Noncommutative Tori, Real Multiplication and Line Bundles

This thesis explores an approach to Hilbert's twelfth problem for real quadratic number elds, concerning the determination of an explicit class eld theory for such elds. The basis for our approach is a paper by Manin proposing a theory of Real Multiplication realising such an explicit theory, analogous to the theory of Complex Multiplication associated to imaginary quadratic elds. Whereas elliptic curves play the leading role in the latter theory, objects known as Noncommutative Tori are the subject of Manin's dream. In this thesis we study a family of topological spaces known as Quantum Tori that arise naturally from Manin's approach. Our aim throughout this thesis is to show that these non-Hausdor spaces have an algebraic character , which is unexpected through their de nition, though entirely consistent with their envisioned role in Real Multiplication. Chapter 1 is a general introduction to the problem, providing a historical and technical background to the motivation behind this thesis. Chapter 2 deals with the problem of de ning continuous maps between Quantum Tori using ideas from Nonstandard Analysis, culminating in a description of the action of a Galois group on certain isomorphism classes of these spaces. Chapter 3 concerns the problem of de ning a nontrivial notion of line bundles over Quantum Tori, while Chapter 4 concerns the existence of sections of these line bundles. We show that such sections have applications to the null values of the derivatives of L-functions attached to real quadratic elds, which in the context of Stark's conjectures is seen to be relevant to Hilbert's problem.