UC Berkeley

The connection between the Langlands correspondence and the co-homology of Rapoport-Zink spaces and Shimura varieties has been the subject of extensive mathematical research over the past few decades. In this thesis, we extend the existing theory in two key ways. Firstly,we give an explicit combinatorial description of the cohomology of Rapoport-Zink spaces of EL-type, building off of earlier work by Harris–Taylor and Shin ([HT01], [Shi12b], [HT01]). Secondly, joint with Alex Youcis, we state a list of axioms for the supercuspidal local Langlands correspondence and prove that they characterize the correspondence in certain cases. The most important of our axioms arises naturally in the study of the cohomology of Shimura varieties and was first stated in work of Scholze and Scholze–Shin ([Sch13b], [SS13]). We verify these axioms in the case of unramified unitary groups.1