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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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##### References:
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