On The Hecke Orbit Conjecture for PEL Type Shimura Varieties

Citation

Xiao, Xiao (Luciena) (2020) On The Hecke Orbit Conjecture for PEL Type Shimura Varieties. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/sjg9-0688. https://resolver.caltech.edu/CaltechThesis:06012020-105456611

Abstract

The Hecke orbit conjecture plays an important role in understanding the geometric structure of Shimura varieties. First postulated by Chai and Oort in 1995, the Hecke orbit conjecture predicts that prime-to-p Hecke correspondences on mod p reductions of Shimura varieties characterize the foliation structure formed by Oort's central leaves. In other words, every prime-to-p Hecke orbit is Zariski dense in the central leaf containing it. Roughly speaking, a central leaf is the locus in a Shimura variety consisting of all points whose corresponding Barsotti-Tate groups belong to a fixed geometric isomorphism class. On the other hand, the prime-to-p Hecke orbit of a closed point x is the (countable) set consisting of all points y such that there is a prime-to-p quasi-isogeny from x to y.

In 2005, Chai and Yu proved the Hecke orbit conjecture for Hilbert modular varieties, followed by a proof for Siegel modular varieties by Chai and Oort in the same year. The major purpose of the present work is to generalize the method of Chai and Oort to Shimura varieties of PEL type. We show that the Hecke orbit conjecture holds for points in certain irreducible components of Newton strata under our assumptions.

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