{
  "abstract": "Let $r\\geq 3$ be an integer, let $q$ be an odd prime power, and set $A=\\mathbb{F}_q[T]$ and $F=\\mathbb{F}_q(T)$. We study the Galois representations associated with rank $r$ Drinfeld modules of the form $ϕ_T=T+g_1τ+\\cdots+g_rτ^r$, where $g_i\\in A$ and $g_r\\neq 0$. Ordering the coefficient tuples $\\vec{g}=(g_1,\\ldots,g_r)$ by their maximum degree, we prove that, for a density-one set of tuples, the $λ$-adic Galois representation is surjective for every prime $λ$ of $A$, and the adelic Galois image contains $\\mathrm{SL}_r(\\widehat A)$. Moreover, for every tuple in this set, the index of the adelic image equals the index of the corresponding rank-one determinant representation and hence divides $q-1$. These results extend Zywina's theorem in rank $2$ to arbitrary ranks $r\\geq 3$. As a further consequence, we establish the existence of rank $r$ Drinfeld modules with surjective adelic Galois representations for every $r\\geq 3$, extending previous existence results which were known for prime values of $r$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01464",
  "author_total": 1,
  "authors": [
    {
      "name": "Anwesh Ray"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7a57cv3ymsx9k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01464",
  "fulltext_url": null,
  "id": "7a57cv3ymsx9k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01464",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01464",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7a57cv3ymsx9k.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01464",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01464"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Adelic Galois representations are maximal for almost all Drinfeld modules",
  "title_text": "Adelic Galois representations are maximal for almost all Drinfeld modules",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/7a57cv3ymsx9k",
  "venue": "arXiv",
  "versions": []
}