p-adic L-functions and Galois cohomology

本课题主要研究伽罗瓦表示所附的某些算术不变量，特别是上同调不变量塞尔默群，与这些伽罗瓦表示所附的L函数或p-进L函数在整数处的零化阶之间的猜想关系。主要目的是证明酉群的伽罗瓦表示的塞尔默群的秩至少等于相应的p-进L函数在0的零点处的阶(实际上人们期望等式成立)。这个策略包括研究特征簇的几何，它是自同构形式的普遍族，在给定的伽罗瓦表示的一点上，以及在该特征簇上构造一个p-进L函数族。这个项目还有另一个方面，那就是重新表述和推广与p-进L函数和伽罗华上同调有关的猜想。虽然这个项目中使用的自同构方法是非常有前途的，但人们并不希望它们单独解决大量关于p-进L函数和伽罗华上同调之间的关系的猜想和问题。要做到这一点，需要来自数学各个部分的许多工具和想法。PI重新阐述这些猜想的目的之一是更清楚地区分哪些部分可以用哪些方法或方法组合来完成，并促进其他领域(例如，超越论)的数学家更多地参与关于这些猜想的工作。

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.