Abelian surfaces with real multiplication over finite fields
Date
2014
Authors
Freese, Hilary, author
Achter, Jeffrey, advisor
Pries, Rachel, committee member
Peterson, Chris, committee member
Tavani, Daniele, committee member
Journal Title
Journal ISSN
Volume Title
Abstract
Given a simple abelian surface A/Fq, the endomorphism algebra, End(A) ⊗ Q, contains a unique real quadratic subfield. We explore two different but related questions about when a particular real quadratic subfield K+ is the maximal real subfield of the endomorphism algebra. First, we compute the number of principally polarized abelian surfaces A/Fq such that K+ ⊂ End(A) ⊗ Q. Second, we consider an abelian surface A/Q, and its reduction Ap = A mod p, then ask for which primes p is K+ ⊂ End(A) ⊗ Q. The result from the first question leads to a heuristic for the second question, namely that the number of p < χ for which K+ ⊂ End(A) ⊗ Q grows like √χ/log(c).
Description
Rights Access
Subject
algebraic geometry
number theory