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For prime powers q, let s(q) denote the probability that a randomly-chosenprincipally-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 generalizedRiemann hypothesis. If A is a principally-polarized abelian surface over anumber field K, let pi_split(A/K, z) denote the number of prime ideals p of Kof norm at most z such that A has good reduction at p and A_p is not simple. Weconjecture 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 growthof s(q) gives us reason to hope that our conjecture is true.

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