{
  "abstract": "Let $p$ be an odd prime, $ψ$ a nontrivial additive character of $\\mathbb F_p$, $ψ_q=ψ\\circ\\mathrm{Tr}_{\\mathbb F_q/\\mathbb F_p}$ and $λ\\in\\mathbb F_p^\\times$. For $h\\in(\\mathbb F_q^\\times)^n$ let $Σ_n(h)$ be the sum of $ψ_q(u_1^{-1}+\\cdots+u_n^{-1})$ over $u\\in(\\mathbb F_q^\\times)^n$ with $h_1u_1+\\cdots+h_nu_n=-λ$, an additively twisted average of products of Kloosterman sums. For every $n\\ge2$, the cohomology computing $Σ_n(h)$ is concentrated in one degree and has dimension $2^n-1$. It has $r(n)=2^n-\\binom n{\\lfloor n/2\\rfloor}$ eigenvalues of weight $n-1$, and its other eigenvalues are the integers $q^{k-1}$, $1\\le k\\le\\lfloor n/2\\rfloor$, with ballot numbers as multiplicities. The part of top weight forms a lisse sheaf $\\mathcal W_n$ of rank $r(n)$ on the torus $\\mathbb G_m^n$. We determine its local monodromy at the coordinate boundary and its determinant: $\\det\\mathcal W_n$ is the Artin--Schreier sheaf of the sum $-2^{n-1}λ^{-1}(h_1+\\cdots+h_n)$ of the critical values of the phase, twisted by $p^{(n+1)r(n)/2}$. The geometric monodromy group of $\\mathcal W_n$, and the arithmetic one after normalization to weight $0$, are equal to $\\{g\\in\\mathrm{GL}_{r(n)}:\\det(g)^p=1\\}$. Consequently the normalized sums become equidistributed like traces of Haar-random elements of $\\{g\\in\\mathrm U(r(n)):\\det(g)^p=1\\}$. The proofs use Laumon's Fourier transform, middle convolution, the Cayley cubic surface and Larsen's alternative, but no classification of finite groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03081",
  "author_total": 1,
  "authors": [
    {
      "name": "Hamed Ebadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7g4vw79b152jt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03081",
  "fulltext_url": null,
  "id": "7g4vw79b152jt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03081",
  "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.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7220408744",
  "pdf_url": "https://arxiv.org/pdf/2610.03081",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7g4vw79b152jt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.03081",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03081"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Cohomology of Linearly Constrained Kloosterman Families: Newton Polytopes, Weights, and Boundary Monodromy",
  "title_text": "Cohomology of Linearly Constrained Kloosterman Families: Newton Polytopes, Weights, and Boundary Monodromy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/7g4vw79b152jt",
  "venue": "arXiv",
  "versions": []
}