{
  "abstract": "It is well known that an Eisenstein series $E_k$ cannot be written as a product of two lower-weight Eisenstein series, except for $E_{14}=E_4 E_{10}= E_4^2 E_6= E_6 E_8$. A similar result is also known for the Hecke eigenforms. It is also known that equalities among the monomials of the Eisenstein series can be reduced to the above-mentioned identities. Recently, the present author, along with E. Saha studied the possible monomial relations of the Poincaré cusp forms of index $1$. So far, the problem of studying the monomial relations has been restricted to the setup of holomorphic modular forms. For an even integer $k\\ge 4$ and a negative integer $m$, the Poincaré series $G_{k}(z,m)$ of weight $k$ and index $m$ is a weakly holomorphic modular form having a pole of order $-m$ at $ι\\infty$. It is immediate that the Poincaré series $G_k(z,m)$ for $m<0$ can not be written as a product of two lower-weight Poincaré series of the same index $m$. In this article, we investigate the possible equalities among the monomials of the Poincaré series $G_k(z,m)$ for arbitrary $m<0$. In particular, we show that for any $m<0$ such that $0.06\\le\\{-2mπ\\}\\le0.99$, two monomials composed of $G_k(z,m)$ of the weights $k\\ge 50$ are never equal, where $\\{x\\}$ denotes the fractional part of a real number $x$. In view of Weyl's equidistribution criterion, at least $93\\%$ of $m<0$ satisfies the above inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28350",
  "author_total": 1,
  "authors": [
    {
      "name": "Divyanshu Kala"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3gd7sm5v83ydk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28350",
  "fulltext_url": null,
  "id": "3gd7sm5v83ydk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28350",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.28350",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3gd7sm5v83ydk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28350",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28350"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the monomials of Poincaré Series of negative index",
  "title_text": "On the monomials of Poincaré Series of negative index",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/3gd7sm5v83ydk",
  "venue": "arXiv",
  "versions": []
}