Endoscopy and the Cohomology of Shimura Varieties Associated with Unitary Groups

This monograph establishes a rigorous classification of the automorphic discrete spectrum appearing in the intersection cohomology of Shimura varieties associated with unitary groups of signature (p,q). Leveraging the stabilization of the Arthur-Selberg trace formula, we analyze the endoscopic transfer of automorphic representations from endoscopic groups to the quasi-split unitary group. We explicitly describe the decomposition of the cohomology in terms of global Arthur parameters, proving that the packet structure predicted by the Langlands conjectures manifests geometrically. Furthermore, we address the construction of Galois representations associated with these cohomological automorphic forms, extending results of Kottwitz, Rogawski, and Mok. The analysis incorporates the contributions of the stable trace formula at non-archimedean places and the description of the discrete series at the archimedean place, providing a comprehensive arithmetic-geometric synthesis.