{
  "abstract": "Let $X$ be a curve over $\\mathbb{F}_q$. Denote by $N(X)$ and $g(X)$ the number of $\\mathbb{F}_q$-rational points and the genus of $X$ respectively. The Ihara constant \\begin{equation*} A(q)=\\limsup_{g(X)\\to\\infty}\\frac{N(X)}{g(X)} \\end{equation*} measures the asymptotic abundance of rational points on curves over $\\mathbb{F}_q$. It is a central problem in the study of algebraic geometry and has also attracted broader mathematical interest. In particular, the constant $A(2)$ is discussed in the \\emph{Optimization Constants in Mathematics} collection maintained by Terence Tao et al \\cite[Number 33]{DIT26}. In this paper, we study the Zassenhaus filtration of pro-$p$ groups arising from class field towers. By combining Golod--Shafarevich theorem, embedding problems and central class field theory, we establish a new criterion for the infinitude of class field towers. As an application, we obtain an improvement on the best known lower bounds of Ihara constant $A(2)$ and $A(3)$ which are explicitly given by \\[ A(2)\\geq 0.317779\\cdots \\text{ and } A(3)\\geq 0.496074\\cdots. \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33649",
  "author_total": 3,
  "authors": [
    {
      "name": "Liming Ma"
    },
    {
      "name": "Chaoping Xing"
    },
    {
      "name": "Zugan Xing"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fesrbqg1vn842.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33649",
  "fulltext_url": null,
  "id": "fesrbqg1vn842",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33649",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223791393",
  "pdf_url": "https://arxiv.org/pdf/2610.33649",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fesrbqg1vn842.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.33649",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33649"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Zassenhaus Filtrations and Golod-Shafarevich Methods for Ihara Constants",
  "title_text": "Zassenhaus Filtrations and Golod-Shafarevich Methods for Ihara Constants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/fesrbqg1vn842",
  "venue": "arXiv",
  "versions": []
}