G-valued local deformation rings and global lifts

Abstract

We study G-valued Galois deformation rings with prescribed properties, where G is an arbitrary (not necessarily connected) reductive group over an extension of Z_l for some prime l. In particular, for the Galois groups of p-adic local fields (with p possibly equal to l) we prove that these rings are generically smooth, compute their dimensions, and show that functorial operations on Galois representations give rise to well-defined maps between the sets of irreducible components of the corresponding deformation rings. We use these local results to prove lower bounds on the dimension of global deformation rings with prescribed local properties. Applying our results to unitary groups, we improve results in the literature on the existence of lifts of mod l Galois representations, and on the weight part of Serre's conjecture.

41 pages; refereed version; to appear in Algebra & Number Theory

References in corpus (5)

• Nilpotent orbits over ground fields of good characteristic
• Serre Weights for U(n)
• Minimally Ramified Deformations when $\ell \ne p$
• Producing Geometric Deformations of Orthogonal and Symplectic Galois Representations
• Dimension theory and components of algebraic stacks

Cited by in corpus (14)

• Geometrization of the local Langlands correspondence
• Coherent sheaves on the stack of Langlands parameters
• Coherent Springer theory and the categorical Deligne-Langlands correspondence
• P-adic L-functions in universal deformation families
• The Jacobson--Morozov morphism for Langlands parameters in the relative setting
• Serre weight conjectures for $p$-adic unitary groups of rank 2
• Serre weights for $GS{p}_4$ over totally real fields
• Potential automorphy of ${\mathrm{GSpin}}_{2n+1}$-valued Galois representations
• Irreducible components of the moduli space of Langlands parameters
• Globally realizable components of local deformation rings
• Deformation of rigid conjugate self-dual Galois representations
• Lyndon-Demushkin method and crystalline lifts of $G_2$-valued Galois representations
• From $p$-modular to $p$-adic Langlands correspondences for $U(1,1)({\mathbb{Q}}_{p^2}/{\mathbb{Q}}_p)$: deformations in the non-supercuspidal case
• Motivic Galois representations valued in Spin groups