Perfect powers in the product of denominators of elliptic curves

Abstract

We use sieving arguments to estimate the frequency of $s$-tuples of rational points $$(P_1,\dots,P_s)\in E_1(\mathbb{Q})\times\cdots\times E_s(\mathbb{Q}),$$ where $E_1,\dots,E_s$ are (not necessarily distinct) elliptic curves over $\mathbb{Q}$, for which the product of their denominators is a perfect $\ell$th power for a fixed prime $\ell$. We consider two settings: one in which the points are of the form $n_iP_i+Q_i$ with $n_i$ ranging over an interval, and another in which we take arbitrary points of bounded canonical height. In the special case where all $Q_i$ are the points at infinity, we also obtain better estimates by using a version of the elliptic sieve with elliptic divisibility sequences. Consequently, we derive analogues of these results for various rational functions, providing elliptic analogs of R. de la Bretèche, P. Kurlberg and I. E. Shparlinski (2021).