Noncommutative Geometry and Arithmetic

Details Noncommutative Geometry and Arithmetic Noncommutative Geometry and Arithmetic

2010-03-18

arxiv.org/abs/1003.3662v1

Physics

Mathematical Physics

16 pages, LaTeX

Scientific paper

This is an overview of recent results aimed at developing a geometry of noncommutative tori with real multiplication, with the purpose of providing a parallel, for real quadratic fields, of the classical theory of elliptic curves with complex multiplication for imaginary quadratic fields. This talk concentrates on two main aspects: the relation of Stark numbers to the geometry of noncommutative tori with real multiplication, and the shadows of modular forms on the noncommutative boundary of modular curves, that is, the moduli space of noncommutative tori. To appear in Proc. ICM 2010.

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