{
  "abstract": "Let $\\Gz$ be a cofinite Fuchsian group whose quotient $\\Gz\\backslash\\HH$ has genus one and a single cusp, normalized to be at $\\infty$ with width one. Let $\\Xg$ be its smooth compactification, which is an elliptic curve over $\\CC$. Let $x,y$ be canonical generators of the function field on $\\Xg$, which have poles of order $2,3$ respectively at the cusp and which {satisfy} a generalized Weierstrass relation $y^2+(a_1x+a_3)y=x^3+a_2x^2+a_4x+a_6$. Let $f$ be the unique (up to scale) weight two cusp form for $\\Gz$, normalized so that the associated holomorphic differential is $ω=dx/(2y+a_1x+a_3)$. With all this, we prove that the generating function identity \\[ \\frac{\\bigl(y(τ)+y(z)+a_1x(z)+a_3\\bigr)f(z)}{x(z)-x(τ)}=\\sum_{m\\ge0}Φ_m(τ)q_z^m \\] defines a family $\\{Φ_m\\}_{m\\ge0}$ of weakly holomorphic modular functions on $\\Xg$ such that the set $\\{Φ_0\\}\\cup\\{Φ_m\\}_{m\\ge2}$ is a canonical basis of the space $M_{0,\\Gz}^{!,\\infty}$ of weakly holomorphic modular functions on $\\Xg$ with poles supported only at the cusp. As an application, by comparing the generating function of the family $\\{Φ_m\\}_{m\\ge0}$ with the generating function of the Niebur--Poincaré series, we show that the generating function of the special values of the Kloosterman zeta function at $1$ is the holomorphic part of a certain weight two harmonic Maass form up to an additive constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35382",
  "author_total": 3,
  "authors": [
    {
      "name": "Joshua S. Friedman"
    },
    {
      "name": "Jay Jorgenson"
    },
    {
      "name": "Lejla Smajlović"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f65eha6wzhart.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35382",
  "fulltext_url": null,
  "id": "f65eha6wzhart",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35382",
  "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",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214795321",
  "pdf_url": "https://arxiv.org/pdf/2609.35382",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f65eha6wzhart.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35382",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35382"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On a generating function of the basis of weakly holomorphic functions on an elliptic curve",
  "title_text": "On a generating function of the basis of weakly holomorphic functions on an elliptic curve",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/f65eha6wzhart",
  "venue": "arXiv",
  "versions": []
}