Counting rational points on Hirzebruch-Kleinschmidt varieties over global function fields

TL;DR

This paper investigates the distribution of rational points of large height on certain split toric varieties over global function fields, providing asymptotic formulas with explicit constants and error bounds.

Contribution

It extends the study of height zeta functions to split toric varieties with Picard rank 2 over global function fields, including a decomposition approach and detailed asymptotic analysis.

Findings

Decomposition of varieties into finitely many subvarieties.

Explicit asymptotic formulas for rational points of large height.

Controlled error terms in the asymptotic estimates.

Abstract

Inspired by Bourqui's work on anticanonical height zeta functions on Hirzebruch surfaces, we study height zeta functions of split toric varieties with Picard rank 2 over global function fields, with respect to height functions associated with big metrized line bundles. We show that these varieties can be naturally decomposed into a finite disjoint union of subvarieties, where precise analytic properties of the corresponding height zeta functions can be given. As application, we obtain asymptotic formulas for the number of rational points of large height on each subvariety, with explicit leading constants and controlled error terms.