Split abelian surfaces over finite fields and reductions of genus-2 curves

Abstract

For prime powers q, let s(q) denote the probability that a randomly-chosen principally-polarized abelian surface over the finite field F_q is not simple. We show that there are positive constants B and C such that for all q, B (log q)^{-3}(log log q)^{-4} < s(q)sqrt(q) < C (log q)^4(log log q)^2, and we obtain better estimates under the assumption of the generalized Riemann hypothesis. If A is a principally-polarized abelian surface over a number field K, let pi_split(A/K, z) denote the number of prime ideals p of K of norm at most z such that A has good reduction at p and A_p is not simple. We conjecture that for sufficiently general A, the counting function pi_split(A/K, z) grows like sqrt(z)/log z. We indicate why our theorem on the rate of growth of s(q) gives us reason to hope that our conjecture is true.