On r-isogenies over ℚ(ζ_r) of elliptic curves with rational j-invariants

2023/07/26 by Filip Najman, Najman, Filip
Arts and Humanities · Mathematics · #Historical Studies and Socio-cultural Analysis #Algebraic Geometry and Number Theory #French Historical and Cultural Studies

paper · pdf · doi:10.48550/arxiv.2307.14131

Abstract

The main goal of this paper is to determine for which prime numbers r≥ 3 can an elliptic curve~E defined over \mathbb Q have an r-isogeny over \mathbb Q(ζ_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.

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