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

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.

Corrects an error in the proofs of Lemma 3.2 and Theorem 3.6; 32 pages

References in corpus (4)

• Modular curves of prime-power level with infinitely many rational points
• On the possible images of the mod ell representations associated to elliptic curves over Q
• Torsion of rational elliptic curves over quartic Galois number fields
• Possible indices for the Galois image of elliptic curves over Q

Cited by in corpus (9)

• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$
• On the torsion of rational elliptic curves over quartic fields
• Serre's constant of elliptic curves over the rationals
• Computing isomorphisms and embeddings of finite fields
• Torsion of rational elliptic curves over the maximal abelian extension of Q
• An algorithm for determining torsion growth of elliptic curves
• Groups of generalized $G$-type and applications to torsion subgroups of rational elliptic curves over infinite extensions of $\mathbb{Q}$
• Growth of torsion of elliptic curves with odd-order torsion over quadratic cyclotomic fields
• Torsion groups of elliptic curves over some infinite abelian extensions of $\mathbb{Q}$