  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.
