Iwasawa theory of p-adic Lie extensions

岩泽 p 进李扩展理论

• 批准号: 48575220
• 负责人: Professor Dr. Otmar Venjakob
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2007
• 资助国家: 德国
• 起止时间: 2006-12-31 至 2010-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/48575220?language=en
• 关键词: Iwasawa; theory; adic; Lie; extensions

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.

l函数的特殊值与伽罗瓦上同调之间的神秘关系是现代数论中最具挑战性的课题之一。它们是同一个几何对象的复数分析和p进分析这两个完全不同的世界的影子，即给定的整数上的丢芬图方程的解空间，或者更一般地说，一个动机M. Iwasawa理论的主要思想是研究这一原理的表现，如类数公式或Birch和Swinnerton Dyer猜想，同时用于这类动机的整个p进族。例如，通过考虑数塔域或（Hida）模形式族而产生。这个项目的目的是为i提供进一步的证据。（非交换）Iwasawa理论中p进l函数的存在性及主要猜想，2。Fukaya和Kato的（等变）e猜想以及asIII。Hida族的2变量主猜想。特别是，我们希望构造第一个真正的非交换的p进l函数，并找到（非交换的）例子，满足由相应l函数的泛函方程决定的e常数本身构建p进族的期望。第三部分系统地研究了亲环和大伽罗瓦表示上的李群及其在希达族算法中的应用。