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

TL;DR

This paper investigates the probability that a randomly chosen abelian surface over a finite field is not simple, providing bounds and conjectures related to reductions of genus-2 curves and their properties over number fields.

Contribution

The authors establish bounds on the probability s(q) that a principally-polarized abelian surface over F_q is not simple, and connect these results to conjectures on reductions of genus-2 curves over number fields.

Findings

Bounds on s(q) involving logarithmic factors

Improved estimates under the generalized Riemann hypothesis

Conjecture on the growth rate of prime reductions of abelian surfaces

Abstract

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.