Dissertation/Thesis Abstract

Commutative endomorphism rings of simple abelian varieties over finite fields
by Bradford, Jeremy, Ph.D., University of Maryland, College Park, 2012, 78; 3557641
Abstract (Summary)

In this thesis we look at simple abelian varieties defined over a finite field k = [special characters omitted]pn with Endk( A) commutative. We derive a formula that connects the p -rank r(A) with the splitting behavior of p in E = [special characters omitted](π), where π is a root of the characteristic polynomial of the Frobenius endomorphism. We show how this formula can be used to explicitly list all possible splitting behaviors of p in [special characters omitted]E, and we do so for abelian varieties of dimension less than or equal to four defined over [special characters omitted]p. We then look for when p divides [[special characters omitted]E : [special characters omitted][π, π]]. This allows us to prove that the endomorphism ring of an absolutely simple abelian surface is maximal at p when p ≥ 3. We also derive a condition that guarantees that p divides [[special characters omitted]E: [special characters omitted][π, π]]. Last, we explicitly describe the structure of some intermediate subrings of p-power index between [special characters omitted][π, π] and [special characters omitted]E when A is an abelian 3-fold with r(A) = 1.

Indexing (document details)
Advisor: Washington, Lawrence C.
Commitee: Brosnan, Patrick, Gasarch, William, Ramachandran, Niranjan, Tamvakis, Harry
School: University of Maryland, College Park
Department: Mathematics
School Location: United States -- Maryland
Source: DAI-B 74/08(E), Dissertation Abstracts International
Subjects: Mathematics
Keywords: Abelian varieties, Endomorphism rings, Finite fields
Publication Number: 3557641
ISBN: 9781303010651
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