Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields

TL;DR

This paper explicitly characterizes how the torsion subgroup of rational elliptic curves with complex multiplication expands over quadratic fields, linking growth patterns to specific invariants of the curves.

Contribution

It provides a detailed, explicit description of quadratic fields where torsion growth occurs for CM elliptic curves, advancing understanding of torsion behavior over quadratic extensions.

Findings

Explicit criteria for torsion growth over quadratic fields

Identification of invariants linked to torsion expansion

Classification of quadratic fields based on torsion growth

Abstract

In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve.