{
  "id": "1812.05993",
  "version": "v1",
  "published": "2018-12-14T15:54:06.000Z",
  "updated": "2018-12-14T15:54:06.000Z",
  "title": "Galois extensions and a Conjecture of Ogg",
  "authors": [
    "Krzysztof Klosin",
    "Mihran Papikian"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $N=pq$, where $p=2,3,5,7,13$ and $q\\neq p$ is another prime. We propose a general strategy for proving a conjecture of Ogg about isogenies from the new quotient of $J_0(N)$ to the Jacobian of the Shimura curve attached to the indefinite quaternion algebra over $\\mathbf{Q}$ of discriminant $N$. This strategy is based on the results of Helm and Ribet. Using this strategy, we prove a general conditional result toward Ogg's conjecture and discuss in detail the case when $N=65$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-14T15:54:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G18"
    ],
    "keywords": [
      "galois extensions",
      "indefinite quaternion algebra",
      "general conditional result",
      "general strategy",
      "shimura curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}