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arXiv · 2606.00466

Perfect powers in the product of denominators of elliptic curves

Subham Bhakta
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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 analogues of R. de la Bret\`eche, P. Kurlberg and I. E. Shparlinski (2021).

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Subham Bhakta. 2026-05-30. Perfect powers in the product of denominators of elliptic curves. https://arxiv.org/abs/2606.00466

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