Three examples of noncommutative boundaries of Shimura varieties

Abstract

We study the noncommutative modular curve (which was already studied by Connes, Manin and Marcolli), and the space of geodesics on the usual modular curve, from the viewpoint of algebraic groups, linear algebra and class field theory. This allows us, first, to understand differently some aspects of Manin's real multiplication program, and secondly, to study higher dimensional analogs of the noncommutative modular curve, called irrational or noncommutative boundaries of Shimura varieties. These are double cosets spaces $L\backslash G(\mathbb{R})/P(K)$ where $G$ is a connected reductive algebraic group over $\mathbb{Q}$, $L$ an arithmetic subgroup of $G(\mathbb{Q})$ and $P(K)$ a real parabolic subgroup in $G(\R)$. We study three examples of these general moduli spaces, and construct analogs of universal families for them. These moduli spaces describe degenerations of complex structures on tori in multifoliations, and contain important arithmetic information. Along the way, we use noncommutative geometry ``à la Connes'' for some moduli interpretations.