{
  "abstract": "Let $q=p^a$ with $p$ prime, $λ\\in\\mathbb F_p^\\times$, and $ψ_q=ψ\\circ\\mathrm{Tr}_{\\mathbb F_q/\\mathbb F_p}$ for a nontrivial additive character $ψ$ of $\\mathbb F_p$. For $n\\ge2$ we study $Σ_n(h)=\\sum_{u\\in(\\mathbb F_q^\\times)^n,\\sum_i h_i u_i=-λ}ψ_q(\\sum_i u_i^{-1})$. On $\\mathbb G_m^n$ we construct a lisse sheaf $\\mathcal W_n$, pure of weight $n+1$ and rank $r(n)=2^n-\\binom n{\\lfloor n/2\\rfloor}$, whose Frobenius traces equal $(-1)^{n-1}qΣ_n(h)$ minus an explicit polynomial in $q$, for $h\\in(\\mathbb F_q^\\times)^n$. For odd $p$, its geometric monodromy has identity component $\\mathrm{SL}_{r(n)}$ and cyclic component group of order divisible by $p$. The full group is $\\{g:\\det(g)^p=1\\}$ for $n=2$, and for $n\\ge3$ using boundary unipotence from the companion preprint arXiv:2610.03081. For $p=2$, the sheaf is self-dual and its geometric monodromy is $\\mathrm{SO}_{r(n)}$ for odd $n$ and $\\mathrm{Sp}_{r(n)}$ for even $n$. After normalization to weight zero, the arithmetic and geometric groups coincide, using the same companion input for odd $p$ and $n\\ge3$. As $q\\to\\infty$ with $p$ fixed, we obtain Sato-Tate laws for centered, normalized sums with error $O(q^{-1/2})$ for polynomial tests and formulas for all power moments. For $p=2$, limiting moments are Gaussian through order $r(n)-1$, with an excess of one above the orthogonal value at order $r(n)$ for odd $n$; if $n+1$ is a power of two, every Frobenius on $\\mathcal W_n$ has eigenvalue $q^{(n+1)/2}$. The proof uses $\\sum_h|Σ_n(h)|^4=c_pq^{3n-2}+O(q^{3n-2-δ(n)})$ for $n\\ge3$, where $c_p=2$ for odd $p$, $c_2=3$ and $δ(n)=\\min(1,(n-2)/2)$, plane sections of the Cayley cubic, exact Frobenius-Schur identities, Laumon's local Fourier transforms and Artin-Schreier theory. For $n=2$ we identify twisted classical Kloosterman sums and compute the fourth moment exactly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04099",
  "author_total": 1,
  "authors": [
    {
      "name": "Hamed Ebadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3md9ejfw93dv2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04099",
  "fulltext_url": null,
  "id": "3md9ejfw93dv2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04099",
  "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"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7220673130",
  "pdf_url": "https://arxiv.org/pdf/2610.04099",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3md9ejfw93dv2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.04099",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04099"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Moments and Large Geometric Monodromy for Linearly Constrained Kloosterman Families",
  "title_text": "Moments and Large Geometric Monodromy for Linearly Constrained Kloosterman Families",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/3md9ejfw93dv2",
  "venue": "arXiv",
  "versions": []
}