Counting rational points on weighted projective spaces over number fields

Abstract

Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a morphism between weighted projective spaces, allowing us to count rational points whose image under this morphism has bounded height. This method provides a more general and simpler proof for a result of the first-named author and Najman on counting elliptic curves with prescribed level structures over number fields. We further include some examples of applications to modular curves.