{
  "abstract": "We construct canonical stable Steenrod squares on modified compactly supported étale cohomology of separated finite-type schemes over rings of $S$-integers in number fields with $2$ invertible, extending Feng's absolute Wu classes to arithmetic bases. For a flat projective morphism $f:X\\to B$ of pure relative dimension, with $X$ regular and $B$ such a base, we prove $v_X=\\operatorname{Sq}^{-1}(w_{\\mathrm{et}}(τ_f+\\mathcal O_X^{\\oplus3}))$ in completed mod-$2$ étale cohomology. Here $τ_f$ is the virtual relative tangent bundle, $w_{\\mathrm{et}}$ the total étale Stiefel--Whitney class, and $\\operatorname{Sq}^{-1}$ the inverse total Steenrod square. Over finite fields of odd characteristic, the formula holds without the three trivial summands. The proof uses a modified compactly supported relative Wu formula extending Benoist's theorem. Our generalized Hecke theorem gives universal mod-$2$ relations involving Chern classes and the Kummer class of $-1$, governed by an arithmetic deformation of Hirzebruch's $2$-Todd series. These hold modulo an explicit archimedean ideal and become vanishing identities over finite fields, over totally imaginary arithmetic bases, or when $-1$ is a square on $X$. Nonempty real loci force infinitely many nonzero Wu components. Applications include Hecke's theorem on the different away from $2$, a finite-field analog of Atiyah's theorem on theta characteristics, and new higher-dimensional relations. We also revisit Serre's Riemann--Hurwitz theorem for spin bundles, recover the Shusterman--Sawin theorem for smooth branched covers of closed $3$-manifolds and prove a function-field analog of the Lusztig--Milnor--Peterson formula, expressing the difference between mod-$2$ and $2$-adic semicharacteristics using the middle Wu class and its Tate-twisted Bockstein.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.06008",
  "author_total": 3,
  "authors": [
    {
      "name": "Shachar Carmeli"
    },
    {
      "name": "Mark Shusterman"
    },
    {
      "name": "Sa'ar Zehavi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5gkftm874dh84.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.06008",
  "fulltext_url": null,
  "id": "5gkftm874dh84",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.06008",
  "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.AT"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7163661129",
  "pdf_url": "https://arxiv.org/pdf/2606.06008",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5gkftm874dh84.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2606.06008",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.06008"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Arithmetic Wu Formulas and the Generalized Hecke Theorem",
  "title_text": "Arithmetic Wu Formulas and the Generalized Hecke Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/5gkftm874dh84",
  "venue": "arXiv",
  "versions": []
}