Algebraic Aspects of Noncommutative Tori : the Riemann-Hilbert Correspondence

In this thesis we have tried to figure out some algebraic aspects of noncommutative tori, aiming at generalizing them to arbitrary noncommutative spaces. In the second section all relevant definitions, some examples and motivations have been provided. In the third section we look at the example of noncommutative tori and see how they can be related to similar objects called noncommutative elliptic curves. We extract a suitably well-behaved subcategory of the category of holomorphic bundles over noncommutative tori. This category turns out to admit a Tannakian structure with Z+ΘZ as the fundamental group. The key to this construction is an equivariant version of the classical Riemann–Hilbert correspondence. The aim was to construct homotopy theoretic invariants of noncommutative tori, e.g., fundamental groups and we make a proposal to that end. The last two sections constitute an attempt to rewrite some parts of noncommutative algebraic geometry in the framework of DG categories. We provide a description of the category of noncommutative spaces and their associated noncommutative motives. We had some arithmetic applications in mind, namely, introducing and studying motivic zeta functions of noncommutative tori. We propose a universal motivic measure on the category of noncommutative spaces. In it lies a subcategory consisting of noncommutative Calabi–Yau spaces containing elliptic curves and noncommutative tori. In this setting we introduce a motivic zeta function of noncommutative tori; more generally that of noncommutative Calabi–Yau spaces. Our work should be put in perspective with the Real Multiplication programme of Manin