, after restricting to integral Hecke operators at p on the automorphic side. We allow arbitrary ramification at p; even the PEL data may be ramified. This gives a description of the semisimple local Hasse-Weil zeta function in these cases. We also treat cases of nontrivial endoscopy. For this purpose, we give a general stabilization of the expression given in previous work, following the stabilization given by Kottwitz. This introduces endoscopic transfers of the functions $\phi_{\tau,h}$ which were introduced in previous work via deformation spaces of $p$-divisible groups. We state a general conjecture relating these endoscopic transfers with Langlands parameters. We verify this conjecture in all cases of EL type, and deduce new results about the endoscopic part of the cohomology of Shimura varieties. This allows us to simplify the construction of Galois representations attached to conjugate self-dual regular algebraic cuspidal automorphic representations of $\mathrm{GL}_n$, as previously constructed by one of us.