Noncommutative Iwasawa theory and p-adic automorphic forms.

在过去的15年里，岩泽理论有了两个相当不相关的发展，以及它与算术几何中一些基本问题的关系。一方面，非交换Iwasawa理论中主要猜想的精确表述在Fukaya-Kato的工作中达到了一定的成熟。π证明了非交换主猜想的一个重要例子（Burns-Rtter-Weiss独立证明了这个例子）。另一方面，自同构形式（p进和λ进）理论被Hida、Tilouine、Urban和Skinner等人系统地发展并应用于证明交换Iwasawa理论中超越经典主要猜想的主要猜想。因此，现在是把这两项发展结合起来以证明在两个方向上都取得新成果的适当时机。我们建议在这方面解决三个相互关联的问题。首先，我们想扩展我们用于证明Tate动机的非交换主猜想的方法，以证明非Tate动机的非交换主猜想的新结果。为此，我们提出系统地研究各种全实域上的p进和λ进自同构形式，以及这些自同构形式随域变化的关系。其次，在Fukaya-Kato的猜想中隐含着p进l函数的某些分解。这些因式分解，仅在少数情况下已知，具有深刻的算术含义，例如对格林伯格的l不变猜想。我们提出了一种新的策略，利用为解决第一个问题而开发的工具来解决这些分解问题。最后，我们提出对函数场的主要猜想进行研究。我们开发的代数技术已经在Burns的工作中证明了Iwasawa理论在函数场上的成果。然而，还有另一类关于函数域的主要猜想（例如在Trihan和他的合作者的工作中）。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猜想。

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.