{
  "id": "2105.13152",
  "version": "v1",
  "published": "2021-05-27T13:54:50.000Z",
  "updated": "2021-05-27T13:54:50.000Z",
  "title": "On Ramanujan's Modular Equations and Hecke Groups",
  "authors": [
    "Md. Shafiul Alam"
  ],
  "categories": [
    "math.NT",
    "math.CV"
  ],
  "abstract": "We study the relation between Hecke groups and the modular equations in Ramanujan's theories of signature 2, 3 and 4. The solution $(\\alpha,\\beta)$ to the generalized modular equation satisfies a polynomial equation $P(\\alpha,\\beta)=0$ and we determine the degree in each of $\\alpha$ and $\\beta$ of the polynomial $P(\\alpha,\\beta)$ in terms of the indices of Hecke subgroups. We establish some mutually equivalent statements related to Hecke subgroups and modular equations, and prove that $(\\alpha,\\beta)$ and $(1-\\beta,1-\\alpha)$ satisfy the same polynomial equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-27T13:54:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30F35",
      "11F06",
      "33C05"
    ],
    "keywords": [
      "ramanujans modular equations",
      "hecke groups",
      "hecke subgroups",
      "polynomial equation",
      "generalized modular equation satisfies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}