Local-global compatibility of automorphic Galois representations over CM fields at $p$

Abstract

Let $F$ be a CM number field; then, to any cuspidal, regular algebraic automorphic representation of ${\mathrm{GL}}_n({\mathbf{A}}_F)$ is associated a compatible system of $p$-adic Galois representations of the absolute Galois group of $F$. We prove that these representations are potentially semi-stable, in the sense of $p$-adic Hodge theory, and satisfy compatibility with the local Langlands correspondence, up to semi-simplification.