New Non-Abelian Zeta Functions for Curves over Finite Fields

Abstract

In this paper, we introduce and study two new types of non-abelian zeta functions for curves over finite fields, which are defined by using (moduli spaces of) semi-stable vector bundles and non-stable bundles. A Riemann-Weil type hypothesis is formulated for zeta functions associated to semi-stable bundles, which we think is more canonical than the other one. All this is motivated by (and hence explains in a certain sense) our work on non-abelian zeta functions for number fields.