Perfect powers in the product of denominators of elliptic curves

2026/05/30 by Subham Bhakta
Mathematics · #math.NT

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Abstract

We use sieving arguments to estimate the frequency of s-tuples of rational points (P₁,…,P_s)∈ E₁(ℚ)×⋯× E_s(ℚ), where E₁,…,E_s are (not necessarily distinct) elliptic curves over ℚ, for which the product of their denominators is a perfect ℓth power for a fixed prime ℓ. 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 Bretèche, P. Kurlberg and I. E. Shparlinski (2021).

Related

• Multiplicative dependence in the denominators of points of elliptic curves
• Perfect squares representing the number of rational points on elliptic curves over finite field extensions
• Perfect powers in elliptic divisibility sequences
• Explicitly bounding perfect powers in elliptic divisibility sequences
• The perfect power problem for elliptic curves over function fields