Equivariant line bundles with connection 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 $F$.