10,670 research outputs found
Refined Brill-Noether Locus and Non-Abelian Zeta Functions for Elliptic Curves
New local and global non-abelian zeta functions for elliptic curves are
studied using certain refined Brill-Noether loci in moduli spaces of
semi-stable bundles. Examples of these zeta functions and a justification of
using only semi-stable bundles are given too. We end this paper with an
appendix on the so-called Weierstrass Groups for general curves, which is
motivated by a construction of Euler systems from torsion points (of elliptic
curves).Comment: Plain Te
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 -functions for number fields, based on
geo-arithmetical cohomology, geo-arithmetical truncation and Langlands' theory
of Eisenstein series.Comment: 37 page
A Note on Arithmetic Cohomologies for Number Fields
As a part of our program for Geometric Arithmetic, we develop an arithmetic
cohomology theory for number fields using theory of locally compact groups
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
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
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
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
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 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
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