{
  "id": "1904.06055",
  "version": "v1",
  "published": "2019-04-12T06:25:35.000Z",
  "updated": "2019-04-12T06:25:35.000Z",
  "title": "On some determinants involving cyclotomic units",
  "authors": [
    "Hai-Liang Wu"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For each odd prime $p$, let $\\zeta_p$ denote a primitive $p$-th root of unity. In this paper, we study the determinants of some matrices with cyclotomic unit entries. For instance, we show that when $p\\equiv 3\\pmod4$ and $p>3$ the determinant of the matrix $\\(\\frac{1-\\zeta_p^{j^2k^2}}{1-\\zeta_p^{j^2}}\\)_{1\\le j,k\\le (p-1)/2}$ can be written as $(-1)^{\\frac{h(-p)+1}{2}}(a_p+b_pi\\sqrt{p})$ with $a_p,b_p\\in\\frac12\\Z$ and $$\\begin{cases}\\nu_p(a_p)=\\nu_p(b_p)=\\frac{p-3}{8}&\\mbox{if}\\ p\\equiv 3\\pmod8, \\\\\\nu_p(a_p)=\\nu_p(b_p)+1=\\frac{p+1}{8}&\\mbox{if}\\ p\\equiv 7\\pmod8,\\end{cases}$$ where $\\nu_p(x)$ denotes the $p$-adic order of a $p$-adic integer $x$, and $h(-p)$ denotes the class number of the field $\\Q(\\sqrt{-p})$. Meanwhile, let $(\\frac{\\cdot}{p})$ denote the Legendre symbol. We have $$2^{\\frac{p+1}{2}}a_pb_p=(-1){^\\frac{p+1}{4}}p^{\\frac{p-3}{4}}\\det [S(p)],$$ and $$2^{\\frac{p-1}{2}}(a_p^2-pb_p^2)=\\frac{p-1}{2}(-p)^{\\frac{p-3}{4}}\\det [S(p)],$$ where $\\det [S(p)]$ is the determinant of the $\\frac{p-1}{2}$ by $\\frac{p-1}{2}$ matrix $S(p)$ with entries $S(p)_{j,k}=(\\frac{j^2+k^2}{p})$ for any $1\\le j,k\\le (p-1)/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-12T06:25:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "determinant",
      "cyclotomic unit entries",
      "odd prime",
      "th root",
      "legendre symbol"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}