For prime powers , let denote the probability that a randomly chosen principally polarized abelian surface over the finite field is not simple. We show that there are positive constants and such that, for all ,
and we obtain better estimates under the assumption of the generalized Riemann hypothesis. If is a principally polarized abelian surface over a number field , let denote the number of prime ideals of of norm at most such that has good reduction at and is not simple. We conjecture that, for sufficiently general , the counting function grows like . We indicate why our theorem on the rate of growth of gives us reason to hope that our conjecture is true.
"Split abelian surfaces over finite fields and reductions of genus-2 curves." Algebra Number Theory 11 (1) 39 - 76, 2017. https://doi.org/10.2140/ant.2017.11.39