{
  "id": "1806.08352",
  "version": "v1",
  "published": "2018-06-21T17:56:58.000Z",
  "updated": "2018-06-21T17:56:58.000Z",
  "title": "Cohomology groups of Fermat curves via ray class fields of cyclotomic fields",
  "authors": [
    "Rachel Davis",
    "Rachel Pries"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The absolute Galois group of the cyclotomic field $K={\\mathbb Q}(\\zeta_p)$ acts on the \\'etale homology of the Fermat curve $X$ of exponent $p$. We study a Galois cohomology group which is valuable for measuring an obstruction for $K$-rational points on $X$. The information needed for measuring this obstruction factors through a $2$-nilpotent extension of $K$. By studying Heisenberg extensions of $K$, we determine a large subquotient of this Galois cohomology group. For $p=3$, we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology which determines it completely.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-21T17:56:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11R34",
      "11R37",
      "11Y40",
      "20D15",
      "11R18",
      "13A50",
      "14F20",
      "20J06",
      "55S35"
    ],
    "keywords": [
      "ray class fields",
      "fermat curve",
      "cyclotomic field",
      "galois cohomology group",
      "absolute galois group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}