#### Vol. 302, No. 2, 2019

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Torsion of rational elliptic curves over the maximal abelian extension of $\mathbb Q$

### Michael Chou

Vol. 302 (2019), No. 2, 481–509
##### Abstract

Let $E$ be an elliptic curve defined over $ℚ$, and let ${ℚ}^{ab}$ be the maximal abelian extension of $ℚ$. In this article we classify the groups that can arise as $E{\left({ℚ}^{ab}\right)}_{tors}$ 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 $E{\left({ℚ}^{ab}\right)}_{tors}$ for any elliptic curve $E∕ℚ$.

##### Keywords
elliptic curves, torsion, abelian, extension
Primary: 11G05
Secondary: 14H52
##### Milestones
Received: 12 April 2018
Revised: 4 January 2019
Accepted: 20 January 2019
Published: 27 November 2019
##### Authors
 Michael Chou Department of Mathematics Tufts University Medford, MA United States