Torsion of Elliptic Curves over Quadratic Fields

TL;DR

This paper develops methods to determine the torsion subgroup structure of elliptic curves over quadratic fields, extending classical theorems to identify possible torsion configurations.

Contribution

It generalizes Nagell-Lutz and Mazur's theorems to quadratic fields, enabling explicit classification of torsion subgroups for specific elliptic curves.

Findings

Full torsion subgroup classification for curves with a square coefficient

Extension of classical theorems to quadratic fields

Identification of at most three possible torsion structures

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.