3 problems

Let $X/\mathbb{F}_{q}$ be a smooth variety, and let $\mathcal{F}$ be an irreducible coefficient object on $X$ with algebraic determinant. Companions conjecture. There exists a numb…

Mixedness conjecture. Every $\overline{\mathbb{Q}}_{\ell}$-coefficient object on $X_0$ is $\iota$-mixed.

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…