{
  "abstract": "Let $Ω$ be a lattice in $\\mathbb{C}$ with algebraic invariants and complex multiplication, let $\\mathcal{E}$ be the elliptic curve associated with $Ω$, and let $\\wp$ be the Weierstrass function relative to $Ω$. Set $ k:=\\operatorname{End}(\\mathcal{E}) \\otimes_{\\mathbb{Z}}\\mathbb{Q}.$ We prove that if $t_1,\\dots,t_s$ are $\\mathbb{Q}$-linearly independent algebraic numbers and $p_1,\\dots,p_n$ are $k$-linearly independent algebraic numbers, then the $s+n$ numbers \\[ \\mathrm{e}^{t_1},\\dots,\\mathrm{e}^{t_s}, \\wp(p_1),\\dots,\\wp(p_n) \\] are algebraically independent over $\\overline{\\mathbb{Q}}$. The proof uses the Tannakian description of the Lie algebra of the unipotent radical of the $1$-motive associated with these points.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15294",
  "author_total": 1,
  "authors": [
    {
      "name": "Cristiana Bertolin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/24gnzp7evda3j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15294",
  "fulltext_url": null,
  "id": "24gnzp7evda3j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15294",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213341071",
  "pdf_url": "https://arxiv.org/pdf/2609.15294",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/24gnzp7evda3j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15294",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15294"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Algebraic independence of the exponential and Weierstrass $\\wp$-functions",
  "title_text": "Algebraic independence of the exponential and Weierstrass ℘-functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/24gnzp7evda3j",
  "venue": "arXiv",
  "versions": []
}