Congruences between automorphic forms and lower bounds on Selmer group

长期以来，对数域F的绝对伽罗瓦群的研究，或者从Tannakian的观点来看，它的连续有限维表示的阿贝尔范畴(比方说在p-ady域上)一直被认为是纯数学中的重要挑战之一。在这些表示中，在Fontaine和Mazur意义下的几何表示特别具有算术意义，它们的范畴应该等价于F上的(仍然是猜想的)混合动机范畴。理解该范畴中的第一个Ext群(更高的Ext群应该是零)是很重要的，Bloch和Kato已经做出了将这些群的维度与L函数在变量的整数值的阶数联系起来的精确猜想。这个项目的目的是利用非调和的自同构形的p-进变形在这些Ext群中构造尽可能多的扩张(希望在对应于L函数的函数方程的中心的情况下，与猜想所预测的一样多)。一个重要的步骤应该是研究围绕非调和自同构形式的被称为本征变元的p进自同构形式的模空间的局部几何。许多算术中的老问题，其中一些可以追溯到丢番图，以及一些新的问题，很好地符合伽罗瓦理论的框架：它们经常可以转化为具有规定性质的某些伽罗瓦表示(即有理数域Q的绝对伽罗瓦群G的表示，或G的一些开子群的表示)的存在性或不存在性的问题。然后，有时，它们可以被证明，就像怀尔斯的费马最后定理一样。伽罗瓦表示的研究分为两个部分：寻找不可约的伽罗瓦表示，然后确定它们之间的扩张。即使第一个问题远未得到解决，布洛赫和加藤也对第二个问题做出了精确的猜测。这些项目旨在通过构建一些有趣的扩展来部分回答这些猜想。该方法使用了自同构形式理论，这曾经是一个相当不同的主题，但现在它与朗兰德程序的伽罗瓦表示理论紧密联系在一起。我们的想法是，人们可以通过观察一些非常特殊的自同构形式，即所谓的非调和形式的(p-adi)变形来获得Galois表示的有趣的扩展。变形越多，就应该能够构建越多的延伸。这些变形被编码在一种被称为本征变体的(p-adi)种类的几何中，并开发工具来研究该几何是该项目中的重要部分。

The study of the absolute Galois group of a number field F or, from a Tannakian point of view, of its abelian category of continuous finite dimensional representations (let us say over a p-adic field) has long been recognized as one of important challenge in pure mathematics. Among those representations, the ones that are geometric, in the sense of Fontaine and Mazur, are especially of arithmetic significance, and their categoryshould be equivalent to the (still conjectural) category of mixed motives over F. It is important to understand the first Ext groups in that category(higher Ext groups should be zero) and Bloch and Kato have made precise conjectures relating the dimension of those groups to the order of L-functions at integers values of the variable. The project aims to construct as much extensions as possible in those Ext groups (hopefully as much as predicted by the conjecture, in the case corresponding to the center of the functional equation of the L-function) using p-adic deformations of non-tempered automorphic forms. An important step should be the study of the local geometry of the "moduli space of p-adic automorphic forms" called Eigenvarieties around the non-tempered automorphic forms.Many old problems in arithmetic, some of them going back as far as Diophantes, as well as some new ones, fit well in the framework of Galois theory: they often can be translated into questions about existence, or non-existence, of certain Galois representations (that is representations of the absolute Galois group G of the field Q of rational numbers, or of some open subgroups of G) with prescribed properties. And then, sometimes, they can be proven, as was Fermat's Last Theorem by Wiles. The study of Galois representations splits up into two parts : finding irreducible Galois representations, and then determining extensions between them. Even if the first problem is far from being solved, precise conjectures about the second one were made by Bloch and Kato. The projects aims to give partial answers to those conjectures, by constructing some interesting extensions. The method uses the theory of automorphic forms, which was once quite a different topic, but which is now strongly tied to the theory of Galois representations by theLangland's program. The idea is that one can obtain interesting extensions of Galois representations by looking at (p-adic) deformations of some very special automorphic forms, the so-called non tempered forms. The more deformations there are, the more extensions one should be able to construct. Those deformations are encoded in the geometry of a (p-adic) variety, known as the Eigenvariety, and developing tools to study that geometry is an important part in the project.