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Smoothness and classicality on Eigenvarieties

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Abstract

Let p be a prime number and f an overconvergent p-adic automorphic form on a definite unitary group which is split at p. Assume that f is of “classical weight” and that its Galois representation is crystalline at p, then f is conjectured to be a classical automorphic form. We prove new cases of this conjecture in arbitrary dimensions by making crucial use of the patched eigenvariety constructed in Breuil et al. (Math. Ann. 2016).

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Author information

Authors and Affiliations

  1. C.N.R.S. Laboratoire de Mathématiques d’Orsay, Université Paris-Sud Université Paris-Saclay, 91405, Orsay, France

    Christophe Breuil

  2. Mathematisches Institut, Universität Münster, Einsteinstrasse 62, 48149, Münster, Germany

    Eugen Hellmann

  3. CMLS, École Polytechnique CNRS, Université Paris-Saclay, 91128, Palaiseau Cédex, France

    Benjamin Schraen

Authors
  1. Christophe Breuil
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  2. Eugen Hellmann
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  3. Benjamin Schraen
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Corresponding author

Correspondence to Christophe Breuil.

Additional information

We thank John Bergdall, Laurent Berger, Peter Scholze and Jack Thorne for their answers to our questions. C. B. and B. S. are supported by the C.N.R.S. and by the A.N.R. project ThéHopaD (ANR-2011-BS01-005) and B. S. thanks I.H.É.S. for its hospitality. E. H. is supported by SFB-TR 45 of the D.F.G.

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Breuil, C., Hellmann, E. & Schraen, B. Smoothness and classicality on Eigenvarieties. Invent. math. 209, 197–274 (2017). https://doi.org/10.1007/s00222-016-0708-y

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