# Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

@article{Daniels2015TorsionSO, title={Torsion subgroups of rational elliptic curves over the compositum of all cubic fields}, author={Harris B. Daniels and {\'A}lvaro Lozano-Robledo and Filip Najman and Andrew V. Sutherland}, journal={Math. Comput.}, year={2015}, volume={87}, pages={425-458} }

Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(3^\infty)$ be the compositum of all cubic extensions of $\mathbb{Q}$. In this article we show that the torsion subgroup of $E(\mathbb{Q}(3^\infty))$ is finite and determine 20 possibilities for its structure, along with a complete description of the $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for…

## 19 Citations

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This article is a first step towards a complete classification of torsion growth of over sextic fields.

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Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\rho_E\colon {\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to {\rm GL}(2,\hat{\mathbb{Z}})$ be the adelic representation associated to the…

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The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More…

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- 2016

### $\ell $ -adic images of Galois for elliptic curves over $\mathbb {Q}$ (and an appendix with John Voight)

- MathematicsForum of Mathematics, Sigma
- 2022

Abstract We discuss the
$\ell $
-adic case of Mazur’s ‘Program B’ over
$\mathbb {Q}$
: the problem of classifying the possible images of
$\ell $
-adic Galois representations attached to…

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- 2018

### Torsion of elliptic curves with rational $j$-invariant defined over number fields of prime degree

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Let $[K:\mathbb{Q}]=p$ be a prime number and let $E/K$ be an elliptic curve with $j(E) \in \mathbb{Q}$. We determine the all possibilities for $E(K)_{tors}$. We obtain these results by studying…

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- 2019

Recently, there has been much interest in studying the torsion subgroups of elliptic curves base‐extended to infinite extensions of Q . In this paper, given a finite group G , we study what happens…

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Abstract Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre’s constant…

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