2012 Fiscal Year Final Research Report
Non-commutative class field theory and Shimura varieties
| Project/Area Number |
21340004
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Nagoya University |
|---|
Principal Investigator |
FUJIWARA Kazuhiro 名古屋大学, 大学院・多元数理科学研究科, 教授 (00229064)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
HESSELHOLT Lars 名古屋大学, 大学院・多元数理科学研究科, 教授 (10436991)
KATO Fumiharu 熊本大学, 大学院・自然科学研究科, 教授 (50294880)
TAKAI Yuuki 東京大学, 大学院・数理科学研究科, 研究員 (90599698)
GEISSER Thomas 名古屋大学, 大学院・多元数理科学研究科, 教授 (30571963)
|
|---|
| Co-Investigator(Renkei-kenkyūsha) |
KATO Fumiharu 熊本大学, 大学院・自然科学研究科, 教授 (50294880)
KOBAYASHI Shinichi 東北大学, 大学院・理学研究科, 准教授 (80362226)
KONDO Shigeyuki 名古屋大学, 大学院・多元数理科学研究科, 教授 (50186847)
SAITO Shuji 東京大学, 大学院・数理科学研究科, 教授 (50153804)
SAITO Takeshi 東京大学, 大学院・数理科学研究科, 教授 (70201506)
|
|---|
| Project Period (FY) |
2009 – 2012
|
| Keywords | 数論 / 非可換類体論 / ガロア表現 / 保型表現 |
|---|
| Research Abstract |
Non-abelian class field theory is studied from various aspects, including a geometric viewpoint. As for foundations of rigid geometry, a joint research with F. Kato and O Gabber (IHES) went on based on international collaboration, yielding results on the Hausdorff completions of commutative rings. As a result of this research, the foundation of rigid geometry is now established in a more general framework,giving more flexibility in applications. We have also obtained a clear explanation of the relationships between the notion of R. Huber’s adic spaces and V. Berkovich’s Berkovich spaces.As part of non-abelian class field theory, we provide a new viewpointthat the deformation theory of Galois representations (Galois deformation theory) can be applied directly to number-theoretical problems. The author has studied the indivisibility of relative class numbers of quadratic extensions by a prime number p as a first example. This is established in general. Our collaborator Y. Takai gave a lower bound estimate for the number of such quadratic extensions, when the field is Galois over the rationals and p is sufficiently large.
|
