{
  "abstract": "The space $M(ρ;\\mathcal{E})$ of weakly holomorphic modular forms for a dimension-$r$ complex representation $ρ$ of $\\mathrm{SL}_2(\\mathbb{Z})$ with growth prescribed by $\\mathcal{E}$ at the cusp is a free module over $\\mathbb{C}[E_4,E_6]$. We first prove that a $\\mathbb{C}[E_4,E_6]$-linearly independent homogeneous family $F_1,\\ldots,F_r$ in this module gives a basis if and only if $\\sum_{i=1}^r \\operatorname{wt}(F_i)=12\\operatorname{tr}\\mathcal{E}$. We then extend this criterion to Jacobi forms. As applications, we determine the generator weights of free modules of $\\mathrm{O}(D_{8d}^+)$-invariant Jacobi forms of index $D_{8d}^{+}(p)$, where $D_{8d}^{+}$ is an even unimodular lattice extending $D_{8d}$, $d\\ge2$, and $p=2,3,5$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39270",
  "author_total": 3,
  "authors": [
    {
      "name": "Jialin Li"
    },
    {
      "name": "Haowu Wang"
    },
    {
      "name": "Bohan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c4sdsxvjtdgv1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39270",
  "fulltext_url": null,
  "id": "c4sdsxvjtdgv1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39270",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39270",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c4sdsxvjtdgv1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39270",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39270"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Free modules of vector-valued modular forms and applications to Jacobi forms",
  "title_text": "Free modules of vector-valued modular forms and applications to Jacobi forms",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/c4sdsxvjtdgv1",
  "venue": "arXiv",
  "versions": []
}