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

Abstract

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