2013 Fiscal Year Final Research Report
Construction of Voronoi theory over adele groups
| Project/Area Number |
23540016
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Osaka University |
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Principal Investigator |
WATANABE Takao 大阪大学, 理学(系)研究科(研究院), 教授 (30201198)
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| Co-Investigator(Kenkyū-buntansha) |
HAYATA Takahiro 山形大学, 理工学研究科, 准教授 (50312757)
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| Co-Investigator(Renkei-kenkyūsha) |
ODA Takayuki 東京大学, 数理科学研究科, 教授 (10109415)
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| Project Period (FY) |
2011 – 2013
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| Keywords | 数論 / アデール群 / 簡約理論 / エルミート定数 |
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| Research Abstract |
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.
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