Torsion growth over cubic fields of rational elliptic curves with complex multiplication

TL;DR

This paper classifies how the torsion subgroup of rational elliptic curves with complex multiplication can grow over cubic fields, providing explicit descriptions based on elliptic curve invariants.

Contribution

It offers the first explicit classification of torsion growth for CM elliptic curves over cubic fields, detailing the possible growth patterns and invariants involved.

Findings

Explicit description of torsion growth over cubic fields for CM elliptic curves

Identification of invariants determining torsion growth

Complete classification of possible torsion subgroup expansions

Abstract

This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this particular case, we give a description of the possible torsion growth over cubic fields and a completely explicit description of this growth in terms of some invariants attached to a given elliptic curve.