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 $\mathbb Z[1/N]$?

  • $\begingroup$ 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 $> 84(g-1)$.) $\endgroup$ – Ariyan Javanpeykar Dec 12 '18 at 19:58
  • $\begingroup$ @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. $\endgroup$ – zzy Dec 12 '18 at 20:08
  • $\begingroup$ 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. $\endgroup$ – Ariyan Javanpeykar Dec 12 '18 at 20:15
  • $\begingroup$ 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.) $\endgroup$ – Ariyan Javanpeykar Dec 12 '18 at 20:16
  • $\begingroup$ @AriyanJavanpeykar Thank you!The reference is useful. $\endgroup$ – zzy Dec 12 '18 at 22:33

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