Eigenvarieties for compact reductive groups

自同构形式代表了经典的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，根据该哲学，某些自同构形式(那些是Hecke代数的特征向量，称为本征形式)应该参数化全局域的Galois群的表示。在经典的模形式的情况下，这是2x2可逆矩阵的群GL(2)的自同构形式，众所周知，本征形式随着权重的变化而在p-进数族中移动，并且这种p-进的变化通过由Coleman和Mazur构造的被称为本征曲线的几何对象的存在来反映。我的研究是关于更复杂的代数群的相似对象(本征簇)的构造和性质。我已经集中在群的实点形成紧空间的情况下；我的论文(将于2007年7月提交)给出了一大类紧群的本征簇的构造。本征簇理论中的一个重要问题是给出本征簇上的点何时真正来自经典模形式的一个很好的判据。众所周知，这样的经典点是稠密的，并且已知的准则暗示给定的点是经典的，但它们不是尖锐的(它们无法检测到一些经典点)。Snaith的计算表明，整个图景与Verma模有关，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，给出经典和非经典点的精确描述。我研究的第二个目标是使这些相当抽象的物体实际上是可计算的。在我的论文中，我开发了计算经典自同构形式的算法，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，我打算使用这些程序来表达关于这些形式的算术的精确猜想，这是可能证明的。特别是，这些计算将提供模p局部朗兰兹对应的实际测试；这一重要猜想的正确公式对于比GL(2)更复杂的群是未知的，任何假设都将具有关于自同构本征形的模p约简的直接可测结果，我的程序应该允许我计算。最后，在紧群的情况下，我的构造证明了一种意想不到的额外结构的存在：由抛物线子群索引的低维的中间本征簇，对应于仅允许在权格的某些方向上进行p-进变化。我希望将这种结构推广到非紧凑的群体。事实上，朗兰兹函数原理预测，在许多情况下，这些特征簇之间应该存在映射；这可能允许人们通过使用这些映射之一将其从紧致群转移到非紧致情况下，获得关于非紧致情况下的特征簇的更明确的信息(可用结构要具体得多)。

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.