{
  "abstract": "Using recent advances in the integral canonical models of Shimura varieties using syntomic methods, we give an extension of the Langlands--Kottwitz--Scholze method from Scholze's paper on the Langlands--Kottwitz method for deformation spaces of $p$-divisible groups. Namely, we define analogues $ϕ^{\\mathcal{G},μ}_{τ,h}$ of the local test functions from op. cit. in full generality, show they satisfy reasonable harmonic-analytic properties, and give rise to trace formulae for the Galois-Hecke action on the cohomology of Shimura varieties of bad reduction. Additionally, we use these new local test functions and the work of Fargues--Scholze to state a more robust and unconditional version of the Scholze--Shin conjecture. Along the way we study the $(\\mathcal{G},μ)$-apertures of Drinfeld and Gardner--Madapusi, in particular showing that such $(\\mathcal{G},μ)$-apertures carry no more information than a Tannakian version of Fontaine--Laffaille theory, at least over nice bases, and give a very explicit description of the universal deformations of $(\\mathcal{G},μ)$-apertures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30537",
  "author_total": 1,
  "authors": [
    {
      "name": "Alex Youcis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a5fppcz6v8krm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30537",
  "fulltext_url": null,
  "id": "a5fppcz6v8krm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30537",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214631726",
  "pdf_url": "https://arxiv.org/pdf/2609.30537",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a5fppcz6v8krm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30537",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30537"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Langlands--Kottwitz--Scholze method",
  "title_text": "On the Langlands--Kottwitz--Scholze method",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/a5fppcz6v8krm",
  "venue": "arXiv",
  "versions": []
}