Congruences between automorphic forms and lower bounds on Selmer group

自守形式与 Selmer 群下界之间的同余

• 批准号: 0501023
• 负责人: Joel Bellaiche
• 金额: --
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0501023&HistoricalAwards=false
• 关键词: Congruences; between; automorphic; forms; lower

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.

从Tannakian的角度来看，对其连续有限维表示的Abelian类别（让我们在P-Adic领域上）对绝对的Galois组的研究一直被认为是纯数学中的重要挑战之一。 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变量的值。该项目的目的是在这些EXT组中构建尽可能多的扩展（希望可以使用非敏感自动形式的P-ADIC变形，在与L功能方程相对应的情况下，猜想的预测尽可能多，在对应于L函数方程的情况下。一个重要的步骤应该是研究“ p-阿杜自动形式的模量空间”的局部几何形状，称为非脾气自动形式周围的特征变体。 Galois表示（即有理数Q的绝对GALOIS组G的表示，或具有规定属性的某些开放子组的表征。然后，有时可以证明它们，就像Fermat的《 Willes的最后定理》一样。对Galois表示形式的研究分为两个部分：查找不可还原的Galois表示，然后确定它们之间的扩展。即使第一个问题尚未解决，第二个问题也是由Bloch和Kato提出的。这些项目的目的是通过构建一些有趣的扩展来对这些猜想提供部分答案。该方法使用了自动形式的理论，这曾经是一个完全不同的话题，但现在与Thelangland的计划与Galois代表理论有着密切的联系。这个想法是，人们可以通过查看某些非常特殊的自动形式的（P-ADIC）变形，即所谓的非矫正形式来获得有趣的Galois表示。变形越多，人们应该构建的扩展越多。这些变形是在（P-ADIC）的几何形状（称为特征变量）的几何形状中编码的，并开发了研究几何形状是该项目的重要组成部分。