Submission date: 6 December 2006

Abstract:

This thesis explores an approach to Hilbert's twelfth problem for real quadratic number fields, concerning the determination of an explicit class field theory for such fields.  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 fields.  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 using techniques from logic and number theory.  Our aim throughout this thesis is to show that these non-Hausdorff spaces have an ``algebraic character'', which is unexpected through their definition, though entirely consistent with their envisioned role in Real Multiplication.

Mathematics Subject Classification: 11U10, 03C99, 11R42

Keywords and phrases: Real Multiplication, Line Bundles, Nonstandard Analysis.

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