{
  "id": "2501.19101",
  "version": "v1",
  "published": "2025-01-31T12:50:35.000Z",
  "updated": "2025-01-31T12:50:35.000Z",
  "title": "Exceptional theta correspondence $\\mathbf{F}_{4}\\times\\mathbf{PGL}_{2}$ for level one automorphic representations",
  "authors": [
    "Yi Shan"
  ],
  "comment": "40 pages, 1 table, in English",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $\\mathbf{F}_{4}$ be the unique (up to isomorphism) connected semisimple algebraic group over $\\mathbb{Q}$ of type $\\mathrm{F}_{4}$, with compact real points and split over $\\mathbb{Q}_{p}$ for all primes $p$. A conjectural computation by the author in arxiv:2407.05859 predicts the existence of a family of level one automorphic representations of $\\mathbf{F}_{4}$, which are expected to be functorial lifts of cuspidal representations of $\\mathbf{PGL}_{2}$ associated with Hecke eigenforms. In this paper, we study the exceptional theta correspondence for $\\mathbf{F}_{4}\\times\\mathbf{PGL}_{2}$, and establish the existence of such a family of automorphic representations for $\\mathbf{F}_{4}$. Motivated by the work of Pollack, our main tool is a notion of \"exceptional theta series\" on $\\mathbf{PGL}_{2}$, arising from certain automorphic representations of $\\mathbf{F}_{4}$. These theta series are holomorphic modular forms on $\\mathbf{SL}_{2}(\\mathbb{Z})$, with explicit Fourier expansions, and span the entire space of level one cusp forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-31T12:50:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F27",
      "11F55",
      "11F67",
      "11F70",
      "20G41"
    ],
    "keywords": [
      "exceptional theta correspondence",
      "automorphic representations",
      "explicit fourier expansions",
      "holomorphic modular forms",
      "compact real points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}