{
  "abstract": "We construct period polynomial relations for finite systems of theta products stable under modular transformations and use them to derive identities among critical $L$-values. Our main example is a three-component theta system of weight $5$, whose coupled period polynomials are determined explicitly and yield new cross-form relations among critical values of the associated eta products. These relations are not consequences of the functional equations of the individual forms. We also develop a weight-$4$ system arising from a quadratic twist of conductor $3$ and obtain relations linking critical values of the original and twisted modular forms. Through modular parametrizations by complete elliptic integrals, these $L$-value identities yield corresponding moment identities. The method gives a unified modular-symbol explanation of several previously known elliptic integral moment relations while producing new critical-value relations between distinct modular forms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.26514",
  "author_total": 1,
  "authors": [
    {
      "name": "Dianbin Bao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/52hpgcwvxmayf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.26514",
  "fulltext_url": null,
  "id": "52hpgcwvxmayf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.26514",
  "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": "W7204563953",
  "pdf_url": "https://arxiv.org/pdf/2608.26514",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/52hpgcwvxmayf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2608.26514",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.26514"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Period Relations for Theta Products and Elliptic Integral Moments",
  "title_text": "Period Relations for Theta Products and Elliptic Integral Moments",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/52hpgcwvxmayf",
  "venue": "arXiv",
  "versions": []
}