Automorphic and Galois representations

受A.Wiles的工作启发，我研究了Iwasawa理论和Langlands猜想。在这里，伽罗瓦表示和Hecke代数的变形环的识别（称为Mazur猜想）起着中心作用。证明了GL(2)在全实数域上的Hecke代数是一个局部完全交环，并可以用模p模表示的泛变形环来标识。该工作的关键步骤是，Hecke环上模曲线的上同群的自由性（主要用于Taylor-Wiles工作）是一个更容易验证的公理的结果。我将这些公理命名为泰勒-怀尔斯系统，与欧拉系统类似（我应该注意到，F.Diamond独立发现了一个类似的想法）。通过用Shimura曲线构造一个Taylor-Wiles系统，证明了一般全实域上的Mazur猜想。通过将结果与水平优化论证（Mazur原理的偶数次情况是最困难的）相结合，许多二维1进伽罗瓦表示对应于自同构表示，从而验证了这些情况下的朗兰兹对应。特别地，对谷山-志村猜想的推广作了较为一般的说明。有一份关于这项工作的报告，包括最近的结果。该细节以预印本的形式分发（完全真实情况下的变形环和Hecke代数），提交给期刊，另外2篇文章正在准备中。在可约残差表示的情况下，证明了在全实域固定时，存在无穷多个满足Mazur猜想的可约残差表示。

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.