2 problems

Companion conjecture. Any algebraic $\ell$-adic coefficient object on $X$ admits an $\ell'$-adic companion.

Deligne's conjecture. The following statements hold: $\mathcal{E}_0$ is pure of weight $0$; it is $E$-rational for some number field $E\subseteq\overline{\mathbb{Q}}_{\ell}$; it is…