Arithmetic Structures on Noncommutative Tori with Real Multiplication

We study the homogeneous coordinate rings of real multiplication noncommutative tori as defined in [24]. An explicit presentation of these rings in terms of their natural generators is given. This presentation is used to study rationality properties of these rings and is the starting point for the analysis of the corresponding geometric data. The modularity of this family of rings is proved and used to define algebras which do not depend on the choice of a complex structure on the noncommutative torus.