Isogenies of non-CM elliptic curves with rational $j$-invariants over number fields

TL;DR

This paper characterizes the possible prime degrees of cyclic isogenies of non-CM elliptic curves with rational j-invariants over number fields of degree up to 7, providing exact sets under certain conjectures.

Contribution

It unconditionally determines the set of possible prime degrees of cyclic isogenies for elliptic curves over number fields of degree up to 7, and, assuming Serre's conjecture, does so for all degrees.

Findings

Unconditional determination of isogeny degrees for degree ≤ 7.

Upper bounds for isogeny degrees for degree > 7.

Exact determination of isogeny degrees assuming Serre's conjecture.

Abstract

We unconditionally determine $I_\Q(d)$, the set of possible prime degrees of cyclic $K$-isogneies of elliptic curves with $\Q$-rational $j$-invariants and without complex multiplication over number fields $K$ of degree $\leq d$, for $d\leq 7$, and give an upper bound for $I_\Q(d)$ for $d>7$. Assuming Serre's uniformity conjecture, we determine $I_\Q(d)$ exactly for all positive integers $d$.