Perfect powers in products of denominators of rational points on elliptic curves

Abstract

Given a rational point P on an elliptic curve, the denominator of its x-coordinate carries interesting arithmetic information. A natural question is how often such denominators, or products of such denominators, are perfect powers. This problem is closely related to counting rational points on certain generalized Fermat curves. In this talk, I will explain how sieve methods can be used to estimate the number of tuples of rational points on elliptic curves for which the product of their denominators, or of their x-coordinates, is a perfect power.