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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.

受怀尔斯工作的启发，我研究了数域上的岩泽理论和朗兰兹猜想。在这里，伽罗瓦表示和Hecke代数（称为Mazur猜想）的变形环的识别起着核心作用。本文证明了全真实的域上GL（2）的Hecke代数是局部完备交环，并将其等同于模p模表示的泛变形环，其基本步骤是证明了Hecke环上模曲线的上同调群的自由性（主要用于Taylor-Wiles工作）是公理的一个推论，这些公理更易于验证。我把这些公理命名为泰勒-怀尔斯系统，与欧拉系统类似（我应该指出，F.戴蒙德独立地发现了类似的想法）。利用Shimura曲线构造Taylor-Wiles系统，证明了一般全真实的域上的Mazur猜想.通过将结果与水平优化参数相结合（Mazur原理的偶数度情况是最困难的），许多二维1-adic伽罗瓦表示对应于自守表示，从而验证了这些情况下的朗兰兹对应。特别地，对Taniyama-Shimura猜想作了较一般的推广。有一份关于这项工作的报告，包括最近的成果。详细分发作为预印本（变形环和Hecke代数在完全真实的情况下），提交给一个期刊，和2个其他文章正在筹备中。即使在可约剩余表示的情况下，它表明，有无穷多个可约表示满足Mazur猜想，当完全真实的领域是固定的。