Eigenvarieties for compact reductive groups

自同构形式代表了经典模形式概念的广泛推广。它们在数论的许多领域都有应用，特别是通过朗兰兹哲学，根据某些自同构形式（Hecke代数的特征向量，被称为特征形式）应该参数化全局域伽罗瓦群的表示。对于经典模形式，即2x2可逆矩阵群GL(2)的自同构形式，已知特征形式随着权值的变化在p进族中移动，并且这种p进的变化通过Coleman和Mazur构造的称为特征曲线的几何对象的存在来反映。我的研究涉及更复杂代数群的类似对象（本征变）的构造和性质。我集中讨论了群的实点形成紧空间的情况；我的论文（将于2007年7月提交）给出了一大类紧群的特征变的构造。本征变理论中的一个重要问题是给出一个很好的判据，说明本征变上的一个点实际上是由经典模形式产生的。已知这样的经典点是密集的，并且已知暗示给定点是经典点的标准，但它们不是尖锐的（它们无法检测到某些经典点）。Snaith的计算表明，这一切都与维尔马模块有关，维尔马模块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，以给出经典和非经典点的精确特征。我研究的第二个目标是使这些相当抽象的对象实际上可以计算。在我的论文中，我开发了计算经典自同构形式的算法，并且有可能将这些算法扩展到计算非经典形式，这些非经典形式对应于本征变上的非经典点。然后，我打算用这些程序来形成关于这些形式的算术的精确猜想，这些猜想可能被证明。特别是，这些计算将提供对模p局部朗兰兹对应的实际检验；对于比GL(2)更复杂的群来说，这个重要猜想的正确表述是未知的，任何假设都将有关于自同构特征形式的模p约简的直接可测试的结果，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的中间特征变，它对应于只允许在权格的某些方向上的p进变。我希望将这种构造推广到非紧群。事实上，朗兰兹泛函原理预言，在许多情况下，这些特征变体之间应该存在映射；这可能允许人们通过使用这些映射中的一个从紧群转移过来，在非紧情况下（可用的结构不太具体）获得关于特征变分的更明确的信息。

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.