Perfect powers in elliptic divisibility sequences

Abstract

Let $E$ be an elliptic curve over the rationals given by an integral Weierstrass model and let $P$ be a rational point of infinite order. The multiple $nP$ has the form $(A_n/B_n^2,C_n/B_n^3)$ where $A_n$, $B_n$, $C_n$ are integers with $A_n C_n$ and $B_n$ coprime, and $B_n$ positive. The sequence $(B_n)$ is called the elliptic divisibility sequence generated by $P$. This paper is concerned with a question posed in 2007 by Everest, Reynolds and Stevens: does the sequence $(B_n)$ contain only finitely many perfect powers? We answer this question positively under three additional assumptions: $P$ is non-integral, the discriminant of $E$ is positive, and $P$ belongs to the connected real component of the identity on $E$. Our method attaches to the problem a Frey curve that is defined over a totally real field of degree at most $24$, and then makes use of modularity and level lowering arguments. We can deduce the same theorem without assuming that the discriminant of $E$ is positive, or assuming that $P$ belongs to the connected real component of the identity, provided we assume some standard conjectures from the Langlands programme.