Growth of torsion groups of elliptic curves upon base change from number fields

Abstract

Given a number field $F_0$ that contains no Hilbert class field of any imaginary quadratic field, we show that under GRH there exists an effectively computable constant $B:=B(F_0)\in\mathbb{Z}^+$ for which the following holds: for any finite extension $L/F_0$ whose degree $[L:F_0]$ is coprime to $B$, one has for all elliptic curves $E_{/F_0}$ that the $L$-rational torsion subgroup $E(L)[\textrm{tors}]=E(F_0)[\textrm{tors}]$. This generalizes a previous result of Gonz\'{a}lez-Jim\'{e}nez and Najman over $F_0=\mathbb{Q}$. Towards showing this, we also prove a result on relative uniform divisibility of the index of a mod-$\ell$ Galois representation of an elliptic curve over $F_0$. Additionally, we show that the main result's conclusion fails when we allow $F_0$ to have rationally defined CM, due to the existence of $F_0$-rational isogenies of arbitrarily large prime degrees satisfying certain congruency conditions.