{
  "abstract": "We compare three heights attached to a curve of genus two: the weighted height of its Igusa--Clebsch invariants, the GIT height of a binary sextic defining it, and the Néron--Tate height on its Jacobian. For a morphism from an abelian variety to a weighted projective space we define a weighted Néron--Tate height and show that the weighted height differs from it by a linear form and a bounded function; the linear form vanishes when inversion is induced by an automorphism of the weighted projective space. For binary sextics we prove the explicit inequality $h_{\\inv}\\le h_{\\text{GIT}} + \\frac{1}{2} \\log 302$, with nonnegative local differences at the finite places that vanish at places of good reduction. We then construct a family of semistable binary sextics over $E\\times E$, where $E:y^2=x^3-2$, whose relative Jacobian carries a section of infinite order, and we show that the Jacobian over the base point is absolutely simple. Along every curve in the parameter space through the base point, the Néron--Tate height of the section is a positive multiple of the weighted moduli height up to an error of square-root size, and up to a bounded error when the curve is rational. As applications, we obtain explicit families of genus-two Jacobians of positive rank over $\\mathbf{Q}$, an infinite congruence lattice of fibres of positive height, and canonical-limit formulas along elliptic curves in $E\\times E$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13850",
  "author_total": 1,
  "authors": [
    {
      "name": "Sajad Salami"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/74mvwhtaqrh2m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13850",
  "fulltext_url": null,
  "id": "74mvwhtaqrh2m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13850",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221514744",
  "pdf_url": "https://arxiv.org/pdf/2610.13850",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/74mvwhtaqrh2m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13850",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13850"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Weighted, GIT and Néron--Tate heights on genus-two moduli and Jacobians",
  "title_text": "Weighted, GIT and Néron--Tate heights on genus-two moduli and Jacobians",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/74mvwhtaqrh2m",
  "venue": "arXiv",
  "versions": []
}