Let $X\to S$ be a curve over $S=\mathrm{Spec} \ \mathbf{Z}$. Let $\omega$ be the relative dualizing sheaf of $X\to S$. Let $g$ be the genus of the generic fibre. Assume that $g\geq 2$.

I know that $\omega$ is ample. Even better, a theorem of Deligne and Mumford says that $\omega^{\otimes 3}$ is very ample.

So there is some integer $n $ depending on $X\to S$ such that $H^1(X,\omega^{\otimes m}) =0$ for all $m\geq n$.

Can we find such an $n$? Is it independent of $g$? Does $n=3$ work?

If not, can we find such an $n$ and bound it in terms of $g$?