{"title":"Higher $\\mathcal{L}$-invariants for $\\mathrm{GL}_3 (\\mathbb{Q}_p)$ and local-global compatibility","authors":"C. Breuil, Yiwen Ding","doi":"10.4310/cjm.2020.v8.n4.a2","DOIUrl":null,"url":null,"abstract":"Let $\\rho_p$ be a $3$-dimensional $p$-adic semi-stable representation of $\\mathrm{Gal}(\\overline{\\mathbb{Q}_p}/\\mathbb{Q}_p)$ with Hodge-Tate weights $(0,1,2)$ (up to shift) and such that $N^2\\ne 0$ on $D_{\\mathrm{st}}(\\rho_p)$. When $\\rho_p$ comes from an automorphic representation $\\pi$ of $G(\\mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real field $F^+$ which is compact at infinite places and $\\mathrm{GL}_3$ at $p$-adic places), we show under mild genericity assumptions that the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(\\mathbb{A}_{F^+}^\\infty)$ of arbitrary fixed tame level contain (copies of) a unique admissible finite length locally analytic representation of $\\mathrm{GL}_3(\\mathbb{Q}_p)$ which only depends on and completely determines $\\rho_p$.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"39 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2018-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Cambridge Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/cjm.2020.v8.n4.a2","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
Abstract
Let $\rho_p$ be a $3$-dimensional $p$-adic semi-stable representation of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ with Hodge-Tate weights $(0,1,2)$ (up to shift) and such that $N^2\ne 0$ on $D_{\mathrm{st}}(\rho_p)$. When $\rho_p$ comes from an automorphic representation $\pi$ of $G(\mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real field $F^+$ which is compact at infinite places and $\mathrm{GL}_3$ at $p$-adic places), we show under mild genericity assumptions that the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(\mathbb{A}_{F^+}^\infty)$ of arbitrary fixed tame level contain (copies of) a unique admissible finite length locally analytic representation of $\mathrm{GL}_3(\mathbb{Q}_p)$ which only depends on and completely determines $\rho_p$.