{
  "id": "math/0202109",
  "version": "v1",
  "published": "2002-02-12T14:46:52.000Z",
  "updated": "2002-02-12T14:46:52.000Z",
  "title": "Real Multiplication and noncommutative geometry",
  "authors": [
    "Yuri I. Manin"
  ],
  "comment": "46 pp., amstex file, no figures",
  "categories": [
    "math.AG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-02-12T14:46:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real multiplication",
      "noncommutative geometry",
      "complex multiplication",
      "elliptic curves",
      "real quadratic irrationalities"
    ],
    "note": {
      "typesetting": "AMS-TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......2109M"
    }
  }
}