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

Abstract

For prime powers $q$, let $split\left(q\right)$ denote the probability that a randomly chosen principally polarized abelian surface over the finite field ${\mathbb{F}}_q$ is not simple. We show that there are positive constants $c_1$ and $c_2$ such that, for all $q$,

$$c_1{\left(logq\right)}^{-3}{\left(loglogq\right)}^{-4}<split\left(q\right)\sqrt{q}<c_2{\left(logq\right)}^4{\left(loglogq\right)}^2,$$

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 ${\pi }_{split}\left(A∕K,z\right)$ denote the number of prime ideals $\mathfrak{p}$ of $K$ of norm at most $z$ such that $A$ has good reduction at $\mathfrak{p}$ and $A_{\mathfrak{p}}$ is not simple. We conjecture that, for sufficiently general $A$, the counting function ${\pi }_{split}\left(A∕K,z\right)$ grows like $\sqrt{z}∕logz$. We indicate why our theorem on the rate of growth of $split\left(q\right)$ gives us reason to hope that our conjecture is true.