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We show that the Eigenvariety attached to Hilbert modular forms over atotally real field $F$ is smooth at the points corresponding to certainclassical weight one theta series and we give a precise criterion for etalenessover the weight space at those points. In the case where the theta series hasreal multiplication, we construct a non-classical overconvergent generalisedeigenform and compute its Fourier coefficients in terms of $p$-adic logarithmsof algebraic numbers. Our approach uses deformations and pseudo-deformations ofGalois representations.

Les Vari\'et\'es de Hecke-Hilbert aux points classiques de poids $1$