{
  "id": "1602.06996",
  "version": "v1",
  "published": "2016-02-22T23:51:28.000Z",
  "updated": "2016-02-22T23:51:28.000Z",
  "title": "Arithmetic of singular character varieties and their $E$-polynomials",
  "authors": [
    "David Baraglia",
    "Pedram Hekmati"
  ],
  "comment": "41 pages",
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "abstract": "We calculate the $E$-polynomials of the $SL_3(\\mathbb{C})$ and $GL_3(\\mathbb{C})$-character varieties of compact oriented surfaces of any genus and the $E$-polynomials of the $SL_2(\\mathbb{C})$ and $GL_2(\\mathbb{C})$-character varieties of compact non-orientable surfaces of any Euler characteristic. Our methods also give a new and significantly simpler computation of the $E$-polynomials of the $SL_2(\\mathbb{C})$-character varieties of compact orientable surfaces, which were computed by Logares, Mu\\~noz and Newstead for genus $g=1,2$ and by Martinez and Mu\\~noz for $g \\ge 3$. Our technique is based on the arithmetic of character varieties over finite fields. More specifically, we show how to extend the approach of Hausel and Rodriguez-Villegas used for non-singular (twisted) character varieties to the singular (untwisted) case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-22T23:51:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D20",
      "32S35",
      "20C15",
      "14L30"
    ],
    "keywords": [
      "singular character varieties",
      "polynomials",
      "arithmetic",
      "compact non-orientable surfaces",
      "euler characteristic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160206996B"
    }
  }
}