Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of $\mathbb{Q}$

TL;DR

This paper classifies the possible torsion groups of rational elliptic curves over cyclotomic extensions of the rationals, providing a detailed understanding of torsion structures in these fields and tools to exclude non-occurring cases.

Contribution

It offers a complete classification of torsion groups over cyclotomic extensions and introduces methods to determine which torsion structures can occur.

Findings

Classified all torsion groups over $Q(zeta_p)$ for prime $p$.

Identified conditions when the classification can be sharpened.

Developed tools to eliminate impossible torsion structures in abelian extensions.

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ up to isomorphism for any prime $p$. When $p - 1$ is not divisible by small integers such as $3, 4, 5, 7$, or $11$, we obtain a sharper classification. For any abelian number field $K$, the torsion subgroup $E(K)_{\text{tors}}$ is a subgroup of $E(\mathbb{Q}^{\text{ab}})_{\text{tors}}$. Our methods provide tools to eliminate non-realized torsion structures from the list of possibilities for $E(K)_{\text{tors}}$.