The Mordell conjecture/Falting's Theorem says that any smooth projective curve $X$ of genus $g\geq 2$ over $\mathbb{Q}$ has finitely many integer points (using the valuatlive criterion).

We can of course apply the analytification functor then the sturcture morphism of $X$ becomes $$X(\mathbb{C})^{an}\rightarrow \text{Spec}(\mathbb{Z})(\mathbb{C})^{an}$$ and an integer point corresponds to a section of this morphism. However $\text{Spec}(\mathbb{Z})(\mathbb{C})^{an}$ is just a point, so this is not an interesting geometric object. However, on an intuitive level, the specta of Dedekind domains correspond to $1$-dimensional curves.

My question therefore is if there exists a (complex) geometric interpretation of the Mordell Conjecture?

The most naive interpretation I have and can thus hope for this would be that there exist only finitely many maps $$\mathbb{C}\rightarrow X(\mathbb{C})^{an}.$$ Is there anything in that direction?