Non-commutative class field theory and Shimura varieties

非交换类场论和 Shimura 簇

• 批准号: 21340004
• 负责人: FUJIWARA Kazuhiro
• 金额: $ 10.48万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21340004/
• 关键词: 数論; 非可換類体論; ガロア表現; 保型表現; リジッド幾何学; 志村多様体; Jacquet-Langlands対応; 保型形式

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.

非交换类场论从不同的角度进行了研究，包括几何观点。关于刚体几何的基础，本文与F. Kato和O Gabber（IHES）继续基于国际合作，产生了交换环的Hausdorff完备化的结果。作为这项研究的结果，刚性几何的基础现在建立在一个更一般的框架，在应用中提供更多的灵活性。我们还清楚地解释了R的概念与Huber的adic空间和V.Berkovich的Berkovich空间.作为非交换类场论的一部分，我们提出了一个新的观点，即伽罗瓦表示的变形理论（伽罗瓦变形理论）可以直接应用于数论问题.作为第一个例子，作者研究了二次扩张的相对类数与素数p的不可分性。这是一般性的。我们的合作者Y.高井给出了一个下界估计的数量，这样的二次扩展，当该领域是伽罗瓦的有理数和p是足够大的。

项目成果

On Hausdorff completions of commutative rings in rigid geometry

刚性几何中交换环的豪斯多夫完成

• DOI: 10.1016/j.jalgebra.2011.02.001
• 发表时间: 2011
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Masao Ishikawa;Fumihiko Nakano;Taizo Sadahiro;Hiroyuki Tagawa;野村明人;島倉裕樹;Tomoki Nakanishi;Kazuhiro Fujiwara
• 通讯作者: Kazuhiro Fujiwara

On the relative and bi-relative K-theory of rings of finite characteristic

有限特征环的相对和双相对K理论

• 发表时间: 2011
• 期刊: J. Amer. Math. Soc.
• 作者: Thomas Geisser;Lars Hesselholt
• 通讯作者: Lars Hesselholt

On the vanishing of relative K-groups

关于相关 K 群的消失

• 发表时间: 2010
• 期刊: Math. Ann.
• 作者: Thomas Geisser;Lars Hesselholt
• 通讯作者: Lars Hesselholt

On the vanishing of negative K-groups

关于负 K 群的消失

• 发表时间: 2010
• 期刊: Math.Ann.
• 作者: T.Geissser;L.Hesselholt
• 通讯作者: L.Hesselholt

Indivisibility of relative class numbers of CM quadratic extensions of totally real Galois fields

全实伽罗瓦域CM二次扩张的相对类数的不可分性

• 发表时间: 2013
• 作者: 横川博一;吉田晴世;福智佳代子;生馬裕子;真崎克彦;Yuuki Takai
• 通讯作者: Yuuki Takai