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

Shi Zhan

TL;DR

This work proves an asymptotic Riemann Hypothesis for Weng's non-abelian zeta functions of genus 2 curves over finite fields as the rank $n$ grows. By leveraging the moduli space of semi-stable vector bundles, along with established high-rank zeta results and detailed asymptotics of the $\alpha$- and $\beta$-invariants, the authors derive explicit polynomial data $P_{X/{\Bbb F}_q,n}(T)$ and show its zeros approach the central line $\Re(s)=\tfrac{1}{2}$ in the large-$n$ limit. The genus-2 specialization yields concrete limits for the polynomial coefficients, linking them to point counts $N_1$ and $N_2$ and the renormalized invariant $\widehat{v}_1$. Overall, the paper provides new evidence for the broader RH validity in non-abelian zeta theory and clarifies how high-rank geometry governs the analytic structure of zetas for higher-genus curves.

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.

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

TL;DR

This work proves an asymptotic Riemann Hypothesis for Weng's non-abelian zeta functions of genus 2 curves over finite fields as the rank grows. By leveraging the moduli space of semi-stable vector bundles, along with established high-rank zeta results and detailed asymptotics of the - and -invariants, the authors derive explicit polynomial data and show its zeros approach the central line in the large- limit. The genus-2 specialization yields concrete limits for the polynomial coefficients, linking them to point counts and and the renormalized invariant . Overall, the paper provides new evidence for the broader RH validity in non-abelian zeta theory and clarifies how high-rank geometry governs the analytic structure of zetas for higher-genus curves.

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.

Paper Structure

This paper contains 15 sections, 19 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Non-Abelian Zeta Function for Function Fields
  3. Artin Zeta Function
  4. Weng Zeta Function
  5. Properties of Rank $n$ Zetas
  6. Riemann Hypothesis
  7. Special Uniformity of Zeta Functions
  8. Properties of $\alpha$- and ${\beta}$- Invariants
  9. Counting Miracle
  10. Semi-Stable Mass
  11. Estimates of $\alpha$- and ${\beta}$- Invariants
  12. High Rank Zeta Function over Genus 2 Curves
  13. Multiplicative Structure of Artin Zeta Function
  14. $P_{X/{\Bbb F}_q,n}(T)$ for Genus 2 Curves
  15. Riemann Hypothesis for $\widehat{\zeta}_{X/{\Bbb F}_q,n}$ over Genus 2 Curves

Key Result

Theorem 1

Let $X$ be a smooth projective curve of genus $g$ over a finite filed ${\Bbb F}_q$, and let $Z_{X/{\Bbb F}_q}(t)$ be its Artin zeta function.

Theorems & Definitions (28)

  • Definition 1: Artin Zeta Function
  • Theorem 1: Zeta Properties
  • Definition 2: Rank $n$ Zeta Function, see HRZ1 and HRZ2
  • Theorem 2: High Rank Zeta Properties, see HRZ2
  • Theorem 3: HRZ2
  • Theorem 4: Hasse's Theorem on Elliptic Curves
  • Theorem 5: Hasse-Weil Bound
  • Conjecture 6: Rank $n$ Riemann Hypothesis, see HRZ2 and RH
  • Lemma 7
  • Theorem 8: Weng--Zagier's Theorem on Rank $n$ Riemann Hypothesis for Elliptic Curves, see EC
  • ...and 18 more