Freese, Hilary, authorAchter, Jeffrey, advisorPries, Rachel, committee memberPeterson, Chris, committee memberTavani, Daniele, committee member2007-01-032007-01-032014http://hdl.handle.net/10217/83742Given 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).born digitaldoctoral dissertationsengCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.algebraic geometrynumber theoryText