Construction of Voronoi theory over adele groups

N×n矩阵的么模积分群在正定n×n对称矩阵的对称锥上起到适当的间断作用。Voronoi的约化理论被认为是构造关于这一作用的基本域的经典方法。利用这个理论，可以计算离散子群的生成元和同调群。本项目的目的是将Voronoi的约化理论推广到任意基场和一般代数群。我们得到了如下结果：我们将Voronoi约化理论的一个基域推广到全实数域上。在此基础上，利用算术极小函数引入了Adele群的Ryshkov域，并利用Ryshkov域构造了Adele群的算术商的基本域。

The group of unimodular integral n by n matrices acts properly discontinuous on the symmetric cone of positive definite n by n symmetric matrices. Voronoi's reduction theory is known as a classical method to construct a fundamental domain on this action. By this theory, one can compute generators and homology groups of discrete subgroups. The purpose of this project is to extend Voronoi's reduction theory to arbitraryl base fields and general algebraic groups. We obtained the following results: We extended a base field of Voronoi's reduction theory to totally real number fields. Furthermore, we introduced Ryshkov domains of adele groups by using arithmetical minimum functions, and then constructed fundamental domains of arithmetic quotients of adele gropus via Ryshkov domains.