General Uniformity of Zeta Functions

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

A Program For Geometric Arithmetic

Proposed is a program for what we call Geometric Arithmetic, based on our works on non-abelian zeta functions and non-abelian class field theory. Key words are stability and adelic intersection-cohomology theory.Comment: Plain Te

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.Comment: 30 pages. to appear at Amer. J of Mat

Zeta functions for function fields

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.Comment: References changed to zero my own remissnes

Special Uniformity of Zeta Functions I. Geometric Aspect

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.Comment: This paper contains an appendix written by H. Yoshid

Non-Abelian L Function for Number Fields

This is an integrated part of our Geo-Arithmetic Program. In this paper we introduce and hence study non-abelian zeta functions and more generally non-abelian $L$-functions for number fields, based on geo-arithmetical cohomology, geo-arithmetical truncation and Langlands' theory of Eisenstein series.Comment: 37 page

H^1_ar for arithmetic surface is finite

For an arithmetic surface X and a Weil divisor $D$, there are natural arithmetic cohomology groups $H_{\mathrm{ar}}^i(X, \mathcal O_X (D))$ $(i=0,1,2)$. Using ind-pro topology on adelic space $\mathbb A_{X, 012}^{\mathrm{ar}}$, we show that $H_{\mathrm{ar}}^0(X, \mathcal O_X (D))$ is discrete, $H_{\mathrm{ar}}^1(X, \mathcal O_X (D))$ is finite, and $H_{\mathrm{ar}}^2(X, \mathcal O_X (D))$ is compact. Moreover, we prove that all possible summations of canonical subspaces $\mathbb A_{X,i}^{\mathrm{ar}}(D),$ $\mathbb A_{X, kl}^{\mathrm{ar}}(D)$ $(i,k,l=0,1,2)$ are closed in $\mathbb A_{X,012}^{\mathrm{ar}}$, and hence complete our proof of topological dualities of among $H^i_{\mathrm{ar}}$'s

Zeta functions for G2G_2 and their zeros

The exceptional group $G_2$ has two maximal parabolic subgroups $P_{long}$, $P_{short}$ corresponding to the so-called long root and short root. In this paper, the second author introduces two zeta functions associated to $(G_2,P_{long})$ and $(G_2,P_{short})$ respectively, and the first author proves that these zetas satisfy the Riemann Hypothesis.Comment: 43 page

Counting Bundles

We introduce new genuine zetas. There are two types, i.e., the pure non- abelian zetas defined using semi-stable bundles, and the group zetas defined for reductive groups. Basic properties such as rationality and functional equation are obtained. Moreover, conjectures on their zeros and uniformity are given.Comment: References changed to zero my own remissnes

Riemann-Roch, Stability and New Non-Abelian Zeta Functions for Number Fields

In this paper, we introduce a geometrically stylized arithmetic cohomology for number fields. Based on such a cohomology, we define and study new yet genuine non-abelian zeta functions for number fields, using an intersection stability.Comment: Version