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