Modular Index Invariants of Mumford Curves

Abstract

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.

arXiv comment40 pages, LaTeX, 8 eps figures

References in corpus (7)

• Type III and spectral triples
• Twisted cyclic theory, equivariant KK theory and KMS States
• Local index formula and twisted spectral triples
• Spectral flow invariants and twisted cyclic theory from the Haar state on SU_q(2)
• Zeta functions that hear the shape of a Riemann surface
• Lectures on Arithmetic Noncommutative Geometry
• On the K-theory of graph C*-algebras

Cited by in corpus (5)

• Dynamical Systems on Spectral Metric Spaces
• Partial actions and KMS states on relative graph $C^{\ast }$-algebras
• Twisted cyclic theory, equivariant KK theory and KMS States
• The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podles sphere
• Tate curves and UHF-algebras