{
  "id": "2502.08487",
  "version": "v1",
  "published": "2025-02-12T15:22:23.000Z",
  "updated": "2025-02-12T15:22:23.000Z",
  "title": "Invariants recovering the reduction type of a hyperelliptic curve",
  "authors": [
    "Lilybelle Cowland Kellock",
    "Elisa Lorenzo"
  ],
  "comment": "40 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Tate's algorithm tells us that for an elliptic curve $E$ over a local field $K$ of residue characteristic $\\geq 5$, $E/K$ has potentially good reduction if and only if $\\text{ord}(j_E)\\geq 0$. It also tells us that when $E/K$ is semistable the dual graph of the special fibre of the minimal regular model of $E/K^{\\text{unr}}$ can be recovered from $\\text{ord}(j_E)$. We generalise these results to hyperelliptic curves of genus $g\\geq 2$ over local fields of odd residue characteristic $K$ by defining a list of absolute invariants that determine the potential stable model of a genus $g$ hyperelliptic curve $C$. They also determine the dual graph of the special fibre of the minimal regular model of $C/K^{\\text{unr}}$ if $C/K$ is semistable. This list depends only on the genus of $C$, and the absolute invariants can be written in terms of the coefficients of a Weierstrass equation for $C$. We explicitly describe the method by which the valuations of the invariants recover the dual graphs. Additionally, we show by way of a counterexample that if $g \\geq 2$, there is no list of invariants whose valuations determine the dual graph of the special fibre of the minimal regular model of a genus $g$ hyperelliptic curve $C$ over a local field $K$ of odd residue characteristic when $C$ is not assumed to be semistable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-12T15:22:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "14D10",
      "14G20",
      "14H45",
      "14Q05"
    ],
    "keywords": [
      "hyperelliptic curve",
      "minimal regular model",
      "dual graph",
      "reduction type",
      "odd residue characteristic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}