Riemann Hypothesis for Non-Abelian Zeta Functions of Curves over Finite Fields

TL;DR

This paper advances the understanding of non-abelian zeta functions for curves over finite fields by developing techniques towards the Riemann hypothesis, providing explicit bounds on invariants related to vector bundles, and linking these bounds to the hypothesis.

Contribution

It introduces new techniques for the Riemann hypothesis of higher rank non-abelian zeta functions and derives explicit bounds on associated invariants.

Findings

Established bounds on non-abelian invariants in terms of genus and field size.

Connected bounds in lower ranks to the Riemann hypothesis for higher ranks.

Provided explicit inequalities involving zeta function invariants.

Abstract

In this paper, we develop some basic techniques towards the Riemann hypothesis for higher rank non-abelian zeta functions of an integral regular projective curve of genus $g$ over a finite field $\mathbb F_q$. As an application of the Riemann hypothesis for these genuine zeta functions, we obtain some explicit bounds on the fundamental non-abelian $\alpha$- and $\beta$-invariants of $X/\mathbb F_q$ in terms of $X$ and $n,\, q$ and $g$: $$\alpha_{X,\mathbb F_q;n}(mn) = \sum_{V}\frac{q^{h^0(X,V)}-1}{\#\mathrm{Aut}(V)} \qquad{\rm and}\qquad \beta_{X,\mathbb F_q;n}(mn ):= \sum_{V}\frac{1}{\#{\mathrm Aut}(V)}\qquad(m\in \mathbb Z)$$ where $V$ runs through all rank $n$ semi-stable $\mathbb F_q$-rational vector bundles on $X$ of degree $mn$. In particular, $$ \prod_{k=1}^{n}\frac{\ \big( \sqrt q^k-1\big)^{2g-1}\ }{(\sqrt q^k+1)}\leq q^{-\binom{n}{2}(g-1)} \beta_{X,\mathbb F_q;n}(0) \leq \prod_{k=1}^{n}\frac{\ \big( 1+\sqrt q^k\big)^{2g-1}\ }{(\sqrt q^k-1)}, $$ Finally, we demonstrate that the related bounds in lower ranks in turn play a central role in establishing the Riemann hypothesis for higher rank zetas.