{
  "id": "1407.3550",
  "version": "v2",
  "published": "2014-07-14T07:13:39.000Z",
  "updated": "2014-08-18T03:05:37.000Z",
  "title": "Dedekind $η$-function, Hauptmodul and invariant theory",
  "authors": [
    "Lei Yang"
  ],
  "comment": "46 pages. arXiv admin note: substantial text overlap with arXiv:1209.1783",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We solve a long-standing open problem with its own long history dating back to the celebrated works of Klein and Ramanujan. This problem concerns the invariant decomposition formulas of the Hauptmodul for $\\Gamma_0(p)$ under the action of finite simple groups $PSL(2, p)$ with $p=5, 7, 13$. The cases of $p=5$ and $7$ were solved by Klein and Ramanujan. Little was known about this problem for $p=13$. Using our invariant theory for $PSL(2, 13)$, we solve this problem. This leads to a new expression of the classical elliptic modular function of Klein: $j$-function in terms of theta constants associated with $\\Gamma(13)$. Moreover, we find an exotic modular equation, i.e., it has the same form as Ramanujan's modular equation of degree $13$, but with different kinds of modular parametrizations, which gives the geometry of the classical modular curve $X(13)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-18T03:05:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F20",
      "11F27",
      "14H42",
      "11G18",
      "14G35",
      "14H45",
      "11G16",
      "11P82"
    ],
    "keywords": [
      "invariant theory",
      "hauptmodul",
      "classical elliptic modular function",
      "invariant decomposition formulas",
      "exotic modular equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.3550Y"
    }
  }
}