Abelian surfaces with real multiplication over finite fields
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 ...