{
  "abstract": "Let $F$ be a totally real field of degree $n$ and discriminant $Δ_F$, and let $X_1(η)$ be the cover of the Hilbert modular variety parametrizing abelian varieties with real multiplication by $\\mathcal O_F$, together with a torsion point having annihilator $η$. Let $L=K_{\\overline X_1(η)}+D$ be the log-canonical bundle on a smooth toroidal compactification, and let $H_L$ be an associated multiplicative height. We prove that rational points on $X_1(η)$ become sparser as $|\\mathrm{Nm}(η)|\\to\\infty$, with $(η,Δ_F)=1$. More precisely, for every number field $K$, set $$ N_{η,K}(B)=\\#\\{x\\in X_1(η)(K):H_L(x)\\leq B\\}. $$ If $|\\mathrm{Nm}(η)|\\ge 5^n$ and $(η,Δ_F)=1$, we prove $$ \\limsup_{B\\to\\infty}\\frac{\\log\\max\\{1,N_{η,K}(B)\\}}{\\log B} \\leqδ_{η,K,n},\\qquad δ_{η,K,n}\\ll_{[K:\\mathbb Q],n}|\\mathrm{Nm}(η)|^{-1/(2n)}. $$ In particular, $δ_{η,K,n}\\to0$ uniformly when $n$ and $[K:\\mathbb Q]$ are bounded and $|\\mathrm{Nm}(η)|\\to\\infty$. The main geometric result is a uniform lower bound, growing with the level, for the log-canonical degree of subvarieties of $X_1(η)$. Combining this estimate with recent progress derived from determinant-method, due to Ellenberg--Lawrence--Venkatesh and Brunebarbe--Maculan, we obtain the sparsity result above. We also prove that, for sufficiently large $|\\mathrm{Nm}(η)|$, every subvariety of $X_1(η)$ is of general type, and establish a higher-dimensional generalization of the geometric torsion theorem of Bakker--Tsimerman. Namely, for a family of abelian varieties with real multiplication over a quasi-projective base of arbitrary dimension, we bound the torsion subgroup of its Mordell--Weil group in terms of the canonical volume of the base, uniformly in the totally real multiplication field of fixed degree.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30033",
  "author_total": 1,
  "authors": [
    {
      "name": "Soheil Memariansorkhabi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cpjrkewq6rgbf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30033",
  "fulltext_url": null,
  "id": "cpjrkewq6rgbf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30033",
  "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.CV"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214273909",
  "pdf_url": "https://arxiv.org/pdf/2609.30033",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cpjrkewq6rgbf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.30033",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30033"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sparsity of rational points on torsion level covers of Hilbert modular varieties",
  "title_text": "Sparsity of rational points on torsion level covers of Hilbert modular varieties",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/cpjrkewq6rgbf",
  "venue": "arXiv",
  "versions": []
}