Torsion of rational elliptic curves over quadratic fields II

TL;DR

This paper classifies how the torsion subgroup of rational elliptic curves can change over quadratic fields, providing a comprehensive list of possible torsion groups and examples for each case.

Contribution

It extends previous work by explicitly computing all possible torsion group enlargements over quadratic fields and providing examples for each scenario.

Findings

Identifies all possible torsion group enlargements over quadratic fields.

Provides explicit examples for each possible torsion group change.

Determines the maximum number of quadratic fields where torsion can increase.

Abstract

Let E be an elliptic curve defined over Q and let G=E(Q)_tors be the associated torsion group. In a previous paper, the authors studied, for a given G, which possible groups G\leq H could appear such that H=E(K)_tors, for [K:Q]=2. In the present paper, we go further in this study and compute, under this assumption and for every such G, all the possible situations where G\neq H. The result is optimal, as we also display examples for every situation we state as possible. As a consequence, the maximum number of quadratic number fields K such that E(Q)_tors\neq E(K)_tors is easily obtained.