This is the companion paper to Invent. Math. 124, No. 1-3, 215–241 (1996; Zbl 0851.11030).
From the introduction: The purpose of this article is to use rigid analysis to clarify the relation between classical modular forms and Katz’s overconvergent forms. In particular, the author proves a conjecture of F. Gouvêa [CRM Proc. Lect. Notes 4, 85–99 (1994; Zbl 0829.11026) Conj. 3] which asserts that every overconvergent -adic modular form of sufficiently small slope is classical.
More precisely, let be a prime, a complete subfield of , be a positive integer such that and an integer. Katz has defined the space of overconvergent -adic modular forms of level and weight over and there is a natural map from weight modular forms of level with trivial character at to . These modular forms are called classical modular forms. In addition, there is an operator on these forms such that if is an overconvergent modular form with -expansion then
Then the author proves (Theorem 6.1), that if is a generalized eigenvector for with eigenvalue (i.e., in the kernel of for some positive integer of weight and has -adic valuation strictly less than , then is a classical modular form. In this case the valuation of is called the slope of .
In the case when has slope 0, this is a theorem of H. Hida [Ann. Sci. Ec. Norm. Supér. (4) 19, 231–273 (1986; Zbl 0607.10022)] and, more generally, it implies Gouvêa’s conjecture mentioned above (which is the above conclusion under the additional hypothesis that the slope of is at most . This almost settles the question of which overconvergent eigenforms are classical, as the slope of any classical modular form of weight is at most .
In Section 7, the author investigates the boundary case of overconvergent modular forms of slope one less than the weight. He shows that non-classical forms with this property exist but that any eigenform for the full Hecke algebra of weight is classical if it does not equal where is an overconvergent modular form of weight .
In Section 8, he proves a generalization of Theorem 6.1 (Theorem 8.1) which relates forms of level to what we call overconvergent forms of level and in Section 9, he interprets these latter as certain Serre -adic modular forms with non-integral weight. The central idea in this paper is expressed in Theorem 5.4 which relates overconvergent modular forms to the de Rham cohomology of a coherent sheaf with connection on an algebraic curve. More precisely, the author shows that there is a map for non-negative from modular forms of weight to modular forms of weight which on -expansions is . When , the -th symmetric power of the first relative de Rham cohomology of the universal elliptic curve with a point of order over the modular curve is naturally a sheaf with connection. Theorem 5.4 is the assertion that the cokernel of is the first de Rham cohomology group of the restriction of this sheaf to the complement of the zeros of the modular form on .
The above result, Theorem 6.1, is intimately connected with the conjectures of Gonuvêa and Mazur on families of modular forms. Indeed, in [Invent. Math. 127, No. 3, 417–479 (1997; Zbl 0918.11026)] the author used it to deduce qualitative versions of these conjectures.