{
  "id": "1907.08781",
  "version": "v1",
  "published": "2019-07-20T08:56:27.000Z",
  "updated": "2019-07-20T08:56:27.000Z",
  "title": "Siegel modular forms of weight 13 and the Leech lattice",
  "authors": [
    "Gaëtan Chenevier",
    "Olivier Taïbi"
  ],
  "comment": "1 table, 30 pages",
  "categories": [
    "math.NT",
    "math.AG",
    "math.GR"
  ],
  "abstract": "For $g=8,12,16$ and $24$, there is a nonzero alternating $g$-multilinear form on the ${\\rm Leech}$ lattice, unique up to a scalar, which is invariant by the orthogonal group of ${\\rm Leech}$. The harmonic Siegel theta series built from these alternating forms are Siegel modular cuspforms of weight $13$ for ${\\rm Sp}_{2g}(\\mathbb{Z})$. We prove that they are nonzero eigenforms, determine one of their Fourier coefficients, and give informations about their standard ${\\rm L}$-functions. These forms are interesting since, by a recent work of the authors, they are the only nonzero Siegel modular forms of weight $13$ for ${\\rm Sp}_{2n}(\\mathbb{Z})$, for any $n\\geq 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-20T08:56:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F46",
      "11F30",
      "11F27",
      "20D08",
      "11H55",
      "11H56",
      "11F66"
    ],
    "keywords": [
      "leech lattice",
      "harmonic siegel theta series built",
      "nonzero siegel modular forms",
      "siegel modular cuspforms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}