Classification of Equivariant Line Bundles on the Drinfeld Upper Half Plane

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 subgroups of matrices whose determinant has even (respectively 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.

41 pages. Comments are welcome. v2: Added classification for subgroup of matrices with even-valued determinant, simplified proof of Mayer-Vietoris sequence for condensed group cohomology, minor changes