{
  "id": "2208.12247",
  "version": "v1",
  "published": "2022-08-25T17:47:57.000Z",
  "updated": "2022-08-25T17:47:57.000Z",
  "title": "Chabauty limits of groups of involutions in $SL(2,F)$ for local fields",
  "authors": [
    "Corina Ciobotaru",
    "Arielle Leitner"
  ],
  "comment": "2 figures, 23 pages",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "We classify Chabauty limits of groups fixed by various (abstract) involutions over $SL(2,F)$, where $F$ is a finite field-extension of $\\mathbb{Q}_p$, with $p\\neq 2$. To do so, we first classify abstract involutions over $SL(2,F)$ with $F$ a quadratic extension of $\\mathbb{Q}_p$, and prove $p$-adic polar decompositions with respect to various subgroups of $p$-adic $SL_2$. Then we classify Chabauty limits of: $SL(2, F) \\subset SL(2,E)$ where $E$ is a quadratic extension of $F$, of $SL(2, \\mathbb{R}) \\subset SL(2, \\mathbb{C})$, and of $H_\\theta \\subset SL(2,F)$, where $H_\\theta$ is the fixed point group of an $F$-involution $\\theta$ over $SL(2,F)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-25T17:47:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local fields",
      "classify chabauty limits",
      "quadratic extension",
      "first classify abstract involutions",
      "adic polar decompositions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}