Counting primitive points of bounded height

TL;DR

This paper provides asymptotic estimates for counting points of bounded height in projective space over a number field extension, introducing a generalized adelic Lipschitz height for broader applications.

Contribution

It introduces a new adelic Lipschitz height and derives precise asymptotic estimates for primitive points over extensions, with improved error terms.

Findings

Asymptotic estimates for points of bounded height over extensions

Introduction of a generalized adelic Lipschitz height

Potential applications to points of fixed degree and subfield conditions

Abstract

Let $k$ be a number field and $K$ a finite extension of $k$. We count points of bounded height in projective space over the field $K$ generating the extension $K/k$. As the height gets large we derive asymptotic estimates with a particularly good error term respecting the extension $K/k$. In a future paper we will use these results to get asymptotic estimates for the number of points of fixed degree over $k$. We also introduce the notion of an adelic Lipschitz height generalizing that of Masser and Vaaler. This will lead to further applications involving points of fixed degree on linear varieties and algebraic numbers of fixed degree satisfying certain subfield conditions.