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Classification of Equivariant Line Bundles on the Drinfeld Upper Half Plane

Georg Linden

TL;DR

The paper classifies G-, G0-, and G^0-equivariant line bundles on the Drinfeld upper half plane Omega by identifying Pic^G(Omega), Pic^{G^0}(Omega), and Pic^{G_0}(Omega) with explicit condensed group cohomology data. It combines Van der Put's transform, which relates invertible analytic functions to currents on the Bruhat–Tits tree, with condensed cohomology and Mayer–Vietoris arguments to produce concrete decompositions: Pic^G(Omega) ≅ $Z \,⊕\, ext{Hom}(F^×, K^×)$, Pic^{G^0}(Omega) ≅ $Z_p \,⊕\, Z/(q^2-1) \,⊕\, ext{Hom}(O_F^{××}, O_K^{××})$, and Pic^{G_0}(Omega) ≅ $Z_p \,⊕\, Z/(q^2-1) \,⊕\, ext{Hom}(G_0, O_K^{××})$. The approach yields an explicit description of both torsion and non-torsion classes and provides an alternative proof path to Ardakov–Wadsley’s torsion results via cohomological and geometric constructions. The results illuminate how current data on the Bruhat–Tits tree control equivariant line bundles on Drinfeld spaces, with potential generalizations to higher-dimensional Drinfeld symmetric spaces. The culmination shows that all equivariant line bundles arise from a combination of twisting by powers of O(1), determinant-characters, and principal-unit characters, organized in a precise condensed-cohomology framework.

Abstract

We explicitly determine the group of isomorphism classes of equivariant line bundles on the non-archimedean Drinfeld upper half plane for $\mathrm{GL}_2(F)$, for its subgroup of matrices whose determinant has trivial valuation, and for $\mathrm{GL}_2(\mathcal{O}_F)$. Our results extend a recent classification of torsion equivariant line bundles with connection due to Ardakov and Wadsley, but we use a different approach. A crucial ingredient is a construction due to Van der Put which relates invertible analytic functions on the Drinfeld upper half plane to currents on the Bruhat-Tits tree. Another tool we use is condensed group cohomology.

Classification of Equivariant Line Bundles on the Drinfeld Upper Half Plane

TL;DR

The paper classifies G-, G0-, and G^0-equivariant line bundles on the Drinfeld upper half plane Omega by identifying Pic^G(Omega), Pic^{G^0}(Omega), and Pic^{G_0}(Omega) with explicit condensed group cohomology data. It combines Van der Put's transform, which relates invertible analytic functions to currents on the Bruhat–Tits tree, with condensed cohomology and Mayer–Vietoris arguments to produce concrete decompositions: Pic^G(Omega) ≅ , Pic^{G^0}(Omega) ≅ , and Pic^{G_0}(Omega) ≅ . The approach yields an explicit description of both torsion and non-torsion classes and provides an alternative proof path to Ardakov–Wadsley’s torsion results via cohomological and geometric constructions. The results illuminate how current data on the Bruhat–Tits tree control equivariant line bundles on Drinfeld spaces, with potential generalizations to higher-dimensional Drinfeld symmetric spaces. The culmination shows that all equivariant line bundles arise from a combination of twisting by powers of O(1), determinant-characters, and principal-unit characters, organized in a precise condensed-cohomology framework.

Abstract

We explicitly determine the group of isomorphism classes of equivariant line bundles on the non-archimedean Drinfeld upper half plane for , for its subgroup of matrices whose determinant has trivial valuation, and for . Our results extend a recent classification of torsion equivariant line bundles with connection due to Ardakov and Wadsley, but we use a different approach. A crucial ingredient is a construction due to Van der Put which relates invertible analytic functions on the Drinfeld upper half plane to currents on the Bruhat-Tits tree. Another tool we use is condensed group cohomology.

Paper Structure

This paper contains 20 sections, 55 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Currents on the Bruhat--Tits Tree
  3. The Bruhat--Tits Tree
  4. Currents
  5. Group Cohomology of Currents
  6. The Van der Put Transform
  7. Equivariant Line Bundles
  8. The Drinfeld Upper Half Plane
  9. The Van der Put Transform
  10. First Consequences for Group Cohomology
  11. Group Cohomology for ${\mathrm{GL}}_2({\mathcal{O}}_F)$
  12. Principal Units and Characters
  13. Prime-to-$p$-Torsion Classes
  14. Non-Torsion Classes
  15. Lifting of Cohomology Classes
  16. ...and 5 more sections

Key Result

Theorem A

There are isomorphisms of abelian groups Here, ${\mathcal{L}}$ is a certain $G^0$-equivariant line bundle which satisfies $[{\mathcal{L}}]^{\otimes q+1} = [{\mathcal{O}}_{\widehat{{\mathrm{det}}} {}^{-1}}]$.

Theorems & Definitions (129)

  • Theorem A
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 119 more