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# Ramification of the Eigencurve at classical RM points

Published:2019-01-21

J.Bellaïche and M.Dimitrov have shown that the $p$-adic eigencurve is smooth but not étale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin-Lehner involution of the completed local ring of the eigencurve at these points and an universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly $2$. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the overconvergent cuspidal Eisenstein points, being the base change lift for $\operatorname{GL}(2)_{/F}$ of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.
 MSC Classifications: 11F80 - Galois representations 11F33 - Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 11R23 - Iwasawa theory