{
  "abstract": "Let $n\\geq 1$ and let $\\ell$ be an odd prime. Let $G=\\mathrm{PGSp}_{2n}(\\mathbb{F}_\\ell)$ or $\\mathrm{GSp}_{2n}(\\mathbb{F}_\\ell)$, and fix a faithful transitive permutation representation $π:G\\longrightarrow S_d$. We study degree-$d$ number fields whose Galois closures have Galois group $G$ and whose associated permutation representation is $π$. For $σ\\in S_d$, write $\\operatorname{ind}(σ)$ for its permutation index, namely $\\operatorname{ind}(σ) = d-\\#\\{\\text{orbits of $σ$ on $\\{1,\\ldots,d\\}$}\\}$. If $τ$ denotes a symplectic transvection, or its image in the projective symplectic group, we prove that the number of such fields with absolute discriminant at most $X$ is bounded below by a constant multiple of $X^{1/(2n\\,\\operatorname{ind}(π(τ)))}$. For the natural vector and projective actions, the exponents we obtain are asymptotically $1/(4n)$ of those predicted by the weak form of Malle's conjecture as $\\ell\\to\\infty$. The fields are constructed from the mod-$\\ell$ Galois representations attached to the Jacobians of a one-parameter family of hyperelliptic curves. The proof combines large symplectic monodromy for this family with a squarefree sieve.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23093",
  "author_total": 1,
  "authors": [
    {
      "name": "Anwesh Ray"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c4sg0y4m6nwx4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23093",
  "fulltext_url": null,
  "id": "c4sg0y4m6nwx4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23093",
  "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.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214016833",
  "pdf_url": "https://arxiv.org/pdf/2609.23093",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c4sg0y4m6nwx4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23093",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23093"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Counting number fields with symplectic Galois group",
  "title_text": "Counting number fields with symplectic Galois group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/c4sg0y4m6nwx4",
  "venue": "arXiv",
  "versions": []
}