Counting rational points on weighted projective spaces over number fields

TL;DR

This paper generalizes Deng's asymptotic formula for counting rational points on weighted projective spaces over number fields, using morphisms to extend the counting to images with bounded height, with applications to elliptic curves and modular curves.

Contribution

It introduces a new approach to count rational points via morphisms between weighted projective spaces, simplifying previous proofs and broadening applicability.

Findings

Generalized Deng's asymptotic formula for rational points

Counted points on images under morphisms with bounded height

Applied results to elliptic curves and modular curves

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.