On r-isogenies over $$\mathbb {Q}(\zeta _r)$$ of elliptic curves with rational j-invariants

Abstract

The main goal of this paper is to determine for which odd prime numbers r can an elliptic curve E defined over \(\mathbb {Q}\) have an r-isogeny over \(\mathbb {Q}(\zeta _r)\). We study this question under various assumptions on the 2-torsion of E. Apart from being a natural question itself, the mod r representations attached to such E arise in the Darmon program for the generalized Fermat equation of signature \((r,r,p)\), playing a key role in the proof of modularity of certain Frey varieties in the recent work of Billerey, Chen, Dieulefait and Freitas.