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.

We prove in this paper 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, and one gives also 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 which are the base change lift for GL(2)/F of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.