{
  "abstract": "We prove that every vector of $\\mathrm{II}_{8n+1,1}$, with $n\\ge1$, is a sum of two roots if and only if $n=1$ or $2$. This settles the Borisov--Nuer conjecture and implies the existence of Ulrich line bundles for every very ample polarization on a complex unnodal Enriques surface. For vectors of negative norm, we express the number of root decompositions as a Fourier coefficient of a vector-valued theta series, reducing existence to positivity. The main analytic difficulty is to obtain estimates uniform over lattices of unbounded determinant. We address this by combining a reduction based on primitive sublattices of codimension one and least determinant with explicit bounds for theta norms and Fourier coefficients of cusp forms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38414",
  "author_total": 2,
  "authors": [
    {
      "name": "Haowu Wang"
    },
    {
      "name": "Shaoyun Yi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45870y2p22xy2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38414",
  "fulltext_url": null,
  "id": "45870y2p22xy2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38414",
  "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.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38414",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45870y2p22xy2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38414",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38414"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Borisov--Nuer conjecture and root decompositions in even unimodular Lorentzian lattices",
  "title_text": "The Borisov--Nuer conjecture and root decompositions in even unimodular Lorentzian lattices",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/45870y2p22xy2",
  "venue": "arXiv",
  "versions": []
}