General Uniformity of Zeta Functions

TL;DR

This paper introduces new zeta functions for compact Riemann surfaces using analytic torsion, exploring their properties, degenerations, and symmetries to unify various zeta theories in geometry and number theory.

Contribution

It develops a framework for abelian and non-abelian zeta functions on Riemann surfaces, proposing a conjecture on their symmetry and unifying different zeta theories.

Findings

Construction of symmetric zetas based on abelian zetas and group symmetries

Analysis of degenerations and singularities of analytic torsions

Conjecture that non-abelian zetas coincide with symmetric abelian zetas

Abstract

Using analytic torsion associated to stable bundles, we introduce zeta functions for compact Riemann surfaces. To justify the well-definedness, we analyze the degenerations of analytic torsions at the boundaries of the moduli spaces, the singularities of analytic torsions at Brill-Noether loci, and the asymptotic behaviors of analytic torsions with respect to the degree. These new yet intrinsic zetas, both abelian and non-abelian, are expected to play key roles to understand global analysis and geometry of Riemann surfaces, such as the Tamagawa number conjecture for Riemann surfaces, searched by Atiyah-Bott, and the volumes formula of moduli spaces of Witten. Relating to this, in our theory on special uniformity of zetas, we will first construct a symmetric zetas based on abelian zetas and group symmetries, then conjecture that our non-abelian zetas coincide with these later zetas with symmetries. All this, together with that for zetas of number fields and function fields, then consists of our theory of general uniformity of zetas.