Height zeta functions of projective bundles

TL;DR

This paper develops a new method using Arakelov geometry to analyze height zeta functions of projective spaces and bundles, establishing their analytic properties and extending results to motivic analogues and Hirzebruch surfaces.

Contribution

It introduces a novel approach applying Arakelov Riemann-Roch to study height zeta functions, proving their analytic continuation and functional equations for various cases.

Findings

Proved analytic continuation of height zeta functions.

Established functional equations for these functions.

Extended results to motivic analogues and Hirzebruch surfaces.

Abstract

We introduce a new approach to study height zeta functions of projective spaces and projective bundles. To study height zeta functions of projective spaces $Z(\mathbb{P}^n, H_{\mathcal{O}(1)}; s)$, we apply the Riemann-Roch theorem of Arakelov vector bundles by van der Geer and Schoof to the integrand of an integral expression of $Z(\mathbb{P}^n, H_{\mathcal{O}(1)}; s)$. We give a proof of the analytic continuation and functional equations of height zeta functions of projective spaces with respect to various height functions. Motivic analogues of these results are also proved. We also study height zeta functions of Hirzebruch surfaces.