  We study the problem of defining line bundles over certain non-Hausdorff
spaces known as Quantum Tori, motivated by the proposed theory of Real
Multiplication for real quadratic fields. We draw analogies from the theory of
Line Bundles over Complex Tori to define a non-trivial cohomological notion of
Line Bundles over Quantum Tori. We prove a structure theorem for isomorphism
classes of such line bundles analogous to the Appel-Humbert Theorem for Complex
Tori. In the second half of this text we consider Lines Bundles over Quantum
Tori as topological spaces, and compare this notion with the cohomological
definition. We define the Chern class of a line bundle and link this to an
alternating pairing on a certain subgroup of Quantum Tori. Finally we
investigate how our results are related to the study of Quantum Tori by Zilber
using Model theoretic techniques.
