Let $V$ be a quasi-projective algebraic curve over a field $k$. Are there finitely many morphisms from $S$ to the triply-punctured sphere $\mathbb{P}^1 -\{0,1,\infty\}$? This is true if $k = \mathbb{C}$ as shown in page 478 of this paper http://www.math.harvard.edu/~ctm/papers/home/text/papers/families/families.pdf
I have heard someone say that this is true in the case of arbitrary $k$ and can be proved using Siegel's Theorem for function fields, but I don't have any idea on how to do so. Any help would be appreciated!
Edit: I have been able to reduce this question to the following: are there finitely many points in $\mathbb{P}^1-\{0,1,\infty\}$ defined over $O(V)$? This should be an analog of Siegel's Theorem for genus 0 curves over function fields; however, I am not able to find any sources regarding whether this analog is true. Any help would be appreciated!
Edit: I found out that this is actually false for characteristic non-zero. We can only conclude this using characteristic 0 using the ABC conjecture for function fields, but for characteristic >0, the finiteness statement can only be made for separable morphisms and is in general false for arbitrary morphisms because of composition with powers of the Frobenius.