{
  "id": "1605.04371",
  "version": "v1",
  "published": "2016-05-14T03:31:23.000Z",
  "updated": "2016-05-14T03:31:23.000Z",
  "title": "Reflection principles for class groups",
  "authors": [
    "Jack Klys"
  ],
  "comment": "22 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We present several new examples of reflection principles which apply to both class groups of number fields and picard groups of of curves over $\\mathbb{P}^{1}/\\mathbb{F}_{p}$. This proves a conjecture of Lemmermeyer about equality of 2-rank in subfields of $A_{4}$, up to a constant not depending on the discriminant in the number field case, and exactly in the function field case. More generally we prove similar relations for subfields of a Galois extension with group $G$ for the cases when $G$ is $S_{3}$, $S_{4}$, $A_{4}$, $D_{2l}$ and $\\mathbb{Z}/l\\mathbb{Z}\\rtimes\\mathbb{Z}/r\\mathbb{Z}$. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-14T03:31:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "class groups",
      "reflection principles",
      "number field case",
      "function field case",
      "galois module computations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}