Non-Abelian L Functions for Function Fields

This is an integrated part of our Geo-Arithmetic Program. In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields by a weighted count of semi-stable bundles. Basic properties such as rationality and functional equation are established. Examples of rank two zetas over genus two curves are given as well. Based on this and motivated by our study for non-abelian zetas of number fields, general non-abelian $L$ functions for function fields are defined and studied using Langlands and Morris' theory of Eisenstein series.