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We show that the Eigenvariety attached to Hilbert modular forms over a totally real field F is smooth at the points corresponding to certain classical weight one theta series and we give a precise criterion for etaleness over the weight space at those points. In the case where the theta series has real multiplication, we construct a non-classical overconvergent generalised eigenform and compute its Fourier coefficients in terms of p-adic logarithms of algebraic numbers. Our approach uses deformations of Galois representations.

Les variétés de Hecke–Hilbert aux points classiques de poids parallèle 1