{
  "id": "1008.0473",
  "version": "v2",
  "published": "2010-08-03T07:29:52.000Z",
  "updated": "2010-08-08T18:55:21.000Z",
  "title": "Algebraic integers as special values of modular units",
  "authors": [
    "Ja Kyung Koo",
    "Dong Hwa Shin",
    "Dong Sung Yoon"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\varphi(\\tau)=\\eta((\\tau+1)/2)^2/\\sqrt{2\\pi}e^\\frac{\\pi i}{4}\\eta(\\tau+1)$ where $\\eta(\\tau)$ is the Dedekind eta-function. We show that if $\\tau_0$ is an imaginary quadratic number with $\\mathrm{Im}(\\tau_0)>0$ and $m$ is an odd integer, then $\\sqrt{m}\\varphi(m\\tau_0)/\\varphi(\\tau_0)$ is an algebraic integer dividing $\\sqrt{m}$. This is a generalization of Theorem 4.4 given in [B. C. Berndt, H. H. Chan and L. C. Zhang, Ramanujan's remarkable product of theta-functions, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 3, 583-612]. On the other hand, let $K$ be an imaginary quadratic field and $\\theta_K$ be an element of $K$ with $\\mathrm{Im}(\\theta_K)>0$ which generators the ring of integers of $K$ over $\\mathbb{Z}$. We develop a sufficient condition of $m$ for $\\sqrt{m}\\varphi(m\\theta_K)/\\varphi(\\theta_K)$ to become a unit.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-08-08T18:55:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F03",
      "11F20"
    ],
    "keywords": [
      "algebraic integer",
      "special values",
      "modular units",
      "imaginary quadratic number",
      "imaginary quadratic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.0473K"
    }
  }
}