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

TL;DR

This paper investigates which prime numbers allow elliptic curves over  to have an r-isogeny over (r) and explores implications for the Darmon program related to generalized Fermat equations.

Contribution

It determines the primes r for which elliptic curves over  have r-isogenies over (r), considering various 2-torsion conditions, with applications to modularity proofs.

Findings

Identifies primes r permitting r-isogenies over (r)

Connects the study to the Darmon program and generalized Fermat equations

Provides criteria based on 2-torsion properties of elliptic curves

Abstract

The main goal of this paper is to determine for which prime numbers $r\geq 3$ 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.