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

Daniels, Harris; Lozano-Robledo, Álvaro; Najman, Filip; Sutherland, Andrew

doi:10.1090/mcom/3213

Torsion subgroups of rational elliptic curves over the compositum of all cubic fields
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by Harris B. Daniels, Álvaro Lozano-Robledo, Filip Najman and Andrew V. Sutherland PDF

Math. Comp. 87 (2018), 425-458 Request permission

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 we 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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