Vol. 11, No. 1, 2017

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 8, 1739–1965
Issue 7, 1489–1738
Issue 6, 1243–1488
Issue 5, 1009–1241
Issue 4, 767–1007
Issue 3, 505–765
Issue 2, 253–503
Issue 1, 1–252

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editors' Addresses
Editors' Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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