Dihedral universal deformations
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RESEARCH
Dihedral universal deformations Shaunak V. Deo1* and Gabor Wiese2 * Correspondence:
[email protected] School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India 1
2 Department of Mathematics, University of Luxembourg, Maison du nombre, 6, avenue de la Fonte, 4364 Esch-sur-Alzette, Luxembourg Full list of author information is available at the end of the article
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 = T’ for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral. Keywords: Deformations of Galois representations, Dihedral representations, Modularity lifting Mathematics Subject Classification: 11F80 (primary), 11F41, 11R29, 11R37
1 Introduction The basic object in this article is a continuous absolutely irreducible representation ρ : G → GL2 (F) that is dihedral in the sense that it is induced from a character, where G is a profinite group and F is a finite field of characteristic p. We consider a deformation ρ : G → GL2 (R) of ρ for any complete local Noetherian algebra R over W (F), the ring of Witt vectors of F, with residue field F. We prove results in the following three settings: (1) Representation theory results: We prove useful alternate characterisations to the dihedral property of a deformation ρ of ρ as above. We also prove that being dihedral is an infinitesimal property, in the following sense: the universal deformation of ρ is dihedral if and only if all infinitesimal deformations are dihedral. (2) Number theory results: Here we let G be GK = Gal(K /K ), the absolute Galois group of a number field K . We give sufficient conditions, using class field theory, to ensure that the universal deformation of ρ relatively unramified outside a finite set of primes remains dihedral.
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S. V. Deo, G. Wiese Res. Number Theory (2020)6:29
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In those cases, we compute the structure of the corresponding universal deformation ring and discuss in a series of remarks in how far the sufficient conditions are necessary. We apply our results on the one hand to Bos
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