{
  "id": "1601.01433",
  "version": "v1",
  "published": "2016-01-07T08:02:44.000Z",
  "updated": "2016-01-07T08:02:44.000Z",
  "title": "A generalization of Watson transformation and representations of ternary quadratic forms",
  "authors": [
    "Jangwon Ju",
    "Inhwan Lee",
    "Byeong-Kweon Oh"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $L$ be a positive definite (non-classic) ternary $\\z$-lattice and let $p$ be a prime such that a $\\frac 12\\z_p$-modular component of $L_p$ is nonzero isotropic and $4\\cdot dL$ is not divisible by $p$. For a nonnegative integer $m$, let $\\mathcal G_{L,p}(m)$ be the genus with discriminant $p^m\\cdot dL$ on the quadratic space $L^{p^m}\\otimes \\q$ such that for each lattice $T \\in \\mathcal G_{L,p}(m)$, a $\\frac 12\\z_p$-modular component of $T_p$ is nonzero isotropic, and $T_q$ is isometric to $(L^{p^m})_q$ for any prime $q$ different from $p$. Let $r(n,M)$ be the number of representations of an integer $n$ by a $\\z$-lattice $M$. In this article, we show that if $m \\le 2$ and $n$ is divisible by $p$ only when $m=2$, then for any $T \\in \\mathcal G_{L,p}(m)$, $r(n,T)$ can be written as a linear summation of $r(pn,S_i)$ and $r(p^3n,S_i)$ for $S_i \\in \\mathcal G_{L,p}(m+1)$ with an extra term in some special case. We provide a simple criterion on when the extra term is necessary, and we compute the extra term explicitly. We also give a recursive relation to compute $r(n,T)$, for any $T \\in \\mathcal G_{L,p}(m)$, by using the number of representations of some integers by lattices in $\\mathcal G_{L,p}(m+1)$ for an arbitrary integer $m$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-07T08:02:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E12",
      "11E20"
    ],
    "keywords": [
      "ternary quadratic forms",
      "watson transformation",
      "representations",
      "extra term",
      "generalization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160101433J"
    }
  }
}