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

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 $g$: $${\alpha }_{X,{\mathbb{F}}_q;n}(mn)=\sum _V\frac{q^{h^0(X,V)}-1}{\#\mathrm{Aut}(V)}\mathrm{and}{\beta }_{X,{\mathbb{F}}_q;n}(mn):=\sum _V\frac{1}{\#\mathrm{A}ut(V)}(m\in \mathbb{Z})$$ where $V$ runs through all rank $n$ semi-stable ${\mathbb{F}}_q$-rational vector bundles on $X$ of degree $mn$. In particular, $$\prod _{k=1}^n\frac{({\sqrt{q}}^k-1{)}^{2g-1}}{({\sqrt{q}}^k+1)}\le q^{-\left(\genfrac{}{}{0ex}{}{n}{2}\right)(g-1)}{\beta }_{X,{\mathbb{F}}_q;n}(0)\le \prod _{k=1}^n\frac{(1+{\sqrt{q}}^k{)}^{2g-1}}{({\sqrt{q}}^k-1)},$$ 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.

arXiv comment45 pages