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On the eigenvariety of Hilbert modular forms at classical parallel weight one points with dihedral projective image


Author: Shaunak V. Deo
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11F41, 11F80
DOI: https://doi.org/10.1090/tran/7064
Published electronically: December 20, 2017
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Abstract: We show that the $ p$-adic eigenvariety constructed by Andreatta-Iovita-Pilloni, parameterizing cuspidal Hilbert modular eigenforms defined over a totally real field $ F$, is smooth at certain classical parallel weight one points which are regular at every place of $ F$ above $ p$ and also determine whether the map to the weight space at those points is étale or not. We prove these results assuming the Leopoldt conjecture for certain quadratic extensions of $ F$ in some cases, assuming the $ p$-adic Schanuel conjecture in some cases, and unconditionally in some cases, using the deformation theory of Galois representations. As a consequence, we also determine whether the cuspidal part of the $ 1$-dimensional parallel weight eigenvariety, constructed by Kisin-Lai, is smooth or not at those points.


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Additional Information

Shaunak V. Deo
Affiliation: Department of Mathematics, MS 050, Brandeis University, 415 South Street, Waltham, Massachusetts 02453
Address at time of publication: Université du Luxembourg, Maison du Nombre, 6, Avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
Email: deoshaunak@gmail.com, shaunak.deo@uni.lu

DOI: https://doi.org/10.1090/tran/7064
Keywords: Parallel weight one Hilbert modular forms, deformation of Galois representations, eigenvariety
Received by editor(s): June 10, 2016
Received by editor(s) in revised form: August 19, 2016, and August 31, 2016
Published electronically: December 20, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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