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Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows

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Abstract

We prove the weight part of Serre’s conjecture in generic situations for forms of U(3) which are compact at infinity and split at places dividing p as conjectured by Herzig (Duke Math J 149(1):37–116, 2009). We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge–Tate weights (2, 1, 0) for \(K/\mathbb {Q}_p\) unramified combined with patching techniques. Our results show that the (geometric) Breuil–Mézard conjectures hold for these deformation rings.

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Acknowledgements

We would like to thank Matthew Emerton, Toby Gee, and Florian Herzig for many helpful conversations and for comments on an earlier draft of the paper. We thank the referee for the very careful reading and many suggestions on an earlier version which improved the exposition and clarity of the article.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, M5S 2E4, Canada

    Daniel Le

  2. School of Mathematics, Institute for Advanced Studies, 1 Einstein Drive, Princeton, NJ, 08540, USA

    Bao V. Le Hung

  3. Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, P.O. Box 210089, Tucson, AZ, 85721, USA

    Brandon Levin

  4. Institut Montpelliérain A. Grothendieck, Université de Montpellier, Cc 051, Place E. Bataillon, 34095, Montpellier Cedex, France

    Stefano Morra

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  1. Daniel Le
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  2. Bao V. Le Hung
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  4. Stefano Morra
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Corresponding author

Correspondence to Brandon Levin.

Appendix: Tables

Appendix: Tables

See Tables 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Table 1 The (2, 1, 0)-admissible elements
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Table 2 Inertial local Langlands
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Table 3 Jordan–Hölder factors of Deligne–Lusztig \(\mathrm {GL}_3(\mathbb {F}_p)\)-representations
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Table 4 Shapes of Kisin modules over \(\mathbb {F}\)
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Table 5 Deforming \(\overline{\mathfrak {M}}\) by shape (without monodromy)
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Table 6 Monodromy equations
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Table 7 Deformation rings with monodromy
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Table 8 Types with Weyl intersection in the proof of Proposition 7.16(2)
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Table 9 Serre weights for semisimple \(\overline{\rho }\)
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Le, D., Le Hung, B.V., Levin, B. et al. Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows. Invent. math. 212, 1–107 (2018). https://doi.org/10.1007/s00222-017-0762-0

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