p-adic L-functions and Galois cohomology

p 进 L 函数和伽罗瓦上同调

• 批准号: 1101615
• 负责人: Joel Bellaiche
• 金额: $ 25.73万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101615&HistoricalAwards=false
• 关键词: adic; functions; Galois; cohomology

The project focuses on the conjectural relations between certain arithmetic invariants attached to Galois representations, in particular the cohomological invariants called Selmer groups, and the order of vanishing at integers of either the L-functions, or the p-adic L-functions attached to those Galois representations. The main objective is to prove that the rank of the Selmer group of a Galois representation attached to an automorphic form for a unitary group is at least equal to the order of vanishing at 0 of the corresponding p-adic L-function (one actually expects that the equality holds). The strategy consists in a study of the geometry of the eigenvariety, which is the universal family of automorphic forms, at a point attached to the given Galois representation, and in the construction of a family of p-adic L-functions on that eigenvariety. The project has an other aspect, which consists in reformulating and generalizing the conjectures relating p-adic L-functions and Galois cohomology. While the automorphic methods used in this project are very promising, it is not expected that they alone will solve the vast array of conjectures and questions concerning the relations between p-adic L-functions and Galois cohomology. Many tools and ideas from various parts of mathematics will be needed to do so. One of the aims of the PI in reformulating those conjectures is to separate more clearly what part can be done with which methods or combination of methods, and to foster a greater involvement of mathematicians in other areas (e.g., theory of transcendence) in the work on those conjectures.

本课题主要研究伽罗瓦表示上的某些算术不变量，特别是称为Selmer群的上同调不变量，与伽罗瓦表示上的l函数或p进l函数在整数处消失的顺序之间的猜想关系。主要目的是证明伽罗瓦表示附属于酉群的自同构形式的Selmer群的秩至少等于相应的p进l函数在0处消失的阶（实际上期望这个等式成立）。该策略包括对特征变数的几何研究，特征变数是附属于给定伽罗瓦表示的自同构形式的一般族，以及在该特征变数上构造p进l函数族。本课题的另一个方面是对p进l函数和伽罗瓦上同调的猜想进行了重新表述和推广。虽然本项目中使用的自同构方法非常有前途，但不能指望它们单独解决关于p进l函数与伽罗瓦上同调之间关系的大量猜想和问题。要做到这一点，需要数学各个领域的许多工具和思想。PI重新表述这些猜想的目的之一是更清楚地区分哪些方法或方法组合可以完成哪些部分，并促进其他领域（例如，超越理论）的数学家更多地参与这些猜想的工作。