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A gluing lemma and overconvergent modular forms. (English) Zbl 1112.11020

Let \(p\) be a prime. Let \(K\) be a finite extension of \(\mathbb Q_p\). In [Invent. Math. 124, No. 1–3, 215–241 (1996; Zbl 0851.11030); J. Théor. Nombr. Bordx. 9, No. 2, 395–403 (1997; Zbl 0942.11025)] R. F. Coleman proves the following control theorem.
Theorem: Let \(f\) be an overconvergent \(U_p\)-eigenform of weight \(k\) defined over \(K\) with eigenvalue \(a_p\). If the \(p\)-adic valuation of \(a_p\) is less than \(k-1\), then \(f\) is classical.
Coleman’s method of proving the theorem is based on a cohomological interpretation of the space of overconvergent forms. The author presents a more intrinsic (geometric in nature) proof of (a generalization of) the control theorem. He proves a gluing lemma for sections of line bundles on a rigid analytic variety in §2. The control theorem is then deduced from the Lemma and (a generalization of) K. Buzzard ’s analytic continuation results [J. Am. Math. Soc. 16, No. 1, 29–55 (2003; Zbl 0942.11025)].

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
11G18 Arithmetic aspects of modular and Shimura varieties
14G22 Rigid analytic geometry
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