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Non-generic components of the Emerton-Gee stack for $\mathrm{GL}_2$

Kalyani Kansal, Ben Savoie

TL;DR

This work analyzes the reduced Emerton--Gee stack ${\mathcal X}_{2,\mathrm{red}}$ for ${\mathrm{GL}}_2(K)$ with $K$ unramified over ${\mathbf Q}_p$ and $p>3$, giving a complete geometric classification of non-generic Serre-weight components in terms of smoothness, normality, and Gorenstein properties. It constructs smooth--local charts via loop groups and Breuil--Kisin theory, decomposes local pieces into manageable factors with resolution--rational normalizations, and computes dualizing cohomology to identify singular loci and their codimensions. A detailed combinatorial link between Serre weights, tame types, and shapes yields precise smoothness and normality criteria, including the Steinberg case. These results refine the conjectural categorical $p$-adic Langlands picture by clarifying how components behave under normalization and how line bundles on normalized pieces push forward to the Emerton--Gee stack.

Abstract

Let $K$ be a finite unramified extension of $\mathbb{Q}_p$ with $p > 3$. We study the extremely non--generic irreducible components in the reduced part of the Emerton--Gee stack for $\mathrm{GL}_2$. We show precisely which irreducible components are smooth, which are normal, and which have Gorenstein normalizations. We show that the normalizations of the irreducible components admit smooth--local covers by resolution--rational schemes. We also determine the singular loci on the components, and use our results to update expectations about the conjectural categorical $p$--adic Langlands correspondence.

Non-generic components of the Emerton-Gee stack for $\mathrm{GL}_2$

TL;DR

This work analyzes the reduced Emerton--Gee stack for with unramified over and , giving a complete geometric classification of non-generic Serre-weight components in terms of smoothness, normality, and Gorenstein properties. It constructs smooth--local charts via loop groups and Breuil--Kisin theory, decomposes local pieces into manageable factors with resolution--rational normalizations, and computes dualizing cohomology to identify singular loci and their codimensions. A detailed combinatorial link between Serre weights, tame types, and shapes yields precise smoothness and normality criteria, including the Steinberg case. These results refine the conjectural categorical -adic Langlands picture by clarifying how components behave under normalization and how line bundles on normalized pieces push forward to the Emerton--Gee stack.

Abstract

Let be a finite unramified extension of with . We study the extremely non--generic irreducible components in the reduced part of the Emerton--Gee stack for . We show precisely which irreducible components are smooth, which are normal, and which have Gorenstein normalizations. We show that the normalizations of the irreducible components admit smooth--local covers by resolution--rational schemes. We also determine the singular loci on the components, and use our results to update expectations about the conjectural categorical --adic Langlands correspondence.
Paper Structure (16 sections, 46 theorems, 187 equations, 2 tables)

This paper contains 16 sections, 46 theorems, 187 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Categorical p--adic Langlands
  3. Strategy and outline
  4. Acknowledgements
  5. Notation and background
  6. Tame inertial types
  7. Breuil--Kisin modules
  8. Étale phi-modules and Galois representations
  9. Serre weights and the Emerton--Gee stack
  10. smooth--local charts
  11. Loop groups and torsors over Y-eta-tau and Z-tau-1
  12. Auxiliary schemes
  13. Local geometry
  14. Classification of Baj
  15. A different decomposition of Ba
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $p > 3$, $K$ unramified over ${\mathbf Q}_p$ of degree $f$, and $\sigma_{\mathbf{m}, \mathbf{n}}$ a Serre weight. Then the following are true:

Theorems & Definitions (111)

  • Theorem 1.1: Theorems \ref{['thm:serre-weight-sing']}, \ref{['thm:steinberg']}
  • Theorem 1.2: Theorem \ref{['thm:non--normal-rep']}
  • Theorem 1.3: Theorem \ref{['thm:non-CM-rep']}
  • Theorem 1.4: Corollary \ref{['cor:line-bundle-pushfwd']}
  • Remark 1.5
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • ...and 101 more