Dihedral Universal Deformations

TL;DR

This paper investigates conditions under which universal dihedral deformations of Galois representations are themselves dihedral, with results spanning representation theory, number theory, and modularity, including applications to the Fontaine-Mazur conjecture.

Contribution

It provides new criteria for when universal deformations are dihedral across multiple mathematical settings, extending understanding of Galois representations and modularity.

Findings

Universal deformation is dihedral if all infinitesimal deformations are dihedral.

Sufficient conditions are given for universal deformations to be dihedral in number field Galois representations.

Cases of the unramified Fontaine-Mazur conjecture are obtained and questions on p-adic Galois representations are addressed.

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.