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Irreducible components of the eigencurve of finite degree are finite over the weight space. (English) Zbl 1460.11079
Summary: Let \(p\) be a rational prime and \(N\) a positive integer which is prime to \(p\). Let \(\mathcal{W}\) be the \(p\)-adic weight space for \(\operatorname{GL}_{2,\mathbb{Q}} \). Let \(\mathcal{C}_N\) be the \(p\)-adic Coleman-Mazur eigencurve of tame level \(N\). In this paper, we prove that any irreducible component of \(\mathcal{C}_N\) which is of finite degree over \(\mathcal{W}\) is in fact finite over \(\mathcal{W} \). Combined with an argument of G. Chenevier [Lond. Math. Soc. Lect. Note Ser. 414, 221–285 (2014; Zbl 1350.11063)] and a conjecture of Coleman-Mazur-Buzzard-Kilford (which has been proven in special cases, and for general quaternionic eigencurves) this shows that the only finite degree components of the eigencurve are the ordinary components.
MSC:
11F85 \(p\)-adic theory, local fields
14G22 Rigid analytic geometry
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