Les variétés de Hecke–Hilbert aux points classiques de poids parallèle 1
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 575-607.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1040
Classification: 11F80, 11F33, 11R23
Keywords: Déformations de représentations galoisiennes $p$-adiques, familles de Hida de formes de Hilbert et formes modulaires de Hilbert de poids $1$.
Betina, Adel 1

1 School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom
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Betina, Adel. Les variétés de Hecke–Hilbert aux points classiques de poids parallèle 1. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 575-607. doi : 10.5802/jtnb.1040. http://www.numdam.org/articles/10.5802/jtnb.1040/

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