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?

  • $\begingroup$ 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? $\endgroup$ – user158636 Jul 28 '20 at 16:41
  • $\begingroup$ @crispr Yes, I'm aware of the function field analogy and the Mordell conjecture in that setting. $\endgroup$ – curious math guy Jul 28 '20 at 16:47
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    $\begingroup$ I guess my question could be clarified by asking "What is the complex analogue of a integer point?" $\endgroup$ – curious math guy Jul 28 '20 at 16:48
  • $\begingroup$ possibly something like this mathoverflow.net/q/52419 $\endgroup$ – user158636 Jul 28 '20 at 17:02

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