Construction of Voronoi theory over adele groups

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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.