{
  "abstract": "We prove an equidistribution result for Galois characters in the graded Lie components $L_n$ associated to the $\\mathbf{Q}_p$-fundamental group of a projective hyperbolic curve $X$ over a number field $k$ with geometrically CM Jacobian. That is, we consider the $p$-adic completion of a fixed character lattice, where we define measures $μ_n$ counting normalized character multiplicities in $L_n$, and show that $(μ_n)_n$ weakly converges to the Haar measure. The key idea to prove convergence is to write the Fourier coefficients of $μ_n$ in terms of traces of operators on $L_n$ induced by symplectic operators on the Tate module, and control them with a known symplectic character formula, followed by Weyl's criterion for compact abelian groups. Our main arithmetic application is to combine equidistribution with the Iwasawa-theoretic method of Coates--Kim and a compactness argument for moving analytic loci, giving a dimension inequality for unipotent Selmer schemes: $\\dim_{\\mathbf{Q}_p}H^1_f(G_{k,T},U_n)< \\dim_{\\mathbf{Q}_p}\\prod_{v\\mid p}H^1_f(G_v,U_n)$ for all sufficiently large $n,$ with the quotient having limit superior at most $1/2.$ This answers a particular case of a conjecture by Kim, from which the finiteness of $X(\\mathbf{Q})$ can be recovered when $k=\\mathbf{Q}.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12013",
  "author_total": 1,
  "authors": [
    {
      "name": "Ander Martin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d5wtw9dfkskzf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12013",
  "fulltext_url": null,
  "id": "d5wtw9dfkskzf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12013",
  "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",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7221177876",
  "pdf_url": "https://arxiv.org/pdf/2610.12013",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d5wtw9dfkskzf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.12013",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12013"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Lie weights for CM curves and unipotent Selmer dimensions",
  "title_text": "Lie weights for CM curves and unipotent Selmer dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/d5wtw9dfkskzf",
  "venue": "arXiv",
  "versions": []
}