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Ramification of the Eigencurve at classical RM points

  • Adel Betina,
    School of Mathematics and Statistics, The University of Sheffield, Hicks building, Hounsfield Road, Sheffield S3 7RH, UK
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Abstract

J.Bella├»che and M.Dimitrov have shown that the $p$-adic eigencurve is smooth but not ├ętale 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.
Keywords: weight one RM modular form, eigencurve, pseudo-deformation, deformation of reducible representation weight one RM modular form, eigencurve, pseudo-deformation, deformation of reducible representation
MSC Classifications: 11F80, 11F33, 11R23 show english descriptions Galois representations
Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
Iwasawa theory
11F80 - Galois representations
11F33 - Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
11R23 - Iwasawa theory
 

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