Comparing Frobenius weights with Mixed Hodge theory

Question

For a variety over a finite field Deligne define weights by looking at the eigenvalues of the Frobenius. On the other hand, if we take a variety over $\mathbb{Q}$, at least for its constant sheaf we have the weight filtration which gives the mixed Hodge structure. How are these two comparable? Can someone point out some references?

Say we have a variety $X$ defined over $\mathbb{Z}[\frac{1}{p}]$. We can consider $p$-etale sheaves on $X$, or rather, the bounded derived category of constructible sheaves. If $F$ belongs to this category, for each prime $q\neq p$ we get a complex in the derived category of the fiber, which is a variety over a finite field, for which we have a weight filtration. Could we patch together these weights theories to get a weight filtration on $F$? Then I would like this weight filtration to induce the filtration of Mixed Hodge theory after restriction to generic fiber and extension of scalars.

Thank you very much.

EDIT: I should specify I want $F$ to induce a mixed sheaf on the fiber over each $q\neq p$.

Answer 1

For constant coefficients, the comparison statement, along with a sketch, appears in Deligne's ICM talk, Poids dans la cohomologie....

For things to work the way you seem to want in your second paragraph, you're going to need a lot more structure for $F$ than what you've given. At the very least, when $F$ is a local system (lisse), you need the weights with respect to the various Frobenius's, at different primes, to be compatible, and you would also need to specify an admissible variation of mixed Hodge structures on $X(\mathbb{C})$ plus appropriate compatibilities with the Hodge theoretic weights. Take a look at Saito's paper Arithmetic mixed sheaves in Inventiones for a general framework.