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Equivariant line bundles with connection on the p-adic upper half plane

Konstantin Ardakov, Simon J. Wadsley

Abstract

Let $F$ be a finite extension of $\mathbb{Q}_p$, let $Ω_F$ be Drinfeld's upper half-plane over $F$ and let $G^0$ the subgroup of $GL_2(F)$ consisting of elements whose determinant has norm $1$. By working locally on $Ω_F$, we construct and classify the torsion $G^0$-equivariant line bundles with integrable connection on $Ω$ in terms of the smooth linear characters of the units of the maximal order of the quaternion algebra over $F$.

Equivariant line bundles with connection on the p-adic upper half plane

Abstract

Let be a finite extension of , let be Drinfeld's upper half-plane over and let the subgroup of consisting of elements whose determinant has norm . By working locally on , we construct and classify the torsion -equivariant line bundles with integrable connection on in terms of the smooth linear characters of the units of the maximal order of the quaternion algebra over .
Paper Structure (21 sections, 86 theorems, 209 equations)

This paper contains 21 sections, 86 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The main result
  4. Motivation
  5. An overview of some related works
  6. Acknowledgements
  7. Conventions and Notation
  8. Background from algebra
  9. Measures on profinite sets
  10. Some stabilisers in and their linear characters
  11. Quaternions
  12. Equivariant sheaves and amalgamated products
  13. Topics in rigid analytic geometry
  14. Line bundles with flat connection on smooth rigid spaces
  15. Equivariant line bundles with flat connections
  16. ...and 6 more sections

Key Result

Theorem A

Suppose that $K$ contains the quadratic unramified extension $L$ of $F$. Then there is an isomorphism of abelian groups

Theorems & Definitions (207)

  • Theorem A
  • Definition 2.1.1
  • Proposition 2.1.3
  • proof
  • Definition 2.1.4
  • Remark 2.1.5
  • Lemma 2.1.6
  • proof
  • Lemma 2.1.7
  • proof
  • ...and 197 more