Classical and overconvergent modular forms

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, we prove a conjecture of F. Gouvêa [G, Conj. 3] which asserts that every overconvergent p-adic modular form of sufficiently small slope is classical. More precisely, let p > 3 be a prime, K a complete subfield of Cp, N be a positive integer such that (N, p) = 1 and k an integer. Katz [K-pMF] has defined the spaceMk(Γ1(N)) of overconvergent p-adic modular forms of level Γ1(N) and weight k over K (see §2) and there is a natural map from weight k modular forms of level Γ1(Np) with trivial character at p to Mk(Γ1(N)). We will call these modular forms classical modular forms. In addition, there is an operator U on these forms (see [G-ApM, Chapt. II §3]) such that if F is an overconvergent modular form with q-expansion F (q) = ∑ n≥0 anq n then UF (q) = ∑