Noncommutative Iwasawa theory and p-adic automorphic forms.

在过去的15年里，岩泽理论有两个相当不相交的发展，它与算术几何中的基本问题的关系。一方面，非对易岩泽理论中主要定理的精确表述在谷-加藤的工作中达到了一定的成熟。非对易主猜想的一个重要例子被PI证明（也被Burns-Rtter-韦斯独立证明）。另一方面，希达、Tilouine、Urban和Skinner等作者系统地发展了自守形式（p-adic和p-adic）理论，并将其应用于证明交换岩泽理论中经典主定理以外的主要定理。因此，现在是联合收割机将这两种发展结合起来以证明两个方向的新结果的适当时机。我们建议在这方面处理三个互相关连的问题。首先，我们要扩展我们的方法来证明非交换主结构的Tate动机证明新的结果的非交换主结构的动机比Tate动机。为此，我们建议系统地研究各种全真实的域上的p-adic和p-adic自守形式，以及这些自守形式随域的变化之间的关系。其次，在Kato的结构中隐含着p-adic L-函数的某些因子分解。这些因式分解，只知道在一对夫妇的情况下，有很深的算术影响，如对格林伯格的L-不变代数。我们提出了一个新的策略来攻击这些因子分解问题，使用开发的工具来解决我们的第一个问题。最后，我们提出研究函数域上的主要结构。我们已经开发的代数技术已经证明非常富有成效的岩泽理论在功能领域的工作伯恩斯。然而，函数域上还有另一个主要分支（例如特里汉和他的合作者的工作）。我们的第三个项目是使用我们的代数结果和伯恩斯使用的技术来攻击这些主要的aesthetures。

The last fifteen years have seen two fairly disjoint developments in Iwasawa theory, and its relationship with come of the basic problems in arithmetic geometry. On the one hand, the precise formulation of the main conjectures in noncommutative Iwasawa theory reached a certain maturity in the work of Fukaya-Kato. An important case of the noncommutative main conjecture was proven by the PI (and independently by Burns-Rtter-Weiss). On the other, the theory of automorphic forms (p-adic and lambda-adic) was systematically developed and applied to prove main conjectures in commutative Iwasawa theory beyond the classical main conjectures by several authors including Hida, Tilouine, Urban and Skinner. It is therefore an appropriate time to combine these two developments to prove new results in both directions. We propose to tackle three inter-related problems in this general area. Firstly, we want to extend our methods used to prove the noncommutative main conjectures for Tate motives to prove new results on noncommutative main conjectures for motives other than Tate motives. To this end we propose to systematically study p-adic and lambda-adic automorphic forms over various totally real fields and relations between these automorphic forms as the fields vary. Secondly, implicit in the conjectures of Fukaya-Kato are certain factorisations of p-adic L-functions. These factorisations, known only in a couple of cases, have deep arithmetic implications such as towards Greenberg's L-invariant conjectures. We propose a new strategy to attack these factorisation problems using the tools developed to tackle our first question. Lastly, we propose to study main conjectures over function fields. The algebraic techniques we have developed have already proven very fruitful in Iwasawa theory over function fields in the work of Burns. There is, however, another family of main conjectures over function fields (e.g. in the work of Trihan and his collaborators). Our third project is to use our algebraic results and techniques used by Burns to attack these main conjectures.