Dihedral universal deformations

@article{Deo2020DihedralUD,
  title={Dihedral universal deformations},
  author={Shaunak V. Deo and Gabor Wiese},
  journal={Research in Number Theory},
  year={2020},
  volume={6},
  pages={1-37}
}
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… 
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