Real Multiplication and noncommutative geometry
摘要
Classical theory of Complex Multiplication (CM) shows that all abelian extensions of a complex quadratic field are generated by the values of appropriate modular functions at the points of finite order of elliptic curves whose endomorphism rings are orders in . 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.
引用
@article{arxiv.math/0202109,
title = {Real Multiplication and noncommutative geometry},
author = {Yuri I. Manin},
journal= {arXiv preprint arXiv:math/0202109},
year = {2007}
}
备注
46 pp., amstex file, no figures