Ekedahl-Oort and Newton stratifications on unitary Shimura varieties, and on Hodge-Newton reducible local Shimura data of abelian type

Abstract

This thesis consists of two parts. In the first part, we develop techniques to study the interactions between Ekedahl-Oort stratification and BTm stratifications with Newton stratification on unitary Shimura varieties. We focus on the case of a unitary Shimura variety with signature (3,2). This work is in collaboration with Emerald Andrews, Deewang Bhamidipati, Maria Fox, Steven R. Groen, and Heidi Goodson. The second part addresses a new case of the Harris-Viehmann conjecture, which establishes a parabolic induction formula on the cohomology groups associated to non-basic local Shimura data. It follows that all supercuspidal representations on a Shimura variety are concentrated along the basic locus, making the conjecture relevant to the Langlands program. Historically, many cases of the Harris-Viehmann conjecture have been approached with the additional condition of Hodge-Newton reducibility on the underlying local Shimura datum. Building on previous work by E. Mantovan (EL/PEL case) and S. Hong (Hodge case), we extend the proof of the conjecture to unramified non-basic local Shimura data of abelian type under the assumption of Hodge-Newton reducibility. We leverage X. Shen's construction of Rapoport-Zink spaces of abelian type at the hyperspecial level. This is joint work with Xinyu Zhou.