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Classical and overconvergent modular forms. (English) Zbl 0851.11030
Let \(F(q) = \sum_{n \geq 0} a_n q^n\) be an overconvergent \(p\)-adic modular form of level \(N\), \((N,p) = 1\), and let \(U\) be the Atkin operator acting on \(q\)-expansions by \(UF(q) = \sum_{n \geq 0} a_{pn} q^n\). The author proves that if \(F\) is a generalized eigenvector for \(U\) with eigenvalue \(\lambda\) of weight \(k + 2\) and \(\lambda\) has \(p\)-adic valuation less than \(k + 1\), then \(F\) is a classical modular form (Theorem 6.1). This implies Gouvêa’s conjecture that every overconvergent \(p\)-adic modular form of sufficiently small slope is classical. The main ingredient in the proof of Theorem 6.1 is the assertion relating overconvergent modular forms to the de Rham cohomology of a certain coherent sheaf with connection on an algebraic curve (Theorem 5.4). Also, a generalization of Theorem 6.1 to level \(Np\) is proved (Theorem 8.1), and an interpretation of overconvergent forms of level \(Np\) as certain Serre \(p\)-adic modular forms with non-integral weight is given (Theorem 9.1).

11F33 Congruences for modular and \(p\)-adic modular forms
14F40 de Rham cohomology and algebraic geometry
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