Dihedral universal deformations

Abstract

This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic number theory, (3) modularity. As to (1), we prove that the universal deformation is dihedral if all infinitesimal deformations are dihedral. Concerning (2) in the setting of Galois representations of number fields, we give sufficient conditions to ensure that the universal deformation relatively unramified outside a finite set of primes is dihedral, and discuss in how far these conditions are necessary. As side-results, we obtain cases of the unramified Fontaine–Mazur conjecture, and in many cases positively answer a question of Greenberg and Coleman on the splitting behaviour at p of p-adic Galois representations attached to newforms. As to (3), we prove a modularity theorem of the form ‘\(R=\mathbb {T}\)’ for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral.

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Acknowledgements

The authors would like to thank Carl Wang-Erickson for useful discussions and encouraging them to include the equi-characteristic case of Theorem 1.4 as well. They also thank Tarun Dalal for pointing out an error in a previous version of this article. They also thank the referee for insightful and helpful comments and suggestions.

The majority of this work was done when S. D. was a postdoc at the University of Luxembourg. G. W. acknowledges support by the IRP AMFOR at the University of Luxembourg. This research is also supported by the Luxembourg National Research Fund INTER/ANR/18/12589973 GALF.

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Correspondence to Shaunak V. Deo.

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Deo, S.V., Wiese, G. Dihedral universal deformations. Res. number theory 6, 29 (2020). https://doi.org/10.1007/s40993-020-00197-y

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Keywords

  • Deformations of Galois representations
  • Dihedral representations
  • Modularity lifting

Mathematics Subject Classification

  • 11F80 (primary)
  • 11F41
  • 11R29
  • 11R37