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

TL;DR

This paper classifies certain equivariant line bundles with connections on the p-adic upper half-plane, linking them to characters of quaternion algebra units, advancing understanding of p-adic geometry and representation theory.

Contribution

It constructs and classifies torsion G^0-equivariant line bundles with integrable connections on the p-adic upper half-plane using characters of quaternion algebra units.

Findings

Classification of torsion G^0-equivariant line bundles with connection

Connection between line bundles and smooth characters of quaternion units

Local construction method on the p-adic upper half-plane

Abstract

Let $F$ be a finite extension of $\mathbb{Q}_p$, let $\Omega_F$ 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 $\Omega_F$, we construct and classify the torsion $G^0$-equivariant line bundles with integrable connection on $\Omega$ in terms of the smooth linear characters of the units of the maximal order of the quaternion algebra over $F$.