## $$p$$-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

Zbl 0773.11030
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### References:

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