Counting points of fixed degree and given height over function fields

TL;DR

This paper provides asymptotic estimates for counting points of fixed degree and height over function fields, extending understanding of point distribution in algebraic geometry over finite fields.

Contribution

It introduces new asymptotic formulas for counting points with fixed degree and height over function fields, particularly when the dimension exceeds twice the degree plus three.

Findings

Asymptotic estimates for points with fixed degree and height over function fields.

Application to asymptotic counts of decomposable forms.

Results valid for projective spaces with dimension greater than twice the degree plus three.

Abstract

Let $k$ be a finite field extension of the function field $\bfF_p(T)$ and $\bar{k}$ its algebraic closure. We count points in projective space $\Bbb P ^{n-1}(\bar{k})$ with given height and of fixed degree $d$ over the field $k$. If $n>2d+3$ we derive an asymptotic estimate for their number as the height tends to infinity. As an application we also deduce asymptotic estimates for certain decomposable forms.