{
  "abstract": "We study supersingular isogeny graphs of rank-two Drinfeld modules over $A=\\mathbb{F}_q[T]$. For distinct finite primes $\\mathfrak p$ and $\\mathfrak q$ of $A$, we prove that the graph in characteristic $\\mathfrak p$, with edges given by cyclic $\\mathfrak q$-isogenies, is connected. The proof combines Gekeler's ideal-class correspondence with strong approximation to realize the graph as a quotient of the Bruhat--Tits tree of the homothety classes of $A_\\mathfrak q$-lattices in $(\\mathbb{F}_q(T))_\\mathfrak q^2$. We also introduce the completeness number $E(\\mathfrak p)$, the least integer such that the graph is complete for every prime $\\mathfrak q\\neq\\mathfrak p$ with $\\operatorname{deg}\\mathfrak q\\geq E(\\mathfrak p)$, and derive explicit upper bounds using the Ramanujan--Petersson bound for the eigenvalues of operators associated to Brandt matrices over function fields. In particular, if $d=\\operatorname{deg}\\mathfrak p$, then $E(\\mathfrak p)\\leq 2d+2$, with sharper bounds depending on $q$ and the parity of $d$. This confirms a conjecture of Micheli and Papikian that the graph becomes complete once $\\operatorname{deg}\\mathfrak q$ is sufficiently large relative to $\\operatorname{deg}\\mathfrak p$. We further obtain parity-dependent lower bounds and prove that $E(\\mathfrak p)\\geq d+\\log_q d-K$ for an absolute constant $K>0$ and sufficiently large odd $d$, thereby refuting the previously suggested bound $E(\\mathfrak p)\\leq d+1$. Finally, we prove a criterion yielding an algorithm to compute $E(\\mathfrak p)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29812",
  "author_total": 1,
  "authors": [
    {
      "name": "Nikola Veselinov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ybc6nyw9yxw5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29812",
  "fulltext_url": null,
  "id": "7ybc6nyw9yxw5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29812",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204894172",
  "pdf_url": "https://arxiv.org/pdf/2608.29812",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ybc6nyw9yxw5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29812",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29812"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On supersingular isogeny graphs of Drinfeld modules",
  "title_text": "On supersingular isogeny graphs of Drinfeld modules",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/7ybc6nyw9yxw5",
  "venue": "arXiv",
  "versions": []
}