Torsion of Elliptic Curves over Quadratic Fields

2014/11/19 by Sophie De Arment, De Arment, Sophie, Jody Ryker +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1411.5341

openalex publication_date 2014/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By focusing on the family E:y²=x³+a, we present strategies for determining the structure of the torsion subgroup of the Mordell-Weil group of an elliptic curve, E(K), over quadratic field K. Generalizations of the Nagell-Lutz theorem and Mazur's theorem to curves defined over quadratic fields allows us to determine the full torsion subgroup of E(K) as one of at most three possibilities when a is a square.

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