New isogenies of elliptic curves over number fields

Abstract: We analyze the fields of definition of cyclic isogenies on elliptic curves to prove the following uniformity result: for any number field $F_0$ which satisfies an isogeny condition, there exists a constant $B:=B(F_0)\in\mathbb{Z}^+$ such that for any finite extension $L/F_0$ whose degree $[L:F_0]$ is coprime to $B$, one has for all elliptic curves $E_{/F_0}$ with $j$-invariant $\neq 0, 1728$ that any $L$-rational cyclic isogeny on $E$ must be $F_0$-rational. We also prove unconditional results for the mod-$\ell$ Galois representations of non-CM elliptic curves with an $F_0$-rational $\ell$-isogeny when $\ell$ is uniformly large.