Torsion of Elliptic Curves over Quadratic Fields

Sophie De Arment; Jody Ryker

Torsion of Elliptic Curves over Quadratic Fields

Number Theory 2014-11-20 v1

Authors: Sophie De Arment , Jody Ryker

Abstract

By focusing on the family $E:y^2=x^3+a$ , 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.

Keywords

elliptic curves torsion subgroup quadratic fields Mordell-Weil group Nagell-Lutz theorem Mazur's theorem rational points

Cite

@article{arxiv.1411.5341,
  title  = {Torsion of Elliptic Curves over Quadratic Fields},
  author = {Sophie De Arment and Jody Ryker},
  journal= {arXiv preprint arXiv:1411.5341},
  year   = {2014}
}

Comments

7 pages

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