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A categorical p-adic Langlands correspondence for GL_2(Q_p)

Andrea Dotto, Matthew Emerton, Toby Gee

Abstract

Let p at least 5 be prime. We construct a fully faithful functor from the derived category of all smooth p-adic representations of GL_2(Q_p) (with a fixed central character) to a derived category of Ind-coherent sheaves on a stack of (phi,Gamma)-modules.

A categorical p-adic Langlands correspondence for GL_2(Q_p)

Abstract

Let p at least 5 be prime. We construct a fully faithful functor from the derived category of all smooth p-adic representations of GL_2(Q_p) (with a fixed central character) to a derived category of Ind-coherent sheaves on a stack of (phi,Gamma)-modules.

Paper Structure

This paper contains 182 sections, 309 theorems, 943 equations.

Table of Contents

  1. Introduction
  2. Overview of our results
  3. Constructing the functor
  4. Constructing the kernel
  5. Localization
  6. Establishing properties of F
  7. Relationship with other works
  8. Sheaf- and category-theoretic techniques
  9. A guide to the paper
  10. A comparison with the proof outlined in the IHES notes.
  11. Acknowledgements
  12. Notation and conventions
  13. Background
  14. Galois representations
  15. Mod p Galois representations, Serre weights, and the weight part of Serre's conjecture
  16. ...and 167 more sections

Key Result

Theorem 1.1.1

There exists an ${\mathcal{O}}$-linear fully faithful exact functor $\mathrm{F}:\mathop{D}_{\mathrm{fp}}\nolimits^b({\mathcal{A}}) \hookrightarrow D_{\mathrm{coh}}^b({\mathcal{X}}).$

Theorems & Definitions (801)

  • Theorem 1.1.1: Definition \ref{['defn:definition of F']} and Theorem \ref{['thm:F-is-fully-faithful']}
  • Remark 1.1.2
  • Theorem 1.1.4
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.1.5
  • Remark 2.1.6
  • Definition 2.1.7
  • Definition 2.1.8
  • ...and 791 more