{
  "id": "1407.1294",
  "version": "v1",
  "published": "2014-07-04T18:50:49.000Z",
  "updated": "2014-07-04T18:50:49.000Z",
  "title": "Congruence properties of Borcherds product exponents",
  "authors": [
    "Keenan Monks",
    "Sarah Peluse",
    "Lynnelle Ye"
  ],
  "comment": "14 pages; preprint of article published in IJNT",
  "journal": "International Journal of Number Theory (2013), vol. 9 (6), pp. 1563-1578",
  "doi": "10.1142/S1793042113500437",
  "categories": [
    "math.NT"
  ],
  "abstract": "In his striking 1995 paper, Borcherds found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant $-d$ evaluated at the modular $j$-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, $A(n,d)$, are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of $A(n,d)$ modulo a prime $\\ell$ by relating it to a modular representation of the logarithmic derivative of the Hilbert class polynomial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-04T18:50:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F33"
    ],
    "keywords": [
      "borcherds product exponents",
      "congruence properties",
      "hilbert class polynomial",
      "special half-integral weight modular forms",
      "product expansion"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.1294M"
    }
  }
}