Compatibility of canonical $\ell$-adic local systems on Shimura varieties

Abstract

For a Shimura variety $(G,X)$ in the superrigid regime and neat level subgroup $K_0$, we show that the canonical family of $\ell$-adic representations associated to a number field point $y\in {\mathrm{Sh}}_{K_0}(G,X)(F)$, \[ \left\{ ρ_{y, \ell} \colon \mathrm{Gal}(\overline{\mathbb{Q}}/F) \to G^{\mathrm{ad}}(\mathbb{Q}_{\ell}) \right\}_{\ell}, \] form a compatible system of $G^{\mathrm{ad}}({\mathbb{Q}}_{\ell })$-representations: there is an integer $N(y)$ such that for all $\ell$, ${\rho }_{y,\ell }$ is unramified away from $N(y)\ell$, and for all $\ell \ne {\ell }^{\prime }$ and $v\nmid N(y)\ell {\ell }^{\prime }$, the semisimple parts of the conjugacy classes of ${\rho }_{y,\ell }({\mathrm{Frob}}_v)$ and ${\rho }_{y,{\ell }^{\prime }}({\mathrm{Frob}}_v)$ are ($\mathbb{Q}$-rational and) equal. We deduce this from a stronger compatibility result for the canonical $G({\mathbb{Q}}_{\ell })$-valued local systems on connected Shimura varieties inside ${\mathrm{Sh}}_{K_0}(G,X)$. Our theorems apply in particular to Shimura varieties of non-abelian type and represent the first such independence-of-$\ell$ results in non-abelian type.

arXiv commentAt referee's advice, the preliminary section on tame specialization has been deleted, and the abstract results for superrigid local systems have been presented in less generality. A few small corrections (especially the end of Prop 2.3) and other minor expository changes have been made