{
  "id": "math/0008134",
  "version": "v3",
  "published": "2000-08-16T20:12:11.000Z",
  "updated": "2001-03-13T15:53:20.000Z",
  "title": "Cyclic covers of the projective line, their jacobians and endomorphisms",
  "authors": [
    "Yuri G. Zarhin"
  ],
  "comment": "LaTeX2e, 17 pages Some typos were corrected",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "We study the endomorphism ring $End(J(C))$ of the complex jacobian $J(C)$ of a curve $y^p=f(x)$ where $p$ is an odd prime and $f(x)$ is a polynomial with complex coefficiens of degree $n>4$ and without multiple roots. Assume that all the coefficients of $f$ lie in a (sub)field $K$ and the Galois group of $f$ over $K$ is either the full symmetric group $S_n$ or the alternating group $A_n$. Then we prove that $End(J(C))$ is the ring of integers in the in the $p$th cyclotomic field, if $p$ is a Fermat prime (e.g., $p=3,5,17,257$). Similar results for $p=2$ (the case of hyperelliptic curves) were obtained by the author in Math. Res. Lett. 7(2000), 123--132.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2001-03-13T15:53:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H40",
      "14K15",
      "14C25"
    ],
    "keywords": [
      "cyclic covers",
      "projective line",
      "endomorphism",
      "full symmetric group",
      "th cyclotomic field"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......8134Z"
    }
  }
}