General Uniformity of Zeta Functions

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.