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

综上所述，我们在《几何算术程序》一文的指导下，在过去几年里开展了我们的研究工作。(1)在Tate论文的启发下，发展了数域上格的上同调理论。(2)引入了n阶非交换Zeta函数和L函数的新的但仍然真实的定义。(3)建立了这些函数的基本性质。(4)证明了在数域的排名为2的情况下，我们的Zeta函数满足黎曼假设。(5)揭示了几何截断和解析截断之间的关系。(6)在4的基础上引入了非交换L函数，并建立了它们的基本性质：我们利用朗兰兹的爱森斯坦级数基本理论引入了一般的非阿贝尔L函数，并由此建立了它们的基本性质，如亚纯连续性、函数方程和奇性。(7)在(5)和(6)的基础上，经过大量的额外工作，我们能够将我们的非阿贝尔L函数与我们所说的艾森斯坦周期，一种特殊的亚瑟周期联系起来。(8)例子：(1)使用由于Jacquet-Lapid-Rogawski而改进的Rankin-Selberg和Zagier方法，我们写下了与所谓尖点形式相关的L函数的精确表达式。(Ii)作为直接应用，我们与多伦多大学的Henry Kim一起，得到了与数域上分裂半单群相关的基域的Arthur截断域的体积的一般公式。(9)(I)我们找到了半稳定格模空间体积的一般公式。(Ii)揭示了非交换Zeta函数的空间值与经典DedekindZeta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.