p-adic L-functions and Galois cohomology

该项目的重点是某些算术不变量之间的代数关系附加到伽罗瓦表示，特别是上同调不变量称为塞尔默群，和消失的顺序在整数的L-功能，或p-adic L-功能附加到这些伽罗瓦表示。主要目的是证明一个酉群的一个自同构形式上的伽罗瓦表示的塞尔默群的秩至少等于对应的p-adic L-函数在0处消失的阶（人们实际上期望等式成立）。该战略包括在研究几何的特征，这是普遍家庭的自守形式，在一个点上重视给定的伽罗瓦表示，并在建设一个家庭的p-进L-功能的特征。该项目还有另一个方面，包括重新制定和推广有关p进L-函数和伽罗瓦上同调的理论。虽然在这个项目中使用的自守方法是非常有前途的，但并不期望他们单独将解决大量的问题和p-adic L-函数和伽罗瓦上同调之间的关系。要做到这一点，需要数学各个部分的许多工具和思想。PI在重新制定这些体系时的目标之一是更清楚地区分哪些部分可以用哪些方法或方法的组合来完成，并促进数学家更多地参与其他领域（例如，超越论）在这些著作的工作。

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.