{
  "annotation_id": "b364bbfa-ee85-40a7-b83f-a490b21ed1f0",
  "date_created": "2026-02-17T05:53:12.434000Z",
  "date_modified": "2026-02-17T05:53:12.434000Z",
  "file_hash": "d28779f3e579eb9ed707dfd3000119545ac046ea3803e8c732f187b8dac972c5",
  "private": false,
  "record": {
    "abstract": "Let $D$ be a totally definite quaternion algebra over a totally real number field $F$, and $\\mathcal{O}$ be an $O_F$-order (of full rank) in $D$. The type number $t(\\mathcal{O})$ is an important arithmetic invariant of $\\mathcal{O}$ that counts the number of isomorphism classes of orders belonging to the same genus as $\\mathcal{O}$ (i.e. locally isomorphic to $\\mathcal{O}$ at every finite place $\\mathfrak{p}$ of $F$). The type number formula has been studied by Eichler, Peters, Pizer, Vigneras, K\\\"orner and many others. As the genus of $\\mathcal{O}$ further divides into spinor genera, one naturally seeks a finer type number formula for the number of isomorphism classes of orders belonging to the same spinor genus of $\\mathcal{O}$. The main goal of this paper is to provide such a refinement for a large class of quaternion $O_F$-orders $\\mathcal{O}$ that includes all Eichler orders. This enables us to prove that $t(\\mathcal{O})$ is divisible by the order of a quotient group $\\mathrm{WSG}(\\mathcal{O})$ of the Gauss genus group $\\mathrm{Cl}^+(O_F)/\\mathrm{Cl}^+(O_F)^2$ naturally attached to $\\mathcal{O}$. Similarly, we show that the trace of the $\\mathfrak{n}$-Brandt matrix $\\mathfrak{B}(\\mathcal{O}, \\mathfrak{n})$ is divisible by the class number $h(F)$ for any nonzero integral $O_F$-ideal $\\mathfrak{n}$. In particular, the class number $h(\\mathcal{O})=\\mathrm{Tr}(\\mathfrak{B}(\\mathcal{O}, O_F))$ is always divisible by $h(F)$ for such quaternion orders. This generalizes the divisibility result of $h(\\mathcal{O})$ proved in a different way by Chia-Fu Yu and the second named author [Indiana Univ. Math. J., Vol. 70, No. 2 (2021)] in the case when $\\mathcal{O}$ is a maximal $O_F$-order in a totally definite quaternion algebra unramified at all the finite places.",
    "arxiv_id": "2601.07171",
    "authors": [
      "Yucui Lin",
      "Jiangwei Xue"
    ],
    "categories": [
      "math.NT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "The spinor type number formula for totally definite quaternion orders",
    "url": "https://arxiv.org/abs/2601.07171",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "98fb867f-417b-4369-b200-43e7406501fb",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}