{
  "abstract": "We show that an ir{\\-}reducible cuspidal eta quotient of level~$N$ is uniquely identified by its orders of vanishing at $O((\\log_2 N)^{r-1})$ cusps, where $r$ denotes the number of distinct prime divisors of $N$. In particular, all such eta quotients which are Fricke eigenforms of a prime power level, are uniquely identified only by their orders of vanishing at~$\\infty$. We also prove that for all $k\\in\\N$, there exists a primitive and irreducible cuspidal eta quotient of weight $k/2$. It follows from the exhaustiveness of Zagier's list of the primitive holomorphic eta quotients of weight~$1/2$ that the 1st Mersmann level, $\\cl{M}_1=12$. Here we show that for all $k\\in\\N$, the $k$-th Mersmann level is divisible by $2^{2k}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10576",
  "author_total": 1,
  "authors": [
    {
      "name": "Soumya Bhattacharya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/660591mb7hbn3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10576",
  "fulltext_url": null,
  "id": "660591mb7hbn3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10576",
  "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": "W7221165053",
  "pdf_url": "https://arxiv.org/pdf/2610.10576",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/660591mb7hbn3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10576",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10576"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The structure of irreducible cuspidal eta quotients",
  "title_text": "The structure of irreducible cuspidal eta quotients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/660591mb7hbn3",
  "venue": "arXiv",
  "versions": []
}