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

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 c(1) and c(2) such that for all q, c(1)(log q)(-3)(log log q)(-4) < split(q)root q < c(2) (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 root 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.