Explicitly bounding perfect powers in elliptic divisibility sequences

Abstract

In this paper we consider elliptic divisibility sequences generated by a point on an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$ given by an integral short Weierstrass equation. For several different such elliptic divisibility sequences, we determine explicitly a finite set of primes such that for all primes $l$ outside this set, the elliptic divisibility sequence contains no $l$-th powers. Our approach uses, amongst other ingredients, the modular method for $\mathbb{Q}$-curves. The corresponding explicit computations fit into a general computational framework for $\mathbb{Q}$-curves, for which they provide illustrative examples.