Assume first that $C$ is a curve, say over $\mathbb{Q}$ and $(E, \nabla)$ is a vector bundle with a flat connection. Assume further that $(E, \nabla)$ has regular singularities at $S=\overline{C}-C$. Then the cohomology $H^j_{dR}(C, (E, \nabla))$ is equipped with a Hodge structure.
There are many similarities between connections and $\ell$-adic sheaves. Assume now that $C$ is a curve over $\mathbb{F}_p$ and $F$ is a smooth $\ell$-adic sheaf on $C$ (BTW $\mathcal{F}$ is pretty ugly here...). Then one can look at the cohomology $H^j(C_{\overline{\mathbb{F}}_q},F)$ together with the action of Frobenius. By Deligne there is a notion of weight filtration (and one does not need to assume tame ramification if I am not mistaken...).
Is there something (even vaguely) similar to a Hodge filtration that one can get?
