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2017 Split abelian surfaces over finite fields and reductions of genus-2 curves
Jeffrey Achter, Everett Howe
Algebra Number Theory 11(1): 39-76 (2017). 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 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.

Citation

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Jeffrey Achter. Everett Howe. "Split abelian surfaces over finite fields and reductions of genus-2 curves." Algebra Number Theory 11 (1) 39 - 76, 2017. https://doi.org/10.2140/ant.2017.11.39

Information

Received: 14 October 2015; Revised: 5 October 2016; Accepted: 12 November 2016; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1357.14059
MathSciNet: MR3602766
Digital Object Identifier: 10.2140/ant.2017.11.39

Subjects:
Primary: 14K15
Secondary: 11G10, 11G20, 11G30

Rights: Copyright © 2017 Mathematical Sciences Publishers

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