{
  "id": "2009.03688",
  "version": "v1",
  "published": "2020-09-07T02:00:11.000Z",
  "updated": "2020-09-07T02:00:11.000Z",
  "title": "$E_8$-singularity, invariant theory and modular forms",
  "authors": [
    "Lei Yang"
  ],
  "comment": "40 pages. arXiv admin note: substantial text overlap with arXiv:1704.01735, arXiv:1511.05278",
  "categories": [
    "math.NT",
    "math.AC",
    "math.AG",
    "math.RT"
  ],
  "abstract": "As an algebraic surface, the equation of $E_8$-singularity $x^5+y^3+z^2=0$ can be obtained as a quotient $C_Y/\\text{SL}(2, 13)$ over the modular curve $X(13)$, where $Y \\subset \\mathbb{CP}^5$ is an algebraic curve given by a system of $\\text{SL}(2, 13)$-invariant polynomials and $C_Y$ is a cone over $Y$. It is different from the Kleinian singularity $\\mathbb{C}^2/\\Gamma$, where $\\Gamma$ is the binary icosahedral group. This gives a negative answer to Arnol'd and Brieskorn's questions about the mysterious relation between the icosahedron and $E_8$, i.e., the $E_8$-singularity is not necessarily the Kleinian icosahedral singularity. Hence, the equation of $E_8$-singularity possesses two distinct modular parametrizations. Moreover, three different algebraic surfaces, the equations of $E_8$, $Q_{18}$ and $E_{20}$-singularities can be realized from the same quotients $C_Y/\\text{SL}(2, 13)$ over the modular curve $X(13)$ and have the same modular parametrizations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-07T02:00:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G35",
      "11F27",
      "13A50",
      "14L30",
      "32S25",
      "32S05"
    ],
    "keywords": [
      "invariant theory",
      "modular forms",
      "algebraic surface",
      "modular curve",
      "distinct modular parametrizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}