Title:

Overconvergent algebraic automorphic forms

I present a general theory of overconvergent padic automorphic forms for reductive algebraic groups whose real points are compact (or, more generally, whose arithmetic subgroups are finite, as in the work of Gross). Let G be such a group, p a prime and P a parabolic subgroup of G defined over Qp. Given a fixed representation V of the Levi factor of P, I construct a family of padic locally analytic representations of the parahoric subgroup associated to P by induction from twists of V by characters. By considering functions from G(Af ) to these representations satisfying suitable equivariance conditions, one obtains a padic Banach module over a certain weight space, with a Heeke action, whose fibre over an integer weight naturally contains the Heeke module of classical automorphic forms of that weight. Using Buzzard's eigenvariety machine, this gives a construction of an eigenvariety parametrising finite slope overconvergent eigenforms. I also prove an analogue in this situation of Coleman's theorem that forms of small slope are classical, implying that this eigenvariety contains a dense set of points corresponding to classical eigenforms. A convenient property of the spaces constructed by these methods is that the definition is sufficiently concrete to allow computer calculations. This is also true for classical automorphic forms on groups satisfying the above condition, as has been noted by various authors. I give a detailed description of an algorithm for calculating classical automorphic forms in the case where G is a definite unitary group, and discuss how the results may be interpreted in terms of Galois representations. In future work I intend to extend this to calculate overconvergent automorphic forms on such groups. In addition, the thesis includes a chapter devoted to the original, motivating example where the theory overconvergent automorphic forms was developed, the case of modular forms for congruence subgroups of GL2 . The main result of this section is the proof of an instance of a conjecture due to Gouvea and Mazur, which is that the overconvergent Heeke eigenforms should form a basis for the entire space of overconvergent forms.
