Non-simple abelian varieties in a family: geometric and analytic approaches

2008/04/14 by Jordan S. Ellenberg, Christian Elsholtz, Ellenberg, J. +5
Computer Science · Mathematics · #11G10 (Primary) #11N35 #14D05 (Secondary) #14K15 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.0804.2166

openalex publication_date 2008/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A_t be a family of abelian varieties over a number field k parametrized by a rational coordinate t, and suppose the generic fiber of A_t is geometrically simple. For example, we may take A_t to be the Jacobian of the hyperelliptic curve y² = f(x)(x-t) for some polynomial f. We give two upper bounds for the number of t ∈ k of height at most B such that the fiber A_t is geometrically non-simple. One bound comes from arithmetic geometry, and shows that there are only finitely many such t; but one has very little control over how this finite number varies as f changes. Another bound, from analytic number theory, shows that the number of geometrically non-simple fibers grows quite slowly with B; this bound, by contrast with the arithmetic one, is effective, and is uniform in the coefficients of f. We hope that the paper, besides proving the particular theorems we address, will serve as a good example of the strengths and weaknesses of the two complementary approaches.

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