{
  "id": "2110.10789",
  "version": "v2",
  "published": "2021-10-20T21:33:46.000Z",
  "updated": "2022-06-14T12:39:14.000Z",
  "title": "The Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces",
  "authors": [
    "Frauke M. Bleher",
    "Adam Wood"
  ],
  "comment": "30 pages. The second version has been substantially shortened, highlighting the differences from arXiv:1707.07133",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Suppose $X$ is a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Let $G$ be a finite group acting faithfully on $X$ over $k$ such that $G$ has non-trivial, cyclic Sylow $p$-subgroups. In this paper we show that for $m > 1$, the decomposition of $\\mathrm{H}^0(X,\\Omega_X^{\\otimes m})$ into a direct sum of indecomposable $kG$-modules is uniquely determined by the divisor class of a canonical divisor of $X/G$ together with the lower ramification groups and the fundamental characters of the closed points of $X$ that are ramified in the cover $X\\to X/G$. This extends to arbitrary $m > 1$ the $m = 1$ case treated by the first author with T. Chinburg and A. Kontogeorgis. We discuss some applications to congruences between modular forms in characteristic $0$, to the tangent space of the global deformation functor associated to $(X,G)$, and to the $kG$-module structure of Riemann-Roch spaces associated to divisors on $X$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-06-14T12:39:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "14H05",
      "14G17",
      "20C20"
    ],
    "keywords": [
      "galois module structure",
      "riemann-roch spaces",
      "holomorphic poly-differentials",
      "lower ramification groups",
      "global deformation functor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}