{
  "abstract": "For a smooth projective curve $C/\\mathbb{Q}$ of genus $\\geq 2$ and $L/\\mathbb{Q}$ an extension, we write $C(L)_{\\text{new}}=\\{P\\in C(L):\\mathbb{Q}(P)=L\\}$. Recent work of Khawaja and Siksek conjectures that this set is empty for $100\\%$ of degree $n$ number fields $L$, when ordered by absolute discriminant. Moreover, they bring evidence towards this conjecture when $C$ is a degree $n$ cover of $\\mathbb{P}^1$. We complement their work by proving analogous results for degree $n$ covers $ψ:C\\to E$ of elliptic curves $E$. Our main result shows that, under suitable hypotheses, the number of distinct absolute discriminants at most $X$ of primitive degree $n$ fields $L$ with $C(L)_{\\text{new}}\\neq\\varnothing$ is $O(X^{1/2})$ or $O(X/(\\log X)^α)$, for some $α>0$. In degrees $2,3,4$ and $5$ we show that these fields have density $0$ among all fields of the same degree (in degree $4$, also among the primitive ones). The novelty is for degrees $4$ and $5$, where we use work of Bhargava--Shankar--Wang and McGown--Thorne--Tucker to count fields with specified local constraints. Moreover, we give concrete examples of $8$ bielliptic modular curves $X_0(N)$, for which $X_0(N)(L)_{\\text{new}}=\\emptyset$ for $100 \\%$ of quadratic fields $L$. Lastly, we point out modular covers of degrees $3$ and $5$ in the LMFDB for which similar conclusions hold.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08753",
  "author_total": 2,
  "authors": [
    {
      "name": "Diana Mocanu"
    },
    {
      "name": "George C. Turcas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8efw30xcxasg1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08753",
  "fulltext_url": null,
  "id": "8efw30xcxasg1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08753",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220932641",
  "pdf_url": "https://arxiv.org/pdf/2610.08753",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8efw30xcxasg1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08753",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08753"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "New algebraic points on covers of elliptic curves",
  "title_text": "New algebraic points on covers of elliptic curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/8efw30xcxasg1",
  "venue": "arXiv",
  "versions": []
}