Specializations of elliptic surfaces, and divisibility in the Mordell-Weil group

2008/11/19 by Patrick Ingram, Ingram, Patrick
Mathematics · #14G05 #14J27 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0811.3109

openalex publication_date 2008/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E be an elliptic surface over the curve C, defined over a number field k, let P be a section of E, and let ℓ be a rational prime. For any non-singular fibre E_t, we bound the number of points Q on E_t of (algebraic) degree at most D over k, such that ℓⁿ Q=P_t, for some n≥ 1. The bound obtained depends only on ℓ, the surface and section in question, D, and the degree [k(t):k]; that is, it is uniform across all fibres of bounded degree. In special cases, we obtain more specific, in some instances sharp, bounds.

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