On the Fourier coefficients of modular forms. II

Abstract

In this paper we continue our study of the p-adic valuations of eigenvalues of the Hecke operator Up. In [U2], we proved that the Newton polygon of the characteristic polynomial of Up on certain spaces of cusp forms of level divisible by p is bounded below by an explicitly given (Hodge) polygon. Here, we investigate the extent to which this result is sharp. In particular, we want to find the highest polygon with integer slopes which lies below the Newton polygon of Up (its "contact polygon"). Knowledge of this polygon yields non-trivial upper bounds on dimensions of spaces of forms defined by slope conditions. In some cases, we can go much further, giving formulae for the dimensions of spaces of forms of certain slopes in terms of forms of weight 2. This can be viewed as a generalization to higher slope of well-known results of Hida [HI on the number of ordinary eigenforms, i.e., eigenforms of slope 0. What underlies all of our results is very fine information on a certain crystalline eohomology group associated to modular forms. In a future paper we will exploit this information further and prove congruences between modular forms of various weights and slopes. This allows us to get good control on the Galois representations modulo p attached to certain forms of weight > 2. The first section of the paper gives our results on modular forms and then in Sect. 2 we give the eohomological results underlying these theorems. The main results on modular forms are 1.4-1.8 and the most important technical result is Theorem 2.4. There is a summary of the rest of the paper at the end of Sect. 2.