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On local Galois deformation rings: generalised reductive groups

Vytautas Paškūnas, Julian Quast

TL;DR

This work extends deformation theory of mod $p$ Galois representations to targets in generalised reductive group schemes, establishing that framed deformation rings $R^{\square}_{\bar\rho}$ are local complete intersections of the expected dimension $\dim G_k\,[F:\mathbb Q_p]+1$, and that their special fibres are normal and of the same codimension. The authors develop a robust geometric framework via the space of generic matrices $X^{\mathrm{gen}}_{G,\bar\rho^{\mathrm{ss}}}$ and its GIT quotient, bound dimensions with Levi-subgroup stratifications, and control obstructions through cohomological methods and Tate duality. They extend Lafforgue's $G$-pseudocharacters, prove finite/universal-homeomorphism relations with pinned determinant spaces $X^{\mathrm{ps}}_G$, and establish irreducible-component labelings in favorable cases (e.g. étale $\pi_1(G')$). The results feed into a refined understanding of the Emerton--Gee stack for generalised groups, with implications for Langlands program constructions involving $L$-groups and $C$-groups, and provide a comprehensive, inductive dimension-control strategy for normality and complete-intersection properties. Overall, the paper offers a broad deformation-theoretic toolkit for non-classical coefficients in the p-adic Langlands landscape, connecting cohomological obstructions, geometric invariant theory, and pseudocharacter frameworks.

Abstract

We study deformation theory of mod $p$ Galois representations of $p$-adic fields with values in generalised reductive group schemes, such as $L$-groups and $C$-groups. We show that the corresponding deformation rings are complete intersections of expected dimension. We determine their irreducible components in many cases and show that they and their special fibres are normal and complete intersection.

On local Galois deformation rings: generalised reductive groups

TL;DR

This work extends deformation theory of mod Galois representations to targets in generalised reductive group schemes, establishing that framed deformation rings are local complete intersections of the expected dimension , and that their special fibres are normal and of the same codimension. The authors develop a robust geometric framework via the space of generic matrices and its GIT quotient, bound dimensions with Levi-subgroup stratifications, and control obstructions through cohomological methods and Tate duality. They extend Lafforgue's -pseudocharacters, prove finite/universal-homeomorphism relations with pinned determinant spaces , and establish irreducible-component labelings in favorable cases (e.g. étale ). The results feed into a refined understanding of the Emerton--Gee stack for generalised groups, with implications for Langlands program constructions involving -groups and -groups, and provide a comprehensive, inductive dimension-control strategy for normality and complete-intersection properties. Overall, the paper offers a broad deformation-theoretic toolkit for non-classical coefficients in the p-adic Langlands landscape, connecting cohomological obstructions, geometric invariant theory, and pseudocharacter frameworks.

Abstract

We study deformation theory of mod Galois representations of -adic fields with values in generalised reductive group schemes, such as -groups and -groups. We show that the corresponding deformation rings are complete intersections of expected dimension. We determine their irreducible components in many cases and show that they and their special fibres are normal and complete intersection.
Paper Structure (58 sections, 209 theorems, 213 equations)

This paper contains 58 sections, 209 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. The conjecture of Böckle--Juschka
  3. Deformation problems with fixed partial determinant
  4. Deformation space of Lafforgue's pseudocharacters
  5. Complete intersection
  6. Definition
  7. Bounding the dimension
  8. Irreducible components
  9. Overview by section
  10. Notation
  11. Acknowledgements
  12. O-group schemes
  13. R-parabolics
  14. G-semisimplification
  15. Extensions
  16. ...and 43 more sections

Key Result

Theorem 1.1

The ring $R^{\square}_{\overline{\rho}}$ is a local complete intersection, flat over $\mathcal{O}$ and of relative dimension $\dim G_k ([F:\mathbb {Q}_p]+1)$. In particular, every continuous representation $\overline{\rho}: \Gamma_F\rightarrow G(k)$ has a lift to characteristic zero. Moreover, $R^{\

Theorems & Definitions (453)

  • Theorem 1.1: Corollaries \ref{['complete_intersection']}, \ref{['R_norm_red']}
  • Theorem 1.2: Corollaries \ref{['flat_det']}, \ref{['conj_BJ']}
  • Theorem 1.3: Corollary \ref{['reg_codim_x']}
  • Corollary 1.4: Corollary \ref{['parafactorial']}
  • Theorem 1.5: \ref{['BJ1']}
  • Theorem 1.6: \ref{['BJ2']}
  • Theorem 1.7: Cotner, cotner
  • Proposition 1.8: \ref{['nu_fin_u']}
  • Proposition 1.9: \ref{['cor_one']}
  • Lemma 1.10: \ref{['sic']}
  • ...and 443 more