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On The Hecke Orbit Conjecture for PEL Type Shimura Varieties


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.


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Shimura varieties, Hecke correspondences, Hecke orbits
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Prize for Excellence in Graduate Research and Teaching, 2017
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Mantovan, Elena
Thesis Committee:
  • Ramakrishnan, Dinakar (chair)
  • Zhu, Xinwen
  • Flach, Matthias
  • Mantovan, Elena
Defense Date:27 May 2020
Record Number:CaltechThesis:06012020-105456611
Persistent URL:
Xiao, Xiao (Luciena)0000-0002-1268-9744
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:13757
Deposited By: Xiao Xiao
Deposited On:08 Jun 2020 16:14
Last Modified:15 Jun 2020 18:07

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