{
  "id": "1902.00699",
  "version": "v1",
  "published": "2019-02-02T11:36:31.000Z",
  "updated": "2019-02-02T11:36:31.000Z",
  "title": "Dilogarithm and higher $\\mathscr{L}$-invariants for $\\mathrm{GL}_3(\\mathbb{Q}_p)$",
  "authors": [
    "Zicheng Qian"
  ],
  "comment": "55 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E$ be a sufficiently large finite extension of $\\mathbb{Q}_p$ and $\\rho_p$ be a semi-stable representation $\\mathrm{Gal}(\\overline{\\mathbb{Q}_p}/\\mathbb{Q}_p)\\rightarrow\\mathrm{GL}_3(E)$ with a rank two monodromy operator $N$ and a non-critical Hodge filtration. We know that $\\rho_p$ has three $\\mathscr{L}$-invariants. We construct a family of locally analytic representations of $\\mathrm{GL}_3(\\mathbb{Q}_p)$ depending on three invariants in $E$ with each of them containing the locally algebraic representation determined by $\\rho_p$. When $\\rho_p$ comes from an automorphic representation $\\pi$ of $G(\\mathbb{A}_{\\mathbb{Q}_p})$ for a suitable unitary group $G_{/\\mathbb{Q}}$, we show that there is a unique object in the above family that embeds into the associated Hecke-isotypic subspace in the completed cohomology. We recall that Breuil constructed a family of locally analytic representations depending on four invariants and proved a similar result of local-global compatibility. We prove that if a representation $\\Pi$ in Breuil's family embeds into the completed cohomology, then it must equally embed into an object in our family determined by $\\Pi$. This gives a purely representation theoretic necessary condition for $\\Pi$ to embed into completed cohomology. Moreover, certain natural subquotients of each object in our family give a true complex of locally analytic representations that realizes the derived object $\\Sigma(\\lambda, \\underline{\\mathscr{L}})$ by Schraen for a unique $\\underline{\\mathscr{L}}$ determined by the object. Consequently, the family we construct gives a relation between the higher $\\mathscr{L}$-invariants studied by Breuil and Ding and the $p$-adic dilogarithm function which appears in the construction of $\\Sigma(\\lambda, \\underline{\\mathscr{L}})$ by Schraen.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-02T11:36:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariants",
      "locally analytic representations",
      "completed cohomology",
      "purely representation theoretic necessary condition",
      "sufficiently large finite extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}