# Modular Index Invariants of Mumford Curves - Mathematics > Quantum Algebra

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Abstract: We continue an investigation initiated by Consani-Marcolli of the relationbetween the algebraic geometry of p-adic Mumford curves and the noncommutativegeometry of graph C*-algebras associated to the action of the uniformizingp-adic Schottky group on the Bruhat-Tits tree. We reconstruct invariants ofMumford curves related to valuations of generators of the associated Schottkygroup, by developing a graphical theory for KMS weights on the associated graphC*-algebra, and using modular index theory for KMS weights. We give explicitexamples of the construction of graph weights for low genus Mumford curves. Wethen show that the theta functions of Mumford curves, and the induced currentson the Bruhat-Tits tree, define functions that generalize the graph weights. Weshow that such inhomogeneous graph weights can be constructed from spectralflows, and that one can reconstruct theta functions from such graphical data.

Author: ** Alan Carey ANU, Matilde Marcolli Caltech, Adam Rennie ANU**

Source: https://arxiv.org/