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

On montre que la variété de Hecke associée aux formes de Hilbert sur un corps totalement réel $F$ est lisse aux points correspondant à certaines séries thêta de poids $1$ et on donne aussi un critère pour que le morphisme poids soit étale en ces points. Lorsque les séries thêta sont à multiplication réelle, on construit des formes surconvergentes propres généralisées qui ne sont pas classiques et on exprime leurs coefficients de Fourier à l’aide de logarithmes $p$-adiques de nombres algébriques. Notre approche utilise la théorie des déformations galoisiennes.

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.

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