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(q) denote the probability that a randomly chosen principally polarized abelian surface over the finite field Fq is not simple. We show that there are positive constants c1 and c2 such that, for all q,

c1(logq)3(loglogq)4 < split(q)q < c 2(logq)4(loglogq)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 πsplit(AK,z) denote the number of prime ideals p of K of norm at most z such that A has good reduction at p and Ap is not simple. We conjecture that, for sufficiently general A, the counting function πsplit(AK,z) grows like zlogz. 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