New isogenies of elliptic curves over number fields

TL;DR

This paper proves a uniformity result for rational isogenies of elliptic curves over number fields, showing under GRH that certain isogenies are defined over the base field, with implications for Galois representations.

Contribution

It establishes a uniform bound B for number fields without rational CM, ensuring L-rational isogenies are F-rational when extension degrees are coprime to B, and analyzes mod-ell Galois representations for large primes.

Findings

Existence of an effectively computable constant B(F) under GRH.

L-rational isogenies are F-rational for extensions with degree coprime to B.

Results on mod-ell Galois representations for non-CM elliptic curves with large prime degree isogenies.

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 $\ell$.