Modular Index Invariants of Mumford Curves

We continue an investigation initiated by Consani–Marcolli of the relation between the algebraic geometry of p-adic Mumford curves and the noncommutative geometry of graph C-algebras associated to the action of the uniformizing p-adic Schottky group on the Bruhat–Tits tree. We reconstruct invariants of Mumford curves related to valuations of generators of the associated Schottky group, by developing a graphical theory for KMS weights on the associated graph C-algebra, and using modular index theory for KMS weights. We give explicit examples of the construction of graph weights for low genus Mumford curves. We then show that the theta functions of Mumford curves, and the induced currents on the Bruhat–Tits tree, define functions that generalize the graph weights. We show that such inhomogeneous graph weights can be constructed from spectral flows, and that one can reconstruct theta functions from such graphical data.