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Local model theory for non-generic tame potentially Barsotti--Tate deformation rings

Bao Viet Le Hung, Ariane Mézard, Stefano Morra

Abstract

We develop a local model theory for moduli stacks of $2$-dimensional non-scalar tame potentially Barsotti--Tate Galois representations of the Galois group of an unramified extension of $\mathbb{Q}_p$. We derive from this explicit presentations of potentially Barsotti--Tate deformation rings, allowing us to prove structural results about them, and prove various conjectures formulated by Caruso--David--Mézard.

Local model theory for non-generic tame potentially Barsotti--Tate deformation rings

Abstract

We develop a local model theory for moduli stacks of -dimensional non-scalar tame potentially Barsotti--Tate Galois representations of the Galois group of an unramified extension of . We derive from this explicit presentations of potentially Barsotti--Tate deformation rings, allowing us to prove structural results about them, and prove various conjectures formulated by Caruso--David--Mézard.
Paper Structure (37 sections, 50 theorems, 140 equations, 10 tables)

This paper contains 37 sections, 50 theorems, 140 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Methods
  4. Acknowledgements
  5. Notation
  6. Tame inertial types and Breuil--Kisin modules
  7. Tame inertial types and Galois representations
  8. Tame inertial types
  9. Galois deformation rings
  10. Breuil--Kisin modules and Emerton--Gee stack
  11. Étale $\Phi$-modules.
  12. Models
  13. Loop groups and open charts
  14. Loop groups and moduli of Breuil--Kisin modules
  15. Models for moduli of Breuil--Kisin modules
  16. ...and 22 more sections

Key Result

Theorem 1.1.1

Assume either $p\geq 7$ or $K=\mathbb{Q}_5$. Then the normalization of $\mathcal{Z}^{\tau}$ has rational singularities and is Gorenstein.

Theorems & Definitions (108)

  • Theorem 1.1.1: Theorem \ref{['thm:rational smoothness']}
  • Theorem 1.1.2: Theorem \ref{['thm:non-normal locus']}
  • Remark 1.1.3
  • Theorem 1.1.4: Proposition \ref{['prop:innomable']}, Theorem \ref{['thm:main:model']}
  • Remark 1.1.5
  • Remark 1.1.6
  • Theorem 1.1.7: Theorem \ref{['thm:indepen']}
  • Remark 2.1.4
  • Lemma 2.1.6
  • proof
  • ...and 98 more