Jeffrey D. Achter and Everett W. Howe: **Split abelian surfaces over finite fields and reductions of genus-2 curves**,
*Algebra Number Theory* **11** (2017) 39–76.
**MR** 3349314,
**Zbl** 06679112.
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 **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*) *q*^{1/2}
<
*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 π_{split}(*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 π_{split}(*A*/*K*, *z*) grows like *z*^{1/2}/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.