Mathematics > Number Theory
[Submitted on 14 May 2018 (v1), last revised 9 Apr 2020 (this version, v3)]
Title:Dihedral Universal Deformations
View PDFAbstract: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.
Submission history
From: Gabor Wiese [view email][v1] Mon, 14 May 2018 20:49:44 UTC (39 KB)
[v2] Fri, 26 Apr 2019 08:04:25 UTC (40 KB)
[v3] Thu, 9 Apr 2020 13:38:14 UTC (42 KB)
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