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Eigenvarieties for compact reductive groups

紧约还原群的特征簇

基本信息

批准号：
EP/F04304X/2
负责人：
David Loeffler
金额：
--
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FF04304X%2F2
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.

自守形式代表了模形式的经典概念的广泛推广。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（那些是Hecke代数的特征向量，被称为特征形）应该参数化全局域的伽罗瓦群的表示。在经典模形式的情况下，它是2x2可逆矩阵的群GL（2）的自守形式，已知本征形随着权重变化而在p-adic族中移动，并且该p-adic变化通过由科尔曼和Mazur构造的称为本征曲线的几何对象的存在来反映。我的研究涉及更复杂的代数群的类似对象（特征簇）的构造和性质。我集中讨论了群的真实的点构成紧致空间的情况;我的论文（将于2007年7月提交）给出了一个广泛的紧致群类的本征簇的构造。本征簇理论中的一个重要问题是给出一个很好的准则，判断本征簇上的一个点何时实际上来自于一个经典的模形式。已知这样的经典点是稠密的，并且已知标准暗示给定点是经典的，但它们不是尖锐的（它们不能检测到一些经典点）。Snaith的计算表明，这幅完整的图像与Verma模有关，Verma模是出现在李代数理论中的结构。我研究的第一个主要目标是发展这个理论，以便对经典点和非经典点给出精确的描述;我研究的第二个目标是使这些相当抽象的对象实际上是可计算的。在我的论文中，我开发了计算经典自守形式的算法，并且应该可以将这些算法扩展到计算对应于本征簇上的非经典点的非经典形式。然后，我打算用这些程序来制定关于这些形式的算术的精确公式，这是可能证明的。特别地，这些计算将提供模p局部朗兰兹对应的实际测试;对于比GL（2）更复杂的群，这个重要猜想的正确表述是未知的，并且任何假设都将具有关于自守本征形的模p约化的直接可检验的结果，我的程序应该允许我计算。最后，在紧群的情况下，我的构造证明了一个意想不到的额外结构的存在：由抛物子群索引的较低维度的中间本征簇，其对应于仅允许权格中某些方向上的p-adic变化。我希望把这个构造推广到非紧群。事实上，朗兰兹函数性原理预测，在许多情况下，这些本征簇之间应该存在映射;这可能允许人们通过使用其中一个映射将其从紧群中转移过来，获得有关非紧情况下本征簇（可用的构造不太具体）的更明确的信息。