{
  "abstract": "Let \\(A\\) be the coefficient ring of a smooth projective curve over \\(\\mathbb{F}_q\\) with a closed point at infinity of arbitrary degree \\(\\mathrm{N}=°(\\infty)\\ge 1\\), and let \\(φ\\) be a rank-one Drinfeld \\(A\\)-module. In this paper, we prove three main results concerning the module of special functions and its relation to Pellarin's \\(\\mathcal L(1)\\)-series. First, the module \\(\\mathrm{sf}(φ)\\) of special functions contains a Tate-algebra unit exactly when it is free of rank one, equivalently when its period lattice is isomorphic to the module of regular differentials. This settles the Gazda--Maurischat conjecture. The proof evaluates Cauchy kernels; on the period lattice all such evaluations agree, and their common characteristic residue recovers the period. Second, for suitable isogeny data to a principal-period target \\(ψ\\), a strictly normalized shtuka product, together with one characteristic residue, gives an explicit formula for the fundamental period \\(\\tildeπ\\) (the generator of the free lattice \\(Λ_ψ\\)). The same defect equation also yields the residue formula for the Drinfeld logarithm. Third, the first Frobenius twist of the normalized shtuka differential is identified with a pairing built from finite torsion traces, whose coefficients are finite étale traces over the field of definition. As an application, Pellarin's \\(\\mathcal L(1)\\)-series is expressed exactly as the first Frobenius twist of the generating differential at a fundamental period. A notable feature of these results is that they hold for arbitrary $\\mathrm{N} \\geq 1$, and the source period lattice need not be free.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03134",
  "author_total": 3,
  "authors": [
    {
      "name": "Chuangqiang Hu"
    },
    {
      "name": "Stephen S. -T. Yau"
    },
    {
      "name": "Lishan Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d1320p6m9s6jq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03134",
  "fulltext_url": null,
  "id": "d1320p6m9s6jq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03134",
  "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.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220427370",
  "pdf_url": "https://arxiv.org/pdf/2610.03134",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d1320p6m9s6jq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03134",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03134"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Pellarin's Identity, Carlitz Period, and Anderson Generating Functions over Arbitrary Curves",
  "title_text": "Pellarin's Identity, Carlitz Period, and Anderson Generating Functions over Arbitrary Curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/d1320p6m9s6jq",
  "venue": "arXiv",
  "versions": []
}