Eigenvarieties for compact reductive groups

紧约还原群的特征簇

基本信息

批准号：
EP/F04304X/1
负责人：
David Loeffler
金额：
$ 27.91万
依托单位：
University of Cambridge
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2008
资助国家：
英国
起止时间：
2008 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FF04304X%2F1
关键词：
Eigenvarieties compact reductive groups

项目摘要

Automorphic forms represent a vast generalisation of the classical notion of modular forms. They have applications to many areas of number theory, especially via the Langlands philosophy, according to which certain automorphic forms (those which are eigenvectors for the Hecke algebra, which are known as eigenforms) should parametrize representations of the Galois groups of global fields.In the case of classical modular forms, which are the automorphic forms for the group GL(2) of 2x2 invertible matrices, it is known that eigenforms move in p-adic families as the weight varies, and this p-adic variation is reflected by the existence of a geometric object known as the eigencurve , constructed by Coleman and Mazur. My research concerns the construction and properties of analogous objects (eigenvarieties) for more complicated algebraic groups. I have concentrated on the case where the real points of the group form a compact space; my thesis (to be submitted July 2007) gives a construction of eigenvarieties for a wide class of compact groups.One important problem in the theory of eigenvarieties is to give a good criterion for when a point on an eigenvariety actually arises from a classical modular form. It is known that such classical points are dense, and criteria are known which imply that a given point is classical, but they are not sharp (they fail to detect some classical points). Calculations of Snaith suggest that the full picture is related to Verma modules, which are constructions that appear in the theory of Lie algebras. The first major objective of my research is to develop this theory to give an exact characterisation of classical and non-classical points.The second aim of my research is to make these rather abstract objects practically computable. During my thesis I developed algorithms for calculating the classical automorphic forms, and it should be possible to extend these to calculate the non-classical forms which correspond to non-classical points on the eigenvariety. I intend to then use these programs to formulate precise conjectures regarding the arithmetic of these forms, which it might be possible to prove. In particular, these calculations would provide a practical test of the modulo p local Langlands correspondence; the correct formulation of this important conjecture is not known for groups more complex than GL(2), and any hypothesis would have directly testable consequences regarding the modulo p reduction of automorphic eigenforms, which my programs should allow me to calculate.Finally, in the case of compact groups my construction demonstrates the existence of an unexpected piece of extra structure: intermediate eigenvarieties of lower dimension indexed by parabolic subgroups, which correspond to allowing p-adic variation only in certain directions in the weight lattice. I hope to generalise this construction to non-compact groups. Indeed, the Langlands functoriality principle predicts that there should exist maps between these eigenvarieties in many cases; this might allow one to obtain more explicit information about eigenvarieties in the non-compact case (where the constructions available are much less concrete) by transferring it over from a compact group using one of these maps.

自动形式代表了模块化形式的经典概念的广泛概括。它们在许多数量理论领域，尤其是通过兰兰兹的哲学上都有应用，根据这些理论，某些自动形式（那些是赫克（Hecke）代数的特征向量的形式（称为特征形式的特征向量）应参数化Galois galois galois galois glois群体的表示。特征形式随着重量的变化而在p-adic家族中移动，而这种p- adic变化反映出由科尔曼和马祖尔构建的称为特征库的几何对象的存在反映。我的研究涉及对更复杂的代数组的类似物体（特征值）的构建和特性。我专注于组的实际点形成紧凑空间的情况。我的论文（将于2007年7月提交）为广泛的紧凑型组提供了特征值的构造。特征值理论中的一个重要问题是给出一个良好的标准，以便何时在特征变量上的观点实际上是由经典模块化形式引起的。众所周知，这样的经典点是密集的，标准是已知的，这意味着给定的观点是经典的，但并不尖锐（它们无法检测到某些经典点）。 Snaith的计算表明，完整的图片与Verma模块有关，Verma模块是谎言代数理论中出现的结构。我的研究的第一个主要目标是开发这一理论，以确切地表征经典和非古典观点。我的研究的第二个目的是使这些相当抽象的对象实际上可以计算。在论文期间，我开发了用于计算经典自身形式的算法，应该可以扩展这些算法以计算与特征因子上非经典点相对应的非经典形式。然后，我打算使用这些程序来制定有关这些形式的算术的精确猜想，这可能是可以证明的。特别是，这些计算将提供对局部Langlands对应的实用测试。这一重要猜想的正确表述对于比GL（2）更复杂的群体不知道，任何假设都会在减少自多态特征形式的模量下有可直接测试的后果，而我的程序应该使我能够使我能够计算出来。对应于仅在重量晶格中的某些方向上允许P-ADIC变化。我希望将这种构建概括为非紧缩群体。确实，兰兰兹的功能性原则预测，在许多情况下，应该存在这些特征性之间的图。这可能允许人们使用其中一个地图将其从紧凑的组中转移到非紧凑型情况下（可用的构造的混凝土少得多），以获取有关特征值的更明确的信息。