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Γ(p)-Level Structure on p-Divisible Groups


Frimu, Andrei (2019) Γ(p)-Level Structure on p-Divisible Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8JYH-KT84.


The main result of the thesis is the introduction of a notion of Γ(p)-level structure for p-divisible groups. This generalizes the Drinfeld-Katz-Mazur notion of full level structure for 1-dimensional p-divisible groups. The associated moduli problem has a natural forgetful map to the Γ0(p)-level moduli problem. Exploiting this map and known results about Γ0(p)-level, we show that our notion yields a flat moduli problem. We show that in the case of 1-dimensional p-divisible groups, it coincides with the existing Drinfeld-Katz-Mazur notion.

In the second half of the thesis, we introduce a notion of epipelagic level structure. As part of the task of writing down a local model for the associated moduli problem, one needs to understand commutative finite flat group schemes G of order p2 killed by p, equipped with an extension structures 0→ H1→ G→ H2→ 0, where H1,H2 are finite flat of order p. We investigate a particular class of extensions, namely extensions of Z/pZ by μp over Zp-algebras. These can be classified using Kummer theory. We present a different approach, which leads to a more explicit classification.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:arithmetic geometry; algebraic geometry; p-divisible groups; level structures; finite flat group schemes; shimura varieties
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Apostol Award for Excellence in Teaching in Mathematics, 2015.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Zhu, Xinwen
Thesis Committee:
  • Mantovan, Elena (chair)
  • Zhu, Xinwen
  • Ramakrishnan, Dinakar
  • Flach, Matthias
Defense Date:22 May 2019
Record Number:CaltechTHESIS:05302019-185557287
Persistent URL:
Frimu, Andrei0000-0002-7782-7204
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11575
Deposited By: Andrei Frimu
Deposited On:10 Jun 2019 22:37
Last Modified:04 Oct 2019 00:26

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