{
  "id": "1704.01735",
  "version": "v1",
  "published": "2017-04-06T07:52:05.000Z",
  "updated": "2017-04-06T07:52:05.000Z",
  "title": "Modular curves, invariant theory and $E_8$",
  "authors": [
    "Lei Yang"
  ],
  "comment": "43 pages. arXiv admin note: text overlap with arXiv:1511.05278",
  "categories": [
    "math.NT",
    "math.AG",
    "math.RT"
  ],
  "abstract": "We find that three different appearances of $E_8$: the equation of $E_8$-singularity (theory of singularities), the exotic del Pezzo surface of degree one (differential topology of $4$-manifolds), and the automorphism group of the configuration of $120$ tritangent planes of Bring's curve (representation theory and classical algebraic geometry), which are well-known to be constructed from the icosahedron, can be constructed from the modular curve $X(13)$ by the theory of invariants for the simple group $\\text{PSL}(2, 13)$. This implies that $E_8$ is not uniquely determined by the icosahedron and solves a problem of Brieskorn in his talk at ICM 1970 on the mysterious relation between the icosahedron and $E_8$. As applications, we give an explicit construction of the modular curve $X(13)$, which is a classical problem studied by Klein, and find an explicit example to Serre's question on the finite subgroups in the Cremona group of large rank.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-06T07:52:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G35",
      "11F27",
      "13A50",
      "32Sxx",
      "57R55",
      "14H45",
      "14J25",
      "14E07"
    ],
    "keywords": [
      "modular curve",
      "invariant theory",
      "exotic del pezzo surface",
      "icosahedron",
      "serres question"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}