On local Galois deformation rings: generalised tori

TL;DR

This work develops a deformation theory for mod $p$ Galois representations valued in generalised tori, giving explicit presentations for framed deformation rings and $G$-pseudocharacter deformation rings and proving formal smoothness over group algebras of finite abelian $p$-groups. It builds an abstract 1-cocycle and Langlands-torus framework to model deformation spaces via $oldsymbol ext{Γ}_1,oldsymbol ext{Γ}_2,oldsymbolΔ$-data, proving representability of representation and pseudocharacter moduli and establishing canonical actions of $ ext{X}(oldsymbol ext{μ})$ on irreducible components. The paper identifies irreducible components with characters of a finite $p$-group $oldsymbol ext{μ}$ and provides dimension formulas linking $ ext{rank}_{Z_p}((oldsymbol ext{Γ}_E^{ ext{ab},p} ens M)^{oldsymbol Δ})$ to deformation dimensions, with formal smoothness preserved under base change. It further connects local torus deformations to a broader program (defG) on generalised reductive groups and translates the results through profinite completion to Galois deformation theory, offering explicit, computable invariants for Langlands-related lifting problems.

Abstract

We study deformation theory of mod $p$ Galois representations of $p$-adic fields with values in generalised tori, such as $L$-groups of (possibly non-split) tori. We show that the corresponding deformation rings are formally smooth over a group algebra of a finite abelian $p$-group. We compute their dimension and the set of irreducible components.

Theorems & Definitions (134)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Definition 2.1