Here is the official version, and a preprint version.
For prime powers q, let s(q) denote the probability that a randomly-chosen principally-polarized abelian surface over the finite field Fq 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) q1/2 < 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 πsplit(A/K, z) denote the number of prime ideals ℘ of K of norm at most z such that A has good reduction at ℘ and A℘ is not simple. We conjecture that for sufficiently general A, the counting function πsplit(A/K, z) grows like z1/2/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.