Iwasawa theory of p-adic Lie extensions

现代数论中最具挑战性的主题之一是L函数的特殊值与伽罗华上同调之间的神秘关系：它们是两个完全不同的世界中的影子，对同一个几何对象进行复分析和p-进分析，即给定的丢番图方程在整数上的解空间，或者更一般的动机M。岩泽理论的主要思想是同时研究这一原理的表现形式，如类数公式或Birch和Swinnerton Dyer猜想对于整个p-进数族的此类动机，例如通过考虑数域的塔或(Hida)模型族而产生的。这个项目的目的是为……提供进一步的证据。P-进L函数的存在性和(非交换)岩泽理论中的主要猜想，II.Fukaya和Kato的(等变)e-猜想以及III.Hida族的2元主要猜想，特别是我们希望构造第一个真正的非交换p元L函数，并找到满足由相应L函数的函数方程所确定的e-常数本身建立p元族的期望的(非对易)例子。在第三项中，我们计划对Pro-p-环和Big Galois表示上的李群进行系统的研究，并将其应用到Hida族的算术中。

One of the most challenging topics in modern number theory is the mysterious relation between special values of L-functions and Galois cohomology: they are the shadows in the two completely different worlds of complex and p-adic analysis of one and the same geometric object, viz the space of solutions for a given diophantine equation over the integral numbers, or more generally a motive M. The main idea of Iwasawa theory is to study manifestations of this principle such as the class number formula or the Birch and Swinnerton Dyer Conjecture simultaneously for whole p-adic families of such motives, which arise e.g. by considering towers of number fields or by (Hida) families of modular forms. The aim of this project is to supply further evidence forI. the existence of p-adic L-functions and for main conjectures in (non-commutative) Iwasawa theory,II. the (equivariant) e-conjecture of Fukaya and Kato as well asIII. the 2-variable main conjecture of Hida families.In particular, we hope to construct the first genuine non-commutative p-adic L-function as well as to find (non-commutative) examples fulfilling the expectation that the e-constants, which are determined by the functional equations of the corresponding L-functions, build p-adic families themselves. In the third item a systematic study of Lie groups over pro-p-rings and Big Galois representations is planned with applications to the arithmetic of Hida families.