# Complex Geometric Interpretation of Mordell conjecture

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?

• it's true that there are finitely many maps from $\mathbb{C}$ to a closed Riemann surface of genus $g\geq 2$. Have you read about the Mordell conjecture over function fields? – user158636 Jul 28 '20 at 16:41
• @crispr Yes, I'm aware of the function field analogy and the Mordell conjecture in that setting. – curious math guy Jul 28 '20 at 16:47
• I guess my question could be clarified by asking "What is the complex analogue of a integer point?" – curious math guy Jul 28 '20 at 16:48
• possibly something like this mathoverflow.net/q/52419 – user158636 Jul 28 '20 at 17:02