Let $C$ be a smooth projective connected curve over $\mathbf{C}$.
Let $X$ be a curve over the function field of $C$.
Arakelov and Parshin proved the Mordell conjecture by considering a model for $X$ over $C$, i.e., a fibered surface $\mathcal{X}\to C$.
Are there any other applications or problems where one considers a model for $X$ over $C$ to obtain deep results about $X$?