Let $X$ be a smooth scheme of finite type over $\mathbb{Z}$ (or let's say a finitely generated $\mathbb{Z}$ algebra). To each prime $p \in \mathbb{Z}$ we can consider the $\mathbb{F}_p$ variety $$X_{\mathbb{F}_p}=X \times_{\mathbb{Z}} \mathbb{F}_p$$ and the $\overline{\mathbb{F}_p}$ variety $$X_{\overline{\mathbb{F}_p}}=X \times_{\mathbb{Z}} \overline{\mathbb{F}_p}=X_{\mathbb{F}_p}\times_{\mathbb{F}_p} \overline{\mathbb{F}_p}$$ (I'm omitting spec everywhere to lighten the notations a little bit).

In this way, the etale cohomology $H^{*}(X_{\overline{\mathbb{F}_p}},\overline{\mathbb{Q}}_{\ell})$ gets endowed with a Frobenius morphism and we can consider the associated increasing weight filtration $W^{i}_m$.

If one now considers $X_{\mathbb{C}}=X \times_{\mathbb{Z}}\mathbb{C}$ this is a smooth complex algebraic variety. We now from comparison theorems that $$H^{*}_{etale}(X_{\mathbb{C}},\overline{\mathbb{Q}}_{\ell})=H^{*}_{simp}(X_{\mathbb{C}}(\mathbb{C}),\overline{\mathbb{Q}}_{\ell}) $$ where on the right side we have the usual simplicial cohomology.

We moreover know that for $p >>>1$ we will have $$H^{*}_{etale}(X_{\mathbb{C}},\overline{\mathbb{Q}}_{\ell}) \cong H^{*}(X_{\overline{\mathbb{F}_p}},\overline{\mathbb{Q}}_{\ell}) .$$ In this way, we can endow the cohomology $H^{*}_{simp}(X_{\mathbb{C}}(\mathbb{C}),\overline{\mathbb{Q}}_{\ell}) $ and so the cohomology with complex coefficients $H^{*}_{simp}(X_{\mathbb{C}}(\mathbb{C}),\mathbb{C}) $ with a weight filtration which I'm denoting $W^{*}_{m,p}$.

On the other side, Deligne's theory of mixed Hodge structures, endowes $H^{*}_{simp}(X_{\mathbb{C}}(\mathbb{C}),\mathbb{C})$ with another weight filtration $W^{*}_{m}$. Do these two filtrations coincide in general? Does the filtration $W_{m,p}$ depends on the prime chosen?

I know that there are some invariants which relate the two filtrations. One can define the E-polynomial for example $$E_p(X_{\mathbb{C}},q)=\sum_{m,k}(-1)^k dim \frac{W^{k}_{m,p}}{W^{k}_{m-1,p}}q^{m} $$ and analogously $$E(X_{\mathbb{C}},q)=\sum_{k}(-1)^k\sum_{i+j=r} h^{i,j;k}q^r $$ where $$h^{i,j;k}=dim Gr_W^{i+j} Gr_F^{i} H^{k}(X_{\mathbb{C}},\mathbb{C})$$ are the Hodge numbers.

One can show that $E_p(X_{\mathbb{C}},q)=E(X_{\mathbb{C}},q)$ using the additivity with respect to locally closed decompisition of both polynomials and the statement for projective varieties which is true as everything is pure.