Abstract

I review some aspects of the theory of noncommutative two-tori with real multiplication, focusing on the role played by Heisenberg groups in the definition of algebraic structures associated to these noncommutative spaces.

Introduction

Noncommutative tori have played a central role in noncommutative geometry since the early stages of the theory. They arise naturally in various contexts and have provided a good testing ground for many of the techniques from which noncommutative geometry has developed [1,14]. Noncommutative tori are defined in terms of their algebras of functions. The study of projective modules over these algebras and the corresponding theory of Morita equivalences leads to the existence of a class of noncommutative tori related to real quadratic extensions of ℚ. These real multiplication noncommutative tori are conjectured to provide the correct geometric setting under which to attack the explicit class field theory problem for real quadratic fields [7]. The right understanding of the algebraic structures underlying these spaces is important for these applications.

The study of connections on vector bundles over noncommutative tori gives rise to a rich theory, which has been recast recently in the context of complex algebraic geometry [1, 3, 5, 11, 12, 16]. The study of categories of holomorphic bundles has thrown light on some algebraic structures related to real multiplication noncommutative tori [9, 10, 18]. Some of these results arise in a natural way from the interplay between Heisenberg groups and noncommutative tori.

Noncommutative tori and their morphisms

In many situations arising in various geometric settings it is possible to characterize spaces and some of their structural properties in terms of appropriate rings of functions.