Non-abelian zeta functions for function fields

Abstract

In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields. More precisely, we first introduce new yet genuine non-abelian zeta functions for curves defined over finite fields, by a "weighted count" on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. Then we define non-abelian L -functions for curves over finite fields using integrations of Eisenstein series associated to L 2 -automorphic forms over certain generalized moduli spaces.