Special Uniformity of Zeta Functions I. Geometric Aspect

TL;DR

This paper explores the geometric structures underlying special uniformity of zeta functions by counting semi-stable bundles over finite fields, linking non-abelian zetas to abelian Artin zetas, and discusses related conjectures and analogues.

Contribution

It reveals geometric interpretations of non-abelian zeta functions through counting semi-stable bundles and connects these to abelian Artin zetas, advancing understanding of their intrinsic structures.

Findings

Semi-stable bundle counting relates to Artin zetas.

Geometric structures underpin special uniformity of zeta functions.

Appendix proves the Riemann Hypothesis for a new group zeta analogue.

Abstract

The special uniformity of zeta functions claims that pure non-abelian zeta functions coincide with group zeta functions associated to the special linear groups. Naturally associated are three aspects, namely, the analytic, arithmetic, and geometric aspects. In the first paper of this series, we expose intrinsic geometric structures of our zetas by counting semi-stable bundles on curves defined over finite fields in terms of their automorphism groups and global sections. We show that such a counting maybe read from Artin zetas which are abelian in nature. This paper also contains an appendix written by H. Yoshida, one of the driving forces for us to seek group zetas. In this appendix, Yoshida introduces a new zeta as a function field analogue of the group zeta for SL2 for number fields and establishes the Riemann Hypothesis for it.