Riemann Hypothesis for Non-Abelian Zeta Functions of Genus 2 Curves

TL;DR

This paper proves that the Riemann Hypothesis holds for Weng zeta functions associated with genus 2 curves over finite fields as the rank tends to infinity, using geometric and asymptotic analysis.

Contribution

It establishes the Riemann Hypothesis for non-abelian zeta functions of genus 2 curves in the high-rank limit, advancing understanding of their analytic properties.

Findings

Riemann Hypothesis holds for high-rank Weng zeta functions

Uses geometric properties of moduli spaces of bundles

Provides new evidence for non-abelian zeta functions

Abstract

In this paper, we investigate Weng zeta functions associated with curves of genus 2 over finite fields. Building upon Weng's framework for non-abelian zeta functions, we establish that, as the rank n tends to infinity, the Riemann Hypothesis holds for these zeta functions. Our proof relies on the geometric properties of the moduli space of semi-stable bundles, together with several established results for high rank zeta functions, complemented by detailed asymptotic analysis. This result provides new evidence supporting the general validity of the Riemann Hypothesis for Weng zeta functions and offers insight into the analytic structure of non-abelian zeta functions associated with higher-genus curves.