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Noncommutative tori, real multiplication and line bundles
, 2006
"... 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 the ..."
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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. Our aim throughout this thesis is to show that these nonHausdorff spaces have an algebraic character, which is unexpected through their definition, 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 defining continuous maps between Quantum Tori using ideas from