Eigenvarieties for compact reductive groups

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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.