Real Multiplication and noncommutative geometry
Abstract
Classical theory of Complex Multiplication (CM) shows that all abelian extensions of a complex quadratic field $K$ are generated by the values of appropriate modular functions at the points of finite order of elliptic curves whose endomorphism rings are orders in $K$. For real quadratic fields, a similar description is not known. However, the relevant (still unproved) case of Stark conjectures ([St1]) strongly suggests that such a description must exist. In this paper we propose to use two--dimensional quantum tori corresponding to real quadratic irrationalities as a replacement of elliptic curves with complex multiplication. We discuss some basic constructions of the theory of quantum tori from the perspective of this Real Multiplication (RM) research project.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- February 2002
- DOI:
- 10.48550/arXiv.math/0202109
- arXiv:
- arXiv:math/0202109
- Bibcode:
- 2002math......2109M
- Keywords:
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- Algebraic Geometry
- E-Print:
- 46 pp., amstex file, no figures