{
  "abstract": "For a matrix $γ\\in\\mathrm{SL}_2({\\mathbb Z})$, we define \\[ {\\mathcal R}(γ)=a_1^2+a_2^2+a_3^2+a_4^2, \\qquad \\text{where} \\ γ=\\begin{pmatrix}a_1&a_2\\\\ a_3&a_4\\end{pmatrix}, \\] and let $S_{\\mathrm{sq}}(X)$ count the number of matrices $γ$ with $\\|γ\\|_\\infty = \\max\\{|a_1|, |a_2|,|a_3|,|a_4|\\} \\leq X$ and such that ${\\mathcal R}(γ)$ is squarefree. We prove that \\[ S_{\\mathrm{sq}}(X) = {\\mathfrak S}_{\\mathcal R}^{\\mathrm{sq}}N(X) +O(X^{19/10+o(1)}), \\quad \\text{as}\\ X\\to \\infty, \\] where $N(X)=\\#\\{γ\\in{\\mathrm{SL}}_2({\\mathbb{Z}}):\\|γ\\|_\\infty\\leq X\\}$ and ${\\mathfrak{S}}_{\\mathcal{R}}^{\\mathrm{sq}}$ is an explicit positive Euler product of local $p^2$-densities. The proof combines the $δ$-method for small moduli with a sum-of-two-squares estimate for large square divisors. This complements a result of J. B. Friedlander and H. Iwaniec (2009) on prime values of ${\\mathcal R}(γ)$, which, however, is conditional on a very strong form of the Elliott--Halberstam conjecture. We also show that ${\\mathcal R}(γ)$ is squarefree and has at most $9$ prime divisors for at least $cN(X)/\\log X$ matrices $γ\\in\\mathrm{SL}_2({\\mathbb Z})$ with $\\|γ\\|_\\infty\\le X$, where $c>0$ is an absolute constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21460",
  "author_total": 2,
  "authors": [
    {
      "name": "Igor E. Shparlinski"
    },
    {
      "name": "Yixiu Xiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/17d1jb9re9jd3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21460",
  "fulltext_url": null,
  "id": "17d1jb9re9jd3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21460",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213926403",
  "pdf_url": "https://arxiv.org/pdf/2609.21460",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/17d1jb9re9jd3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.21460",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21460"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Arithmetic structure of $L_2$-norms of ${\\mathrm{SL}}_2(\\mathbb{Z})$ matrices",
  "title_text": "Arithmetic structure of L_2-norms of SL_2(ℤ) matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/17d1jb9re9jd3",
  "venue": "arXiv",
  "versions": []
}