From Stability, Riemann-Roch to Non-Abelian Zeta Functions

总的来说，我们在过去的几年里，以我们的论文《A Program on Geometric Arithmetic》为指导，继续我们的研究。(1)在Tate论文的推动下，发展了一种新的数域上格的上同调理论。(2)引入n阶非阿贝尔ζ函数和L函数的新的真实定义。(3)建立了这些函数的基本性质(4)证明了在数域的秩为二的情况下，我们的ζ函数满足黎曼假设。(5)揭示几何截断与解析截断的关系(6)引入非阿贝尔L函数并建立其基本性质：在4的基础上，利用Langlands的爱森斯坦级数基本理论引入一般非阿贝尔L函数，并由此建立其亚纯延拓、泛函方程、奇异性等基本性质。(7)基于(5)和(6)，通过大量额外的努力，我们能够将我们的非阿贝尔L函数与我们称之为爱森斯坦周期的东西联系起来，爱森斯坦周期是一种特殊的阿瑟周期。(8)举例：(1)利用Jacquet-Lapid-Rogawski的Rankin-Selberg和Zagier方法的一个高级版本，我们写出了与所谓尖端形式相关的L函数的精确表达式。（ii）作为直接应用，利用(i)，我们与多伦多大学的Henry Kim （University of Toronto）一起得到了数域上分裂半简单群所关联的基本域的Arthur截断域体积的一般公式。当参数趋于无穷大时，这个结果恢复了（Siegel和）Langlands在与（SLn和）一般分裂半单群相关的基本域的体积上的著名公式。(9)(i)得到了半稳定格模空间体积的一般公式。（ii）揭示了非阿贝尔zeta函数的空间值与经典Dedekind zeta函数的特殊值之间的基本关系。

Over all, we, guided by our paper of A Program on Geometric Arithmetic, carry on our researches in the past a few years.(1)Develop a new cohomology theory for lattices over number fields, motivated by Tate's thesis.(2)Introduce New yet genuine defnition of rank n non-abelian zeta and L functions.(3)Establish fundamental properties of these functions(4)Show that in case of rank two for number fields, our zeta functions satisfies the Riemann Hypothesis.(5)Expose the relation between Geometric Truncation and Analytic Truncations(6)Introduce non-abelian L functions and establish their basic properties : Based on 4, we introduce general non-abelian L functions using Langlands' fundamental theory of Eisenstein series and hence also establish their basic properties such as meromorphic continuation, functional equations and singularities.(7)Based on (5) and (6),with substantial additional hard work, we are able to relate our non-abelian L functions and what we call Eisenstein periods, a special kind of Arthur's period.(8)Examples :(i)Using an advanced version of Rankin-Selberg and Zagier method due to Jacquet-Lapid-Rogawski, we write down the precise expression for our L functions associated to the so-caleld cusp forms.(ii) As a direct application, using (i),we together with Henry Kim (University of Toronto) obtain a general formula for volumes of Arthur's truncated domains of fundemantal domains associated to split semi-simple groups over number fields. This result when letting the parameter go to infinity then recovers the famous formula of (Siegel and) Langlands on the volumes of fundamental domains associated to (SLn and) general split semi-simple groups over number fields.(9)(i)We find a general formula for the volume of moduli space of semi-stable lattices.(ii)We exposes a fundamental relation between spaciel values of non-abelian zeta functions and special values of classical Dedekind zeta functions.