Heights and morphisms in number fields

TL;DR

This paper derives explicit formulas with error estimates for counting rational points of bounded height on projective varieties over number fields, focusing on morphisms and different height functions.

Contribution

It provides new explicit counting formulas with error terms for rational points under morphisms, connecting Weil and canonical heights over number fields.

Findings

Explicit counting formulas with error terms for rational points.

Connections between Weil height and Call-Silverman canonical height.

Asymptotic behavior of point counts as height bound grows.

Abstract

We give a formula with explicit error term for the number of $K$-rational points $P$ satisfying $H(f(P)) \le X$ as $X \to \infty$, where $f$ is a nonconstant morphism between projective spaces defined over a number field $K$ and $H$ is the absolute multiplicative Weil height. This yields formulae for the counting functions of $f(\mathbb{P}^m(K))$ with respect to the Weil height as well as of $\mathbb{P}^m(K)$ with respect to the Call-Silverman canonical height.