Arithmetic study of Shimura varieties

首席研究员一直在研究非阿贝尔类场理论（朗兰兹计划）通过算术几何志村品种。将非交换类域理论应用到代数数论中，对代数数论中的Leopoldt猜想有了新的认识。特别是发现了A.Wiles关于椭圆曲线的Taniyama-Shimura猜想的工作与W.Thurston的三维双曲几何之间的联系，这种联系与B.Mazer和M.Morishita关于纽结和素数的类比密切相关，显示了相关研究领域的多样性。在巴黎13（2002年）和名古屋（2003年）举行的国际会议上报告了这一点。摘要已发表，更详细的文章正在整理中。2.在研究期间，志村品种的p-adic描述在世界范围内得到了发展。受此启发，作者正试图与F.加藤（京都）一起建立刚性几何的新基础。这一工作推广了刚体几何的基本框架，为刚体几何的应用奠定了坚实的基础。3.首席调查员从2002年5月起在巴黎第七大学呆了一个半月，并于2002年9月在巴黎巴黎第十三大学呆了一个半月。有许多研究活动，包括讨论与MF Vigneras（巴黎7），JP Labesse（巴黎7），M.哈里斯，（巴黎7），AM Aubert（高等师范）J.Tilouine（巴黎13），特别是自守形式和代表性理论的p进群。我们的团队还使用p-adic方法从更多数论角度与P.Colmez（巴黎6）和J.Nekovar（巴黎7）进行合作。在这些合作中，赠款得到了有效利用。

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.