### Abstract

This is joint work, in progress, with A. Connes on the complex geometry of the quotient space of rank 2 Q-lattices modulo commensurability. It builds on our prior work on modular Hecke algebras and their Hopf symmetry, and on the Connes-Marcolli C*-algebraic framework for Q-lattice spaces.

The emerging spectral-geometric picture, modeled on the transverse geometry of a generic codimension 1 foliation, has notable arithmetic overtones.