Zeta functions for function fields

TL;DR

This paper introduces new non-abelian zeta functions for algebraic curves over finite fields, exploring their properties, conjectures, and non-abelian nature, extending the theory of zeta functions beyond classical abelian cases.

Contribution

It defines two types of non-abelian zeta functions for curves, establishes their fundamental properties, and discusses their conjectured zeros and uniformity, advancing the understanding of non-abelian zeta functions.

Findings

Proved rationality and functional equations for the new zeta functions

Formulated conjectures on zeros and uniformity of these zetas

Explained the non-abelian nature using parabolic reduction and stability

Abstract

We introduce new non-abelian zeta functions for curves defined over finite fields. There are two types, i.e., pure non-abelian zetas defined using semi-stable bundles, and group zetas defined for pairs consisting of (reductive group, maximal parabolic subgroup). Basic properties such as rationality and functional equation are obtained. Moreover, conjectures on their zeros and uniformity are given. We end this paper with an explanation on why these zetas are non-abelian in nature, using our up-coming works on 'parabolic reduction, stability and the mass'. The constructions and results were announced in our paper on 'Counting Bundles' arXiv:1202.0869.