Is Hom scheme between projective curves of large genus finite etale?

Let $$K$$ be a number field, $$T$$ be a finite set containing some finite places of $$K$$, and $$S=\operatorname{Spec} O_{K,T}$$. If $$X,Y$$ are two projective smooth curves over $$S$$ with genus large than $$1$$ and $$0$$ respectively. Then is the Hom scheme $$Hom(X,Y)$$ finite etale over $$S$$?

Motivation: Let $$E$$ be an elliptic curve over $$\mathbb Q$$, then $$E$$ can be defined over $$\mathbb Z[1/N]$$ where $$N$$ is the conductor of $$E$$. We know there is a non-constant map $$\phi:X_0(N) \rightarrow E$$ over $$\mathbb Q$$ where the modular curve $$X_0(N)$$ is also defined over $$\mathbb Z[1/N]$$, then can $$\phi$$ also be defined over $$\mathbb Z[1/N]$$?

• If $Y$ is an elliptic curve, then this is false. Let $f:X\to Y$ be a morphism. Now, any isogeny $\phi:Y\to Y$ defines a new morphism $\phi \circ f:X\to Y$ and the degree of this map increases. Thus, $Hom(X,Y)$ is not even finite etale over $\mathbb{C}$. Anyway, if genus(Y) > 1, then $Hom(X,Y)$ is finite (assuming $X$ and $Y$ are projective smooth curves of genus >1). It is not etale in general, as there are maps $X\to Y$ over $\mathbb{F}_p$ which can't be lifted in general. (Hint: find a curve $X$ and an automorphism over $\mathbb{F}_p$ of order $> 84(g-1)$.) – Ariyan Javanpeykar Dec 12 '18 at 19:58
• @AriyanJavanpeykar Oh, thank you! What I thought before is if $Hom$ is finite etale over $O_{K,T}$, the existence of a $K$ point will imply $Hom$ has an irreducible component isomorphic to $O_{K,T}$ thus $Hom$ has a $O_{K,T}$ point. – zzy Dec 12 '18 at 20:08
• You're welcome. I think that the hom-scheme will complicate things here a bit. It is not of finite type over $O_{K,T}$. However, the hom-scheme satisfies the valuative criterion for properness. Indeed, any morphism $X_K\to Y_K$ with $X$ and $Y$ smooth proper curves of genus $>0$ over $O_{K,T}$ extends (uniquely) to a morphism $X\to Y$. You can read Liu-Tong's paper on Neron models for a proof. – Ariyan Javanpeykar Dec 12 '18 at 20:15
• What you ask in your motivation has nothing to do with the "finite etaleness" of the Hom-scheme. The extension you desire exists by the Neron mapping property of a smooth proper model of E over ℤ[1/N] (and the fact that X0(N) is smooth over ℤ[1/N]). (Thus: the hom-scheme satisfies the valuative criterion of properness. It is not necessarily proper though, because it is not of finite type.) – Ariyan Javanpeykar Dec 12 '18 at 20:16
• @AriyanJavanpeykar Thank you！The reference is useful. – zzy Dec 12 '18 at 22:33