{
  "id": "2408.01643",
  "version": "v1",
  "published": "2024-08-03T03:00:52.000Z",
  "updated": "2024-08-03T03:00:52.000Z",
  "title": "Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2)",
  "authors": [
    "Kin Ming Tsang"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider a variant of the strong multiplicity one theorem. Let $\\pi_{1}$ and $\\pi_{2}$ be two unitary cuspidal automorphic representations for $\\mathrm{GL(2)}$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which $\\left\\lvert a_{v}(\\pi_{1}) \\right\\rvert > \\left\\lvert a_{v}(\\pi_{2}) \\right\\rvert$, where $a_{v}(\\pi_{i})$ is the trace of Langlands conjugacy class of $\\pi_{i}$ at $v$. One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which $\\left\\lvert a_{v}(\\pi_{1})\\right\\rvert \\neq \\left\\lvert a_{v}(\\pi_{2}) \\right\\rvert$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-03T03:00:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11F41"
    ],
    "keywords": [
      "hecke eigenvalues",
      "lower dirichlet density",
      "unitary cuspidal automorphic representations",
      "langlands conjugacy class",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}