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arXiv · 1509.00528

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

Harris B. Daniels·Alvaro Lozano-Robledo·Filip Najman·Andrew V. Sutherland
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Abstract

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 infinitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves, and a complete list of $j$-invariants for each of the 4 that do not.

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Harris B. Daniels, Alvaro Lozano-Robledo, Filip Najman, Andrew V. Sutherland. 2015-09-02. Torsion subgroups of rational elliptic curves over the compositum of all cubic fields. https://doi.org/10.1090/mcom/3213

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