{
  "id": "2408.10546",
  "version": "v1",
  "published": "2024-08-20T04:57:03.000Z",
  "updated": "2024-08-20T04:57:03.000Z",
  "title": "Coordinate Transformation in Faltings' Extension",
  "authors": [
    "Shanxiao Huang"
  ],
  "categories": [
    "math.RT",
    "math.AG",
    "math.NT"
  ],
  "abstract": "Analogue to Fontaine's computation for $\\Omega_{\\bar{\\mathbb{Z}}_p/\\mathbb{Z}_p}$, we compute the structure of $\\Omega_{\\mathcal{O}_{\\bar{K}_0}/\\mathcal{O}_{K_0}}$ (here $K_0$ is the completion of $\\mathbb{Q}_p(T)$ at place $p$) and prove that $p^{1-1/p^n}\\mathrm{d}p^{1/p^n}$, $T^{1-1/p^n}\\mathrm{d}T^{1/p^n}$ and $S^{1-1/p^n}\\mathrm{d}S^{1/p^n}$ are linearly dependent (Here $S := 1-T$). The main aim of this article is to find the linear equations for these three differential forms. Then we define a map which is called \"differential version\" of Fontaine's map to express the equations in a computable way. Finally, we prove that the coefficients in the equation can be expressed in some polynomial forms and compute some examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-20T04:57:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "coordinate transformation",
      "main aim",
      "polynomial forms",
      "differential forms",
      "differential version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}