{
  "id": "1108.3131",
  "version": "v1",
  "published": "2011-08-16T01:06:02.000Z",
  "updated": "2011-08-16T01:06:02.000Z",
  "title": "Real components of modular curves",
  "authors": [
    "Andrew Snowden"
  ],
  "comment": "33 pages, 4 tables",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We study the real components of modular curves. Our main result is an abstract group-theoretic description of the real components of a modular curve defined by a congruence subgroup of level N in terms of the corresponding subgroup of SL_2(Z/NZ). We apply this result to several families of modular curves (such as X_0(N), X_1(N), etc.) to obtain formulas for the number of real components. Somewhat surprisingly, the multiplicative order of 2 modulo N has a strong influence in many cases: for instance, if N is an odd prime then the real locus of X_1(N) is connected if and only if -1 and 2 generate (Z/NZ)^*.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-16T01:06:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modular curve",
      "real components",
      "abstract group-theoretic description",
      "congruence subgroup",
      "main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.3131S"
    }
  }
}