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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the eigenvariety of Hilbert modular forms at classical parallel weight one points with dihedral projective image
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by Shaunak V. Deo PDF
Trans. Amer. Math. Soc. 370 (2018), 3885-3912 Request permission

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
  • 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
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 3885-3912
  • MSC (2010): Primary 11F41, 11F80
  • DOI: https://doi.org/10.1090/tran/7064
  • MathSciNet review: 3811513