Stability of torsion subgroups of elliptic curves over non-Galois extensions of odd prime degree

TL;DR

This paper investigates the stability of torsion subgroups of elliptic curves over non-Galois extensions of prime degree, providing Galois-theoretic conditions and refining known results for quintic extensions.

Contribution

It establishes new Galois-theoretic criteria for torsion stability and refines existing classifications for torsion structures over quintic non-Galois extensions.

Findings

Galois-theoretic conditions for torsion stability

Relation between torsion growth and cyclotomic character

Refinement of torsion structure classifications for quintic extensions

Abstract

Let $K$ be a field of characteristic $0$ and $E/K$ an elliptic curve over $K$. For a finite extension $L/K$ and a prime~$\ell$, we provide Galois-theoretic sufficient conditions on $L/K$ under which $E\left(L\right)\left[\ell^{\infty}\right] = E\left(K\right)\left[\ell^{\infty}\right]$. For a non-Galois extension $L/K$ of prime degree, we relate the growth of the $\ell^{\infty}$-torsion subgroup of $E$ under the base change $L/K$ to the image of the mod-$\ell$ cyclotomic character. In particular, In particular, we refine Gonz{\'a}lez-Jim{\'e}nez's result by ruling out certain torsion structures for quintic non-Galois extensions $L/\mathbb{Q}$.