Automorphic and Galois representations

自守和伽罗瓦表示

基本信息

批准号：
08640023
负责人：
FUJIWARA Kazuhiro
金额：
$ 1.41万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至 1997
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08640023/
关键词：
Galois representation Automorphic representation Hecke algebra 楕円曲線

项目摘要

I have studied the Iwasawa theory and Langlands conjecture over number fields, motivated by A.Wiles' work. Here the identification of deformation rings of Galois representations and Hecke algebras (called Mazur conjecture) plays a central role. I have shown that the Hecke algebra of GL (2) over a totally real field is a local complete intersection ring, and is identified with a universal deformation ring of a mod p modular representation.The essential step in the work is, that the freeness property of a cohomology group of a modular curve over a Hecke ring (used essentially in Taylor-Wiles work) is a consequence of axioms which are easier to verify. I have named the axioms Taylor-Wiles system, in analogy with Euler systems (I should note that a similar idea was found independently by F.Diamond). By constructing a Taylor-Wiles system by Shimura curves, the Mazur conjecture over general totally real fields is proved. By combining the result with a level optimization argument (the even degree case of the Mazur principle is most difficult), many two dimensional 1-adic Galois representations correspond to automorphic representations, thus verifying the Langlands correspondence in these cases. Especially, a generalization of Taniyama-Shimura conjecture is shown fairly generally. There is a report on this work, including recent results. The detail is distributed as a preprint (Deformation rings and Hecke algebras in the totally real case), submitted to a journal, and 2 other articles are under preparation.Even in case of reducible residual representations, it is shown that there are infinitely many reducible representations which satisfy the Mazur conjecture, when the totally real field is fixed.

我已经研究了伊瓦沙瓦理论和兰兰兹的猜想，这是由A.Wiles的工作激励的。在这里，鉴定Galois表示和Hecke代数（称为Mazur猜想）的变形环起着核心作用。我已经表明，完全真实的字段上GL（2）的Hecke代数是一个局部完整的相交环，并具有Mod P模块化表示的普遍变形环。工作的重要步骤是，在hecke环上使用模块化曲线的同类单位属性组的同类属性（在Taylor-wiles Workify中使用）是一个轴的轴，它是一个更轻松的轴。我以与Euler系统的类似（我应该指出，F.Diamond独立发现了类似的想法）。通过通过Shimura曲线构建泰勒 - 遗嘱系统的系统，就证明了一般真实领域的Mazur猜想。通过将结果与级别优化参数（Mazur原理的偶数案例最困难）相结合，许多二维1-辅助Galois表示形式对应于自态表示，从而在这些情况下验证了Langlands对应关系。尤其是，taniyama-shimura猜想的概括相当普遍。有一份关于这项工作的报告，包括最近的结果。该细节作为预印本（完全真实的情况中的变形环和Hecke代数）分布，提交给期刊，还有另外2篇文章正在准备中。即使在可还原的残留表示方面，也表明，有许多可靠的可减少的表示，可以满足Mazur的猜测，而当完全真实的现场固定时，则可以满足MAZUR的猜测。