{
  "abstract": "Let $p$ be an odd prime, and let $V_1$ and $V_2$ be symplectic self-dual $p$-adic de Rham representations of $G_{\\mathbb{Q}}$ that are pure (of weight $-1$) and satisfy the Panchishkin condition at $p$. Suppose that $V_1$ and $V_2$ are residually isomorphic, compatibly with their Panchishkin filtrations. We prove the relative $p$-parity conjecture relating the ratio of the global root numbers of $V_1$ and $V_2$ to the difference of their Bloch-Kato Selmer dimensions. We have no restriction on the dimension of the Galois representation, the Hodge-Tate weights or the monodromy operator at $p$, and the residual isomorphism need not be symplectic. The proof expresses ratios of local root numbers in terms of local invariants over the cyclotomic $\\mathbb{Z}_p$-extension: a new invariant introduced at $p$, and local Iwasawa $λ$-invariants at $\\ell\\neq p$. Both are compatible with the arithmetic local constants of Mazur-Rubin and Nekovář. A key input, proved using Sen theory, is that the Bloch-Kato and strict local conditions at $p$ coincide over the cyclotomic $\\mathbb{Z}_p$-extension, generalising results of Coates-Greenberg and Nekovář. As applications, we obtain new cases of the $p$-parity conjecture, including cases of rank greater than two: Artin twists of CM Hida families, orthogonal Artin twists of ordinary modular forms, symmetric cubes along Hida families, and families of abelian surfaces, together with unconditional numerical examples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08316",
  "author_total": 3,
  "authors": [
    {
      "name": "Subham Garai"
    },
    {
      "name": "Somnath Jha"
    },
    {
      "name": "Sudhanshu Shekhar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2t4ynhmrpc31r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08316",
  "fulltext_url": null,
  "id": "2t4ynhmrpc31r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08316",
  "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.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08316",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2t4ynhmrpc31r.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08316",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08316"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$p$-parity conjecture for Panchishkin Galois representations",
  "title_text": "p-parity conjecture for Panchishkin Galois representations",
  "type": "preprint",
  "updated_at": "2026-11-08T18:20:59Z",
  "url": "https://research.draconitic.com/paper/2t4ynhmrpc31r",
  "venue": "arXiv",
  "versions": []
}