{
  "id": "2502.08760",
  "version": "v1",
  "published": "2025-02-12T20:06:26.000Z",
  "updated": "2025-02-12T20:06:26.000Z",
  "title": "Modular Forms and Certain ${}_2F_1(1)$ Hypergeometric Series",
  "authors": [
    "Esme Rosen"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of ${}_2F_1(1)$ hypergeometric series and Ramanujan's theory of alternative bases to compute the exact central $L$-value of these Hecke eigenforms in terms of special beta values. We also show the integral Fourier coefficients can be written in terms of Jacobi sums, reflecting a motivic relation between the hypergeometric series and the modular forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-12T20:06:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67"
    ],
    "keywords": [
      "hypergeometric series",
      "modular forms",
      "hecke eigenforms",
      "framework relating hypergeometric motives",
      "special beta values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}