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

Achter, Jeffrey; Howe, Everett

doi:10.2140/ant.2017.11.39

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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

DOI: 10.2140/ant.2017.11.39

Abstract

For prime powers $q$ , let $split (q)$ 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 $c_{1}$ and $c_{2}$ such that, for all $q$ ,

c_{1} {(log q)}^{- 3} {(log log q)}^{- 4} < split (q) \sqrt{q} < c_{2} {(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 $π_{split} (A ∕ K, z)$ 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 $π_{split} (A ∕ K, z)$ grows like $\sqrt{z} ∕ log z$ . We indicate why our theorem on the rate of growth of $split (q)$ 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

Milestones

Received: 14 October 2015

Revised: 5 October 2016

Accepted: 12 November 2016

Published: 23 January 2017

Authors

Jeffrey D. Achter
Department of Mathematics Colorado State University Weber Building Fort Collins, CO 80523-1874 United States

Everett W. Howe
Center for Communications Research 4320 Westerra Court San Diego, CA 92121-1969 United States

Vol. 11, No. 1, 2017