{
  "abstract": "For a finite family of representations of a reductive group, we define a Laurent polynomial on the group. The exponential sum associated this Laurent polynomial is called a hypergeometric exponential sum. We introduce an arithmetic hypergeometric $\\mathcal D$-module to study the hypergeometric exponential sum. It is an overholonomic arithmetic $\\mathcal D$-module with a Frobenius structure so that the trace of the Frobenius at a rational point is the exponential sum. We prove that over the locus where the Laurent polynomial is nondegenerate, the arithmetic hypergeometric $\\mathcal {D}$-module defines an $F$-isocrystal overconvergent along the degenerate locus. As an application, we get an estimation of the hypergeometric exponential sum.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.00470",
  "author_total": 3,
  "authors": [
    {
      "name": "Lei Fu"
    },
    {
      "name": "Xuanyou Li"
    },
    {
      "name": "Chenhan Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a0sy20nzkwmy0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.00470",
  "fulltext_url": null,
  "id": "a0sy20nzkwmy0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.00470",
  "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",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7172454397",
  "pdf_url": "https://arxiv.org/pdf/2608.00470",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a0sy20nzkwmy0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2608.00470",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.00470"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Arithmetic hypergeometric $\\mathcal {D}$-modules and exponential sums on reductive groups",
  "title_text": "Arithmetic hypergeometric 𝒟-modules and exponential sums on reductive groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/a0sy20nzkwmy0",
  "venue": "arXiv",
  "versions": []
}