{
  "abstract": "Let $E/\\mathbb{Q}$ be an elliptic curve. Zywina's refinement of Koblitz's conjecture predicts that there are infinitely many primes $p$ of good reduction for which $\\#E_p(\\mathbb{F}_p)$ is prime precisely when $E$ has no congruence obstruction. For non-CM elliptic curves over $\\mathbb{Q}$, we classify all primitive congruence obstructions. In particular, we show that a primitive obstruction of composite level can occur only at $6$, $10$, $14$, $15$, or $30$ and determine the corresponding Galois images. As a consequence, for non-CM elliptic curves, the existence of any congruence obstruction is detected by the mod $210$ Galois image, so the positivity of the Koblitz--Zywina constant is determined at level $210$. We also determine which primitive obstruction levels can occur simultaneously. Finally, we prove, assuming GRH, that if $E$ has no congruence obstruction, then for every $0<κ<1/8$ there are infinitely many primes $p$ of good reduction for which the least prime divisor of $\\#E_p(\\mathbb{F}_p)$ is greater than $κ\\log p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30573",
  "author_total": 3,
  "authors": [
    {
      "name": "Sung Min Lee"
    },
    {
      "name": "Jacob Mayle"
    },
    {
      "name": "Rakvi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4y9hzdqr9khw7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30573",
  "fulltext_url": null,
  "id": "4y9hzdqr9khw7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30573",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214593942",
  "pdf_url": "https://arxiv.org/pdf/2609.30573",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4y9hzdqr9khw7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30573",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30573"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Congruence obstructions in the refined Koblitz conjecture",
  "title_text": "Congruence obstructions in the refined Koblitz conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/4y9hzdqr9khw7",
  "venue": "arXiv",
  "versions": []
}