$\hat{G}$-local systems on smooth projective curves are potentially automorphic
$\hat{G}$-local systems on smooth projective curves are potentially automorphic
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DOI：
10.4310/acta.2019.v223.n1.a1
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发表时间：
2016-09
期刊：
影响因子：
3.7
通讯作者：
Gebhard Bockle;M. Harris;Chandrashekhar B. Khare;J. Thorne
中科院分区：
文献类型：
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作者：
Gebhard Bockle;M. Harris;Chandrashekhar B. Khare;J. Thorne
设$X$是有限域$\mathbb{F}_q$上的光滑的、射影的、几何连通的曲线，$G$是$\mathbb{F}_q$上的分裂半单代数群.它的对偶群$\widehat{G}$是$\mathbb{Z}$上的分裂约化群。猜想，$X$(等价于连续同态的任何共轭类$\pi_1(X)\to\widehat{G}({\mathbb{q}_L)$)上的$X$上的任何$X$-adi$\widehat{G}$-局部系统应与群$G$的处处未分枝自同构表示相关联.证明了对于Zariski稠密映象的任一同态{pi_1(X)to宽{G}(overline{mathbb{q}}_L)$，存在一个有限伽罗瓦覆盖$Y\to X$，相应的局部系统在该覆盖上变得自同构.
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.