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.

自同构形式代表了模块化形式的经典概念的广泛概括。它们在数论的许多领域都有应用，特别是通过朗兰兹哲学，根据朗兰兹哲学，某些自守形式（赫克代数的特征向量，称为特征形式）应该参数化全局域的伽罗瓦群的表示。对于经典模形式的情况，即 2x2 可逆矩阵的群 GL(2) 的自同构形式，已知特征形式移入p-adic 族随着权重的变化而变化，这种 p-adic 变化通过称为特征曲线的几何对象的存在来反映，该几何对象由 Coleman 和 Mazur 构建。我的研究涉及更复杂的代数群的类似对象（特征变量）的构造和性质。我集中讨论了群的实点形成一个紧空间的情况；我的论文（将于 2007 年 7 月提交）为一类广泛的紧群给出了特征向量的构造。特征向量理论中的一个重要问题是为特征向量上的点实际上从经典模形式中产生的时间给出一个很好的标准。众所周知，此类经典点是密集的，并且已知暗示给定点是经典点的标准，但它们并不尖锐（它们无法检测到某些经典点）。 Snaith 的计算表明，整个情况与 Verma 模相关，Verma 模是李代数理论中出现的结构。我研究的第一个主要目标是发展这一理论，以准确描述经典和非经典点。我研究的第二个目标是使这些相当抽象的对象实际上可计算。在我的论文期间，我开发了计算经典自同构形式的算法，并且应该可以扩展这些算法来计算与特征向量上的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，并且有可能证明这一点。特别是，这些计算将为模 p 局部朗兰兹对应提供实际测试；对于比 GL(2) 更复杂的群，这个重要猜想的正确表述尚不清楚，并且任何假设都会对自守本征型的模 p 约简产生直接可检验的结果，我的程序应该允许我计算它。最后，在在紧群的情况下，我的构造证明了意外的额外结构的存在：由抛物线子群索引的较低维度的中间特征变量，其对应于仅在权重格中的某些方向上允许p进数变化。我希望将这种构造推广到非紧群。事实上，朗兰兹函子原理预测在许多情况下这些特征变量之间应该存在映射；这可能允许人们通过使用这些映射之一从紧凑组转移来获得有关非紧凑情况下特征变量的更明确的信息（其中可用的构造不太具体）。