{
  "abstract": "Heyer's left adjoint to derived parabolic induction gives rise to higher mod $p$ Satake transforms \\[ \\mathcal S_V^i: \\mathcal H_G(V) \\longrightarrow \\mathcal H_M\\bigl(H^i(K_U,V)\\bigr), \\] whose extreme degrees recover Herzig's invariant and coinvariant transforms. For $G=\\mathbf G(F)$, where $\\mathbf G$ is a connected reductive group defined over a finite extension $F/\\mathbb{Q}_p$, we describe these maps explicitly using Mackey decompositions and corestriction in group cohomology. For $G=\\mathrm{GL}_n(F)$, we evaluate them on the standard Hecke generators, showing that their support lies in a single Weyl orbit and that their values on the individual support cosets are concrete corestriction operators. For $G=\\mathrm{GL}_2(F)$, we determine the image in every cohomological degree for each non-Steinberg Serre weight. The endpoint images are the classical polynomial Satake images, while every interior image is of the form \\[ k[x,y^{\\pm1}]/(x^r) \\] for some positive integer $r$. Thus, every interior image is a finite nilpotent thickening of the Laurent polynomial algebra $k[y^{\\pm1}]$. We give explicit bounds for its thickness, prove that they are sharp, and show that in the unramified case every interior image is either reduced or a square-zero thickening.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13203",
  "author_total": 4,
  "authors": [
    {
      "name": "Hargun Bhatia"
    },
    {
      "name": "Yashi Jain"
    },
    {
      "name": "Casey Perdue"
    },
    {
      "name": "Ben Savoie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2fkmkkjh9gm6t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13203",
  "fulltext_url": null,
  "id": "2fkmkkjh9gm6t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13203",
  "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.NT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7221546333",
  "pdf_url": "https://arxiv.org/pdf/2610.13203",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2fkmkkjh9gm6t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13203",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13203"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Higher mod $p$ Satake transforms: explicit formulas and images for $\\mathrm{GL}_2$",
  "title_text": "Higher mod p Satake transforms: explicit formulas and images for GL_2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/2fkmkkjh9gm6t",
  "venue": "arXiv",
  "versions": []
}