{
  "abstract": "In the direction of Szpiro's conjecture for all elliptic curves $E$ over $\\mathbb{Q}$, the current best bound is $\\log Δ\\ll N \\log N$ where $Δ$ is the absolute value of the minimal discriminant of $E$ and $N$ is the conductor of $E$ (the implicit constant is absolute). This bound dates back to 2013. We obtain the stronger bound $\\log Δ\\ll N\\log \\log N$ which was previously known only under GRH. In addition, for semistable elliptic curves we prove $\\log Δ\\ll N$. Our methods build on the theory developed by the author for approaching the $abc$ conjecture via Shimura curves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17390",
  "author_total": 1,
  "authors": [
    {
      "name": "Hector Pasten"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1xpspcj9c1gvk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17390",
  "fulltext_url": null,
  "id": "1xpspcj9c1gvk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17390",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213391143",
  "pdf_url": "https://arxiv.org/pdf/2609.17390",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1xpspcj9c1gvk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17390",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17390"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Improved bounds for Szpiro's conjecture",
  "title_text": "Improved bounds for Szpiro's conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/1xpspcj9c1gvk",
  "venue": "arXiv",
  "versions": []
}