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\(p\)-adic Banach spaces and families of modular forms. (English) Zbl 0918.11026
In part A, the author shows how Serre’s \(p\)-adic Banach-Fredholm-Riesz theory may be extended over complete, normed rings. He applies these results to elliptic modular forms. He proves that the Fredholm determinants of the \(U\)-operator acting on (integer) weight \(k\) overconvergent modular forms are specializations of a Fredholm determinant of a completely continuous operator over the Banach algebra of rigid analytic functions on any sufficiently large closed disc in the weight space (theorem B). One consequence is a qualitative version of the Gouvêa-Mazur \(R\)-family conjecture (conjecture 3 of the article [F. Gouvêa and B. Mazur, Math. Comput. 58, 793-805 (1992; Zbl 0773.11030)]; section B.5).

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
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