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.

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（让我们说在一个p-adic域上）一直被认为是纯数学中的一个重要挑战。在这些表示中，在方丹和马祖尔意义上的几何表示特别具有算术意义，它们的范畴应该等价于F上的混合动机范畴（仍然是几何的）。重要的是要了解第一个外群在这一类（更高的外群应是零）和布洛赫和加藤作出了精确的approachtures有关的维度这些团体的秩序的L-函数在整数值的变量。该项目的目标是使用非回火自守形式的p-adic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函数的函数方程的中心的情况下）。一个重要的步骤应该是研究局部几何的“模空间的p-adic自守形式”所谓的本征变种周围的非回火自守形式。许多老问题的算术，其中一些可以追溯到丢番，以及一些新的，以及适合的框架伽罗瓦理论：它们通常可以被转化为关于某些伽罗瓦表示的存在或不存在的问题（即有理数域Q的绝对伽罗瓦群G的表示，或G的某些开子群的表示）具有规定的性质。然后，有时候，它们可以被证明，就像怀尔斯的费马大定理一样。伽罗瓦表示的研究分为两个部分：找到不可约的伽罗瓦表示，然后确定它们之间的扩展。即使第一个问题远未解决，Bloch和Kato对第二个问题也作了精确的解释。该项目旨在通过构建一些有趣的扩展来部分回答这些问题。该方法使用理论的自守形式，这曾经是一个相当不同的话题，但现在是密切相关的理论伽罗瓦表示theLangland的计划。这个想法是，人们可以通过观察一些非常特殊的自守形式（所谓的非回火形式）的（p-adic）变形来获得伽罗瓦表示的有趣扩展。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个（p-adic）品种的几何形状中，称为特征品种，开发工具来研究该几何形状是该项目的重要组成部分。