{
  "id": "2211.04708",
  "version": "v1",
  "published": "2022-11-09T07:02:54.000Z",
  "updated": "2022-11-09T07:02:54.000Z",
  "title": "Computation of Hecke eigenvalues (mod $p$) via quaternions",
  "authors": [
    "Yiannis Fam"
  ],
  "comment": "25 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In a 1987 letter, Serre proves that the systems of Hecke eigenvalues arising from mod $p$ modular forms (of fixed level $\\Gamma(N)$ coprime to $p$, and any weight $k$) are the same as those arising from functions $\\Omega(N) \\to \\bar{\\mathbb F}_p$, where $\\Omega(N)$ is some double quotient of $D^\\times (\\mathbb A_f)$ and $D$ is the unique quaternion algebra over $\\mathbb Q$ ramified at $\\{p,\\infty\\}$. We present an algorithm which then computes these Hecke eigenvalues on the quaternion side in a combinatorial manner.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-09T07:02:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R52",
      "11F30"
    ],
    "keywords": [
      "computation",
      "unique quaternion algebra",
      "combinatorial manner",
      "modular forms",
      "quaternion side"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}