Torsion groups of elliptic curves over $\mathbb{Q}(\mu_{p^{\infty}})$

TL;DR

This paper classifies the torsion subgroups of elliptic curves over specific cyclotomic fields, extending known results to infinite p-power cyclotomic extensions for primes 5, 7, 11, and particular cases 16 and 27.

Contribution

It provides a complete classification of torsion groups of elliptic curves over cyclotomic fields generated by p-power roots of unity, including new cases for primes 5, 7, 11, 16, and 27.

Findings

Classified torsion groups over $Q(zeta_{p})$ for p=5,7,11.

Determined torsion possibilities over $Q(zeta_{16})$ and $Q(zeta_{27})$.

Extended classification to $Q(mu_{p^{rown}})$ for primes p=5,7,11.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and $p \in \{5,7,11 \}$ be a prime. We determine the possibilities for $E(\mathbb{Q}(\zeta_{p}))_{tors}$. Additionally, we determine all the possibilities for $E(\mathbb{Q}(\zeta_{16}))_{tors}$ and $E(\mathbb{Q}(\zeta_{27}))_{tors}$. Using these results we are able to determine the possibilities for $E(\mathbb{Q}(\mu_{p^{\infty}}))_{tors}$.