Explicit Kummer Theory for Elliptic Curves

2019/09/11 by Davide Lombardo, Lombardo, Davide, Sebastiano Tronto +1
Mathematics · #11F80 #14K15 #14L10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1909.05376

openalex publication_date 2019/09/11 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

Let E be an elliptic curve defined over a number field K, let α∈ E(K) be a point of infinite order, and let N^-1α be the set of N-division points of α in E(K). We prove strong effective and uniform results for the degrees of the Kummer extensions [K(E[N],N^-1α) : K(E[N])]. When K=ℚ, and under a minimal assumption on α, we show that the inequality [ℚ(E[N],N^-1α) : ℚ(E[N])] ≥ cN² holds with a constant c independent of both E and α.

Citations

• Small rational points on elliptic curves over number fields
• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques

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