On Hodge-Newton Reducible Local Shimura Data of Hodge Type

Abstract

Rapoport-Zink spaces are formal moduli spaces of p -divisible groups which give rise to local analogues of certain Shimura varieties. In particular, one can construct them from purely group theoretic data called local Shimura data . The primary purpose of this dissertation is to study Rapoport-Zink spaces whose underlying local Shimura datum is of Hodge type and Hodge-Newton reducible. Our study consists of two main parts: the study of the l -adic cohomology of Rapoport-Zink spaces in relation to the local Langlands correspondence and the study of deformation spaces of p -divisible groups via the local geometry of Rapoport-Zink spaces. The main result of the first part is a proof of the Harris-Viehmann conjecture in our setting; in particular, we prove that the l -adic cohomology of Rapoport-Zink spaces contains no supercuspidal representations under our assumptions. In the second part, we obtain a generalization of Serre-Tate deformation theory for Shimura varieties of Hodge type.