{
  "abstract": "Motivated by the classical quartic and cubic theta identities, we establish a product formula for Siegel--Jacobi theta series associated with rationally isometric positive-definite quadratic spaces. Our main theorem expresses a product of theta series of arbitrary Siegel degree as a finite double character sum over quotients of commensurable lattices. For the root lattice $A_2$, it yields a finite Fourier identity for cubes of cubic theta functions, a new proof of the two-variable cubic identity, and seven independent relations in degree two. A second specialization gives a new square analogue of the two-variable cubic identity. At the origin it reduces to $a(q)a(q^2)=b(q)b(q^2)+c(q)c(q^2)$; in degree two, the same seven relation patterns recur with each cube replaced by the corresponding product at $τ$ and $2τ$. We further extend the cubic identities on $A_2$ to $\\ell$-th-power identities on $A_{\\ell-1}$ for every $\\ell\\geq3$, and to $2\\ell$-th-power identities on $A_{\\ell-1}$ for every $\\ell\\geq2$, in arbitrary Siegel degree. The doubled-order construction recovers Jacobi's quartic identity and gives its analogues in arbitrary Siegel degree.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27959",
  "author_total": 3,
  "authors": [
    {
      "name": "Rong Chen"
    },
    {
      "name": "Haigang Zhou"
    },
    {
      "name": "Xiao-Jie Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7j4wv46gtvvhh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27959",
  "fulltext_url": null,
  "id": "7j4wv46gtvvhh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27959",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7223086214",
  "pdf_url": "https://arxiv.org/pdf/2610.27959",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7j4wv46gtvvhh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27959",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27959"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Theta Relations",
  "title_text": "Theta Relations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/7j4wv46gtvvhh",
  "venue": "arXiv",
  "versions": []
}