New isogenies of elliptic curves over number fields

Abstract

Using Galois representations, we analyze fields of definition of cyclic isogenies on elliptic curves to prove the following uniformity result: for any number field $F$ which has no rational CM, under GRH there exists an effectively computable constant $B:=B(F)\in\mathbb{Z}^+$ such that for any finite extension $L/F$ whose degree $[L:F]$ is coprime to $B$, one has for all elliptic curves $E_{/F}$ that any $L$-rational isogeny on $E$ is $F$-rational. For any number field $F$, under GRH we also prove results for the mod-$\ell$ Galois representations of non-CM elliptic curves with an $F$-rational isogeny of uniformly large prime degree $\ell$.