Gebhard Bockle, M. Harris, Chandrashekhar B. Khare, J. Thorne
{"title":"$\\hat{G}$-local systems on smooth projective curves are potentially automorphic","authors":"Gebhard Bockle, M. Harris, Chandrashekhar B. Khare, J. Thorne","doi":"10.4310/acta.2019.v223.n1.a1","DOIUrl":null,"url":null,"abstract":"Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\\mathbb{F}_q$. Its dual group $\\widehat{G}$ is a split reductive group over $\\mathbb{Z}$. Conjecturally, any $l$-adic $\\widehat{G}$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $\\pi_1(X) \\to \\widehat{G}(\\overline{\\mathbb{Q}}_l)$) should be associated to an everywhere unramified automorphic representation of the group $G$. \nWe show that for any homomorphism $\\pi_1(X) \\to \\widehat{G}(\\overline{\\mathbb{Q}}_l)$ of Zariski dense image, there exists a finite Galois cover $Y \\to X$ over which the associated local system becomes automorphic.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"1 1","pages":""},"PeriodicalIF":4.9000,"publicationDate":"2016-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"40","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/acta.2019.v223.n1.a1","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 40
Abstract
Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Its dual group $\widehat{G}$ is a split reductive group over $\mathbb{Z}$. Conjecturally, any $l$-adic $\widehat{G}$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $\pi_1(X) \to \widehat{G}(\overline{\mathbb{Q}}_l)$) should be associated to an everywhere unramified automorphic representation of the group $G$.
We show that for any homomorphism $\pi_1(X) \to \widehat{G}(\overline{\mathbb{Q}}_l)$ of Zariski dense image, there exists a finite Galois cover $Y \to X$ over which the associated local system becomes automorphic.