Relation between l-adic and l'-adic geometric monodromy

Question

Suppose $X$ is a smooth family of algebraic varieties over the base $B:=\mathbb{P}^1\backslash\lbrace0,1,\infty\rbrace$ over $\overline{\mathbb{Q}}$; then we can form the relative $l$-adic cohomology on the base, which will be (having fixed some cohomological degree of interest) an $l$-adic sheaf; we can think of it as a $\mathbb{Q}_l$ vector space $V_l$ equipped with a $\pi_1(B)$ action. (Algebraic $\pi_1$, of course.)

We can do the same with $l'$-adic cohomology for some other prime $l'$, getting a vector space $V_{l'}$ over $\mathbb{Q}_{l'}$ equipped with a $\pi_1(B)$ action.

As far as I can see, there ought to be a relatively tight connection between these two groups. For instance, it seems morally like we ought to be able to choose pro-generators $\alpha,\beta$ for $\pi_1(B)$, and find some matrices $M_\alpha,M_\beta$ over $\overline{\mathbb{Q}}$, such that, w.r.t suitable bases of $V_l,V_{l'}$, $\alpha$ acts via $M_\alpha$ on both $V_l$ and $V_{l'}$ (thinking of $M_\alpha$ as determining an $l$- and $l'$- adic matrix in turn) and similarly for $M_\beta$. Can this indeed be done? If not, can you do something close? If it can, what's a good reference for a precisely stated theorem---rather than just intuition---for these things?

Answer 1

As $\pi_1(B)$ is the same over $\overline{\mathbb Q}$ as over $\mathbb C$ we may embed $\overline{\mathbb Q}$ in $\mathbb C$ and then extend scalars to $\mathbb C$ to reduce to the case when $\mathbb C$ is the base field. In that case we can use the comparison theorem to conclude that both systems are obtained by scalar extension from the transcendentally defined system with $\mathbb Q$. Hence you can find the matrices you want even with entries in $\mathbb Q$.