{
  "id": "2410.10882",
  "version": "v1",
  "published": "2024-10-10T06:05:43.000Z",
  "updated": "2024-10-10T06:05:43.000Z",
  "title": "Type number for orders of level (N_1,N_2)",
  "authors": [
    "Yifan Luo",
    "Haigang Zhou"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2402.17443",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $N_1=p_1^{2u_1+1}...p_w^{2u_w+1}$, where the $p_i$ are distinct primes, $u_1,...,u_w$ are nonnegative integers and $w$ is an odd integer, and $N_2$ be a positive integer such that $\\gcd(N_1,N_2)=1$. In this paper, we give an explicit formula for the type number, i.e. the number of isomorphism classes, of orders of level $(N_1, N_2)$. The method of proof involves the Siegel-Weil formula for ternary quadratic forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-10T06:05:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E20",
      "11R52",
      "11F37",
      "11E41"
    ],
    "keywords": [
      "type number",
      "ternary quadratic forms",
      "explicit formula",
      "siegel-weil formula",
      "odd integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}