{
  "abstract": "Adapting a construction of Kurihara in the setting of Drinfeld modular forms, we define Poincaré series on the Drinfeld half-plane $Ω$. These series are built from products of meromorphic $1$-forms that are naturally associated to geodesics on the Bruhat-Tits tree. We establish convergence under a finiteness condition on the geodesics, verify this condition in several cases, and give sufficient conditions for the resulting cusp forms to be nonzero. For the principal congruence subgroup $Γ(\\mathfrak{n})$ of $GL_2(\\mathbb{F}_q[T])$, we construct explicit linearly independent families of Poincaré series by lifting certain $k$-forms from the components of the analytic reduction of $Γ(\\mathfrak{n})\\backslashΩ$. We formulate conjectures on the vanishing orders at cusps, and we prove the first of them for an explicit family of Poincaré series by computing the corresponding expansions at the cusps; as an application, we obtain the Drinfeld modular forms $h$ and $Δ$ as Poincaré series (up to a sign). Finally, for cocompact groups attached to quaternion algebras over $\\mathbb{F}_q(T)$ that split at $\\infty$, we show that the Poincaré series span the whole space of modular forms of given weight and type.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23964",
  "author_total": 2,
  "authors": [
    {
      "name": "Milan Berger-Guesneau"
    },
    {
      "name": "Mihran Papikian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f4xfzp08fk62e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23964",
  "fulltext_url": null,
  "id": "f4xfzp08fk62e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23964",
  "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",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214040465",
  "pdf_url": "https://arxiv.org/pdf/2609.23964",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f4xfzp08fk62e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2609.23964",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23964"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Non-Archimedean Poincaré series and geodesics on the Bruhat-Tits tree",
  "title_text": "Non-Archimedean Poincaré series and geodesics on the Bruhat-Tits tree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/f4xfzp08fk62e",
  "venue": "arXiv",
  "versions": []
}