{
  "abstract": "In this paper, we establish a new $Ω$-result for the values $|L(σ,χ_P)|$ of quadratic Dirichlet $L$-functions at and near the central point, where \\[ σ=\\frac12+\\fracκ{\\ln g}, \\qquad κ\\geq 0, \\] and $P$ ranges over monic irreducible polynomials associated with hyperelliptic curves of genus $g$ over a fixed finite field $\\mathbb{F}_q$. We consider the asymptotic setting in which $q$ is fixed and $g\\to\\infty$. More precisely, for every $ε\\in (0,1/2)$ andevery fixed $κ\\geq 0$, we prove that \\[ \\max_{P \\in \\mathcal{P}_{2g+1}} |L(σ,χ_P)| \\gg \\exp \\left(\\left(e^{-κ} \\ln q \\,\\sqrt{\\frac{\\sqrt{q}+1}{\\sqrt{q}-1}(1/2-ε)} +o(1)\\right)\\sqrt{\\frac{g\\ln_2 g}{\\ln g}} \\right), \\] where $\\mathcal{P}_{2g+1}$ is the set of all monic irreducible polynomials of degree $2g+1$ in $\\mathbb{F}_q[t]$. In particular, when $κ=0$, this gives the corresponding result at the central point $σ=1/2$. Our result extends the recent work of Darbar and Maiti (2024) and yields an improved lower bound for the extreme values in this family. In addition, for $1/2<σ<1$ and sufficiently large $n$, we study the extreme values of $L(σ,χ_P)$, where $P\\in\\mathcal{P}_n$, and obtain an improved lower bound compared with the result of Lumley (2021).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18452",
  "author_total": 1,
  "authors": [
    {
      "name": "Ahammad Mostafa Hossain"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/23ykd5mxjh4nv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18452",
  "fulltext_url": null,
  "id": "23ykd5mxjh4nv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18452",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213502234",
  "pdf_url": "https://arxiv.org/pdf/2609.18452",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/23ykd5mxjh4nv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2609.18452",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18452"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Extreme Values of Quadratic Dirichlet $L$-functions over Monic Irreducible Polynomials in $\\mathbb{F}_q[t]$",
  "title_text": "Extreme Values of Quadratic Dirichlet L-functions over Monic Irreducible Polynomials in 𝔽_q[t]",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/23ykd5mxjh4nv",
  "venue": "arXiv",
  "versions": []
}