Congruences between automorphic forms and lower bounds on Selmer group

从Tannakian的观点来看，对数字域F的绝对伽罗瓦群的研究，即它的连续有限维表示的阿贝尔范畴（比方说在p进域上）的研究，一直被认为是纯数学中的一个重要挑战。那些表示,那些几何,在铺满的感觉和Mazur,尤其的算术意义,及其categoryshould相当于(仍推测的)一类混合动机在f .重要的是要理解第一Ext组织类别(更高的Ext组应该是零)和布洛赫和加藤准确推测相关团体的维度的顺序L-functions整数变量的值。该项目旨在使用非调质自同构形式的p进变形在这些Ext群中构造尽可能多的扩展（希望与猜想预测的一样多，在对应于l函数的泛函方程中心的情况下）。一个重要的步骤应该是研究“p进自同构形式的模空间”的局部几何，称为非调质自同构形式周围的特征变分。许多算术上的老问题，有些可以追溯到丢芬忒斯时代，还有一些新问题，都很适合于伽罗瓦理论的框架。这些问题往往可以转化为关于某些具有规定性质的伽罗瓦表象的存在或不存在的问题（即有理数领域Q的伽罗瓦绝对群G的表象，或G的某些开放子群的表象）。然后，有时候，它们可以被证明，就像怀尔斯的费马大定理。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展。即使第一个问题还远没有解决，布洛赫和加藤对第二个问题做出了精确的推测。该项目旨在通过构建一些有趣的扩展来给出这些猜想的部分答案。该方法使用了自同构形式理论，这曾经是一个完全不同的主题，但现在它与兰兰德程序的伽罗瓦表示理论紧密联系在一起。这个想法是，人们可以通过观察一些非常特殊的自同构形式的（p进）变形，即所谓的非调质形式，来获得伽罗瓦表示的有趣扩展。变形越多，可以构造的扩展就越多。这些变形被编码为一种（p进的）几何变化，称为特征变化，开发工具来研究该几何变化是该项目的重要组成部分。

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.