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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 $X$?

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