Torsion of rational elliptic curves over the maximal abelian extension of Q

TL;DR

This paper classifies the possible torsion groups of rational elliptic curves over the maximal abelian extension of Q, providing explicit models and an algorithm for computation.

Contribution

It introduces a classification of torsion groups over , along with methods for explicit modeling and an algorithm for computing torsion subgroups.

Findings

Classification of torsion groups over 

Explicit models of modular curves of mixed level structure

Algorithm for computing torsion groups

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\mathbb{Q}^{ab}$ be the maximal abelian extension of $\mathbb{Q}$. In this article we classify the groups that can arise as $E(\mathbb{Q}^{ab})_{\text{tors}}$ up to isomorphism. The method illustrates techniques for finding explicit models of modular curves of mixed level structure. Moreover we provide an explicit algorithm to compute $E(\mathbb{Q}^{ab})_{\text{tors}}$ for any elliptic curve $E/\mathbb{Q}$.