Lin Weng

Volume of truncated fundamental domains

For T > 0, by applying Arthurs analytic truncation AT to the constant function 1 on G(F)\G(A)1, we obtain a characteristic function of a compact subset i(T) of G(F)\G(A)1. The aim of this paper is to give a general formula for the volume of a(T), which is an exponential polynomial function of T. Consequently, by letting T -* oo, we obtain a new way to evaluate the volume of fundamental domain for G(F)\G(A)1. While a(T) seems to be quite arbitrary, its volume admits beautiful structures: (i) Surprisingly enough, the coefficients of T are special values of zeta functions. (ii) The volume is an alternating sum associated to standard parabolic subgroups. ((ii) is suggested by geometry after all, a(T) is obtained from G(F)\G(A)1 by truncating off cuspidal regions which are known to be parametrized by standard parabolic subgroups.) Besides all basic properties of Arthurs analytic truncation, our method is based on a fundamental formula of Jacquet-Lapid-Rogawski on an integration of truncated Eisenstein series associated to cusp forms. Their formula, an advanced version of the Rankin-Selberg method, is obtained using the so-called Bernstein principle via regularized integration over cones.

The Conference on L-Functions

In this paper, we introduce multi-variable zeta-functions of roots, and prove the analytic continuation of them. For the root systems associated with Lie algebras, these functions are also called Witten zeta-functions associated with Lie algebras which can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case of type Ar, we have already studied some analytic properties in our previous paper. In the present paper, we prove certain functional relations among these functions of types Ar (r = 1, 2, 3) which include what is called Witten’s volume formulas. Moreover we mention some structural background of the theory of functional relations in terms of Weyl groups.

Arithmetic geometry and number theory

On Local -Factors (D H Jiang) Deligne Pairings over Moduli Spaces of Punctured Riemann Surfaces (K Obitsu et al.) Vector Bundles on Curves over p (A Werner) Absolute CM-periods -- Complex and p-Adic (H Yoshida) On Special Zeta Values in Positive Characteristic (J Yu) Automorphic Forms, Eisenstein Series and Spectral Decompositions (L Weng) Geometric Arithmetic: A Program (L Weng).

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).

Zeta-Functions Defined by Two Polynomials

The analytic continuation of certain multiple zeta-functions is shown. In particular, the analytic continuation of the zeta-function ζ(s; P, Q), defined by two polynomials P and Q, follows. Then the holomorphy of ζ(s; P, Q) at non-positive integers is proved, and explicit formulas for the values ζ(0; P, Q) and ζ′(0; P, Q) are given. The latter formula gives a generalization of an explicit formula for the regularized determinant of the Laplacian on the high-dimensional sphere.

L 2-metrics, projective flatness and families of polarized abelian varieties

We compute the curvature of the L 2 -metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kahler manifolds. As an application, we show that the L 2 -metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kahler manifold, this leads to the poly-stability of the direct image with respect to any Kahler form on the parameter space.

Green's Functions for Quasi-Hyperbolic Metrics on Degenerating Riemann Surfaces with a Separating Node

In this article, we consider a family of compact Riemann surfaces of genus q ≥ 2 degenerating to a Riemann surface with a separating node and many non-separating nodes. We obtain the asymptotic behavior of Greens functions associated to a continuous family of quasi-hyperbolic metrics on such degenerating Riemann surfaces.