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The eigencurve is proper. (English) Zbl 1404.11050

Summary: We prove in this article that, for any prime \(p\) and tame level \(N\), the projection from the eigencurve to the weight space satisfies a rigid analytic version of the valuative criterion for properness introduced by K. Buzzard and F. Calegari [Doc. Math. Extra Vol., 211–232 (2007; Zbl 1138.11015)]. This gives a negative answer to a question of R. Coleman and B. Mazur [Lond. Math. Soc. Lect. Note Ser. 254, 1–113 (1998; Zbl 0932.11030)].

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
11S20 Galois theory
14F30 \(p\)-adic cohomology, crystalline cohomology
14G20 Local ground fields in algebraic geometry
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References:

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