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.

现代数字理论中最具挑战性的话题之一是特殊值的特殊价值与galois的共同体之间的神秘关系：它们是两个完全不同的世界的阴影，对一个几何对象进行了复杂和padic对象的分析，即解决方案的解决方案的空间，用于整体数字，或者是一般的理论。同时，这些动机的整个P-Adic家族同时，类别的班级公式或Birch和Swinnerton Dyer的猜想，例如通过考虑数字塔的塔或（HIDA）模块化形式的家族。该项目的目的是提供进一步的证据。 P-ADIC L功能的存在以及（非共同）Iwasawa理论中的主要猜想，ii。福卡亚和加藤的（均等）电子注射也是Asiii。尤其是，我们希望构建第一个真正的非共同的p-Adic L功能，并找到（非交通性的）示例，以满足对E-Constants的期望，这些示例是由相应的L功能的功能方程式确定的，建立P-Adadic家族，这是p-aadadig的家族。在第三项中，计划对谎言组进行谎言组对Pro-p-Cond和Big Galois表示，并计划在HIDA家族的算术中应用。