{
  "id": "2407.18258",
  "version": "v1",
  "published": "2024-07-11T15:38:21.000Z",
  "updated": "2024-07-11T15:38:21.000Z",
  "title": "On Galois covers of curves and arithmetic of Jacobians",
  "authors": [
    "Alexandros Konstantinou",
    "Adam Morgan"
  ],
  "comment": "A significant part of this paper has been split out of the original version of arXiv:2211.06357. 27 pages, comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the arithmetic of curves and Jacobians endowed with the action of a finite group $G$. This includes a study of the basic properties, as $G$-modules, of their $\\ell$-adic representations, Selmer groups, rational points and Shafarevich-Tate groups. In particular, we show that $p^\\infty$-Selmer groups are self-dual $G$-modules, and give various `$G$-descent' results for Selmer groups and rational points. Along the way we revisit, and slightly refine, a construction going back to Kani and Rosen for associating isogenies to homomorphisms between permutation representations. With a view to future applications, it is convenient to work throughout with curves that are not assumed to be geometrically connected (or even connected); such curves arise naturally when taking Galois closures of covers of curves. For lack of a suitable reference, we carefully detail how to deduce the relevant properties of such curves and their Jacobians from the more standard geometrically connected case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-11T15:38:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G30",
      "11G10",
      "11G20",
      "14H25",
      "14H30",
      "14H40",
      "14K02"
    ],
    "keywords": [
      "galois covers",
      "selmer groups",
      "arithmetic",
      "rational points",
      "shafarevich-tate groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}