Perfect powers in elliptic divisibility sequences

Abstract

AbstractLet be an elliptic curve given by an integral Weierstrass equation. Let be a point of infinite order, and let be the elliptic divisibility sequence generated by . This paper is concerned with a question posed in 2007 by Everest, Reynolds and Stevens: does contain only finitely many perfect powers? We answer this question positively under the following three additional assumptions: is non‐integral; , where is the discriminant of ; , where denotes the connected component of identity. Our method makes use of Galois representations attached to elliptic curves defined over totally real fields, and their modularity. We can deduce the same theorem without assumptions (ii) and (iii), provided that we assume some standard conjectures from the Langlands programme.