Slopes of overconvergent modular forms

In the first part of this thesis, we consider the slopes of the U2 operator acting on spaces of 2-adic overconvergent modular forms with nontrivial weight-character of tame level 1. We establish a sufficient criterion for these slopes to be given by a simple formula, and prove this criterion in several cases. This allows us to write down all of the slopes for certain weight-characters of small level. This criterion can be written quite simply in terms of modular functions. These results on overconvergent modular forms also imply similar results for certain spaces of classical modular forms, which also allows us to prove results about the field over which the Fourier expansions of the normalised classical cuspidal modular eigenforms are defined. These calculations provide evidence for an analogue of the Gouvea-Mazur conjecture that the slopes of classical cuspidal modular eigenforms vary smoothly as the weight varies. We also present new conjectures of the same form for p = 3 and p = 5, and present some numerical evidence for them. The second part of this thesis considers the Hecke algebras attached to certain spaces of classical cuspidal modular forms of prime level. It proves that some of these Hecke algebras are not Gorenstein when localised at a prime ideal above 2. This shows that the methods developed by Mazur, Gross and Edixhoven for proving that localisations of a Hecke algebra are Gorenstein fail in some cases, because we have exhibited explicit localisations which are not Gorenstein. The computations in both parts of the thesis show the power of computational methods when applied to number theory, in solving problems which can be described concretely. It also shows that computational methods can be useful in identifying patterns and generating data. These can then be investigated by more theoretical methods.