{
  "abstract": "We investigate Miller's bias conjecture for general one-parameter families of elliptic curves over $\\mathbb{Q}$ with non-constant $j$-invariants. Utilizing methods developed by Michel, we view the second moment as the sum of distinct cohomological components: a main $p^2$ term, a $p^{3/2}$ term arising from the first cohomology over the projective line, and terms of order $p$ or lower corresponding to singular fibers. By assuming the generalized Sato-Tate conjecture, we employ representation and motive theory to demonstrate that the coefficient of the $p^{3/2}$ term averages to zero in the limit. We then independently analyze the lower-order contributions without assuming Sato-Tate. By applying the Chebotarev density theorem and analyzing Galois representations, we show that the contributions to the order $p$ term from additive singular fibers also average to zero. Consequently, the only remaining average contribution to the $p$ term is $-B_pp$, where $B_p$ denotes the number of ``bad'' points where the discriminant vanishes. Because $B_p$ averages to a strictly positive value, the overall coefficient of the $p$ term averages to a strictly negative value in the limit, thereby establishing the bias conjecture for these families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38664",
  "author_total": 4,
  "authors": [
    {
      "name": "Lucas Chen"
    },
    {
      "name": "Joshua Im"
    },
    {
      "name": "Steven J. Miller"
    },
    {
      "name": "Devayani Pradhan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eh7kpy40kykkv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38664",
  "fulltext_url": null,
  "id": "eh7kpy40kykkv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38664",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7216595923",
  "pdf_url": "https://arxiv.org/pdf/2609.38664",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eh7kpy40kykkv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.38664",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38664"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Bias Conjecture for One-Parameter, Non-Isotrivial Elliptic Curve Families",
  "title_text": "The Bias Conjecture for One-Parameter, Non-Isotrivial Elliptic Curve Families",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/eh7kpy40kykkv",
  "venue": "arXiv",
  "versions": []
}