#### Vol. 11, No. 1, 2017

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

### Jeffrey D. Achter and Everett W. Howe

Vol. 11 (2017), No. 1, 39–76
##### 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}

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.

 Dedicated to the memory of Professor Tom M. Apostol
##### Keywords
abelian surface, curve, Jacobian, reduction, simplicity, reducibility, counting function
##### Mathematical Subject Classification 2010
Primary: 14K15
Secondary: 11G10, 11G20, 11G30