Arithmetic study of Shimura varieties

志村品种的算术研究

基本信息

批准号：
13440004
负责人：
FUJIWARA Kazuhiro
金额：
$ 9.28万
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：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

The head investigator has been studying a non-abelian class field theory (Langlands program) via arithmetic geometry of Shimura varieties. Here is a brief summary of the research during the period.1.By applying the non-abelian class field theory to algebraic number theory, we have got a new insight on Leopoldt conjecture in algebraic number theory. Especially, a connection between A.Wiles' work on Taniyama-Shimura conjecture for elliptic curves and W.Thurston's three dimensional hyperbolic geometry was found. This connection is closely related to an analogy between knots and primes dur to B.Mazer and M.Morishita, showing the variety of the related research areas. This is reported at international conferences hold at Paris 13(2002), Nagoya(2003). The abstract is published, and more detailed article is under preparation.2.During the research period, p-adic description of Shimura varieties has been developed worldwide. Motivated by this, the author is trying to establish a new foundation of rigid geometry with F.Kato(Kyoto). This work generalizes the framework of rigid geometry, giving more solid foundation for the application. The detail is under preparation, and will be published as a book in near future.3.The head investigator stayed one month and half at University Paris 7 from May 2002,and also University Paris 13 in September 2002. There were many research activities including discussions with M.F.Vigneras(Paris 7), J.P.Labesse(Paris 7), M.Harris, (Paris 7), A.M.Aubert(Ecole Normal)J.Tilouine(Paris 13), especially on automorphic forms and representation theory of p-adic groups. Our group is also making collaboration with P.Colmez(Paris 6) and J.Nekovar(Paris 7) from more number theoretical viewpoint using p-adic methods. For these collaborations, the grant is used effectively.

首席研究员一直在研究非亚伯类阶级田地理论（Langlands计划），该杂志的算术几何形状。这是该期间研究的简要摘要。1。通过将非亚伯阶级字段理论应用于代数数理论，我们对代数数理论中的Leopoldt猜想有了新的见解。尤其是，发现了A.Wiles在Taniyama-shimura猜想的椭圆曲线和W.Thurston的三维双曲几何形状之间的联系。这种联系与B.Mazer和M.Morishita之间的结和Primes Dur之间的类比密切相关，显示了相关研究领域的种类。这在国际会议上在巴黎举行的13（2002），名古屋（2003）举行。摘要已发表，并且正在准备更详细的文章。在研究期间，全世界已经开发了有关Shimura品种的P-Adic描述。由此激励，作者正试图与F.Kato（Kyoto）建立新的刚性几何形状基础。这项工作概括了刚性几何形状的框架，为应用程序赋予了更多坚实的基础。 The detail is under preparation, and will be published as a book in near future.3.The head investigator stayed one month and half at University Paris 7 from May 2002,and also University Paris 13 in September 2002. There were many research activities including discussions with M.F.Vigneras(Paris 7), J.P.Labesse(Paris 7), M.Harris, (Paris 7), A.M.Aubert(Ecole Normal)J.Tilouine(Paris 13), especially关于P-ADIC群体的自称形式和表示理论。我们的小组还使用P-ADIC方法从更多的理论观点中与P.Colmez（Paris 6）和J.Nekovar（巴黎7）合作。对于这些合作，该赠款将有效地使用。