{
  "id": "math/0301186",
  "version": "v1",
  "published": "2003-01-17T17:41:08.000Z",
  "updated": "2003-01-17T17:41:08.000Z",
  "title": "On the Zeta Function of Forms of Fermat Equations",
  "authors": [
    "Lars Bruenjes"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We study ``forms of the Fermat equation'' over an arbitrary field $k$, i.e. homogenous equations of degree $m$ in $n$ unknowns that can be transformed into the Fermat equation $X_1^m+...+X_n^m$ by a suitable linear change of variables over an algebraic closure of $k$. Using the method of Galois descent, we classify all such forms. In the case that $k$ is a finite field of characteristic greater than $m$ that contains the $m$-th roots of unity, we compute the Galois representation on $l$-adic cohomology (and so in particular the zeta function) of the hypersurface associated to an arbitrary form of the Fermat equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-01-17T17:41:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11E76",
      "19F27"
    ],
    "keywords": [
      "fermat equation",
      "zeta function",
      "arbitrary field",
      "suitable linear change",
      "algebraic closure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......1186B"
    }
  }
}