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

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 $L/\mathbb{Q}$.