Is Hom scheme between projective curves of large genus finite etale?
2
$begingroup$
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 $mathbb Z[1/N]$?
number-theory algebraic-geometry arithmetic-geometry
asked Dec 9 '18 at 19:08
zzyzzy
2,6431420
$endgroup$
add a comment |
2
$begingroup$
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 $mathbb Z[1/N]$?
number-theory algebraic-geometry arithmetic-geometry
asked Dec 9 '18 at 19:08
zzyzzy
2,6431420
$endgroup$
$begingroup$
If $Y$ is an elliptic curve, then this is false. Let $f:Xto Y$ be a morphism. Now, any isogeny $phi:Yto Y$ defines a new morphism $phi circ f:Xto Y$ 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 $Xto Y$ 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 $X_Kto Y_K$ with $X$ and $Y$ smooth proper curves of genus $>0$ over $O_{K,T}$ extends (uniquely) to a morphism $Xto Y$. 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
add a comment |
2
2
2
1
$begingroup$
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 $mathbb Z[1/N]$?
number-theory algebraic-geometry arithmetic-geometry
asked Dec 9 '18 at 19:08
zzyzzy
2,6431420
$endgroup$
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 $mathbb Z[1/N]$?
number-theory algebraic-geometry arithmetic-geometry
number-theory algebraic-geometry arithmetic-geometry
asked Dec 9 '18 at 19:08
zzyzzy
2,6431420
asked Dec 9 '18 at 19:08
zzyzzy
2,6431420
edited Dec 10 '18 at 20:52
zzy
asked Dec 9 '18 at 19:08
zzyzzy
2,6431420
asked Dec 9 '18 at 19:08
zzyzzy
2,6431420
asked Dec 9 '18 at 19:08
zzyzzy
2,6431420
2,6431420
$begingroup$
If $Y$ is an elliptic curve, then this is false. Let $f:Xto Y$ be a morphism. Now, any isogeny $phi:Yto Y$ defines a new morphism $phi circ f:Xto Y$ 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 $Xto Y$ 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 $X_Kto Y_K$ with $X$ and $Y$ smooth proper curves of genus $>0$ over $O_{K,T}$ extends (uniquely) to a morphism $Xto Y$. 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
add a comment |
$begingroup$
If $Y$ is an elliptic curve, then this is false. Let $f:Xto Y$ be a morphism. Now, any isogeny $phi:Yto Y$ defines a new morphism $phi circ f:Xto Y$ 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 $Xto Y$ 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 $X_Kto Y_K$ with $X$ and $Y$ smooth proper curves of genus $>0$ over $O_{K,T}$ extends (uniquely) to a morphism $Xto Y$. 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
$begingroup$
If $Y$ is an elliptic curve, then this is false. Let $f:Xto Y$ be a morphism. Now, any isogeny $phi:Yto Y$ defines a new morphism $phi circ f:Xto Y$ 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 $Xto Y$ 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
$begingroup$
If $Y$ is an elliptic curve, then this is false. Let $f:Xto Y$ be a morphism. Now, any isogeny $phi:Yto Y$ defines a new morphism $phi circ f:Xto Y$ 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 $Xto Y$ 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
$begingroup$
@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 $X_Kto Y_K$ with $X$ and $Y$ smooth proper curves of genus $>0$ over $O_{K,T}$ extends (uniquely) to a morphism $Xto Y$. You can read Liu-Tong's paper on Neron models for a proof.
$endgroup$
– Ariyan Javanpeykar
Dec 12 '18 at 20:15
$begingroup$
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 $X_Kto Y_K$ with $X$ and $Y$ smooth proper curves of genus $>0$ over $O_{K,T}$ extends (uniquely) to a morphism $Xto Y$. 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
$begingroup$
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
$begingroup$
@AriyanJavanpeykar Thank you！The reference is useful.
$endgroup$
– zzy
Dec 12 '18 at 22:33
add a comment |
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$begingroup$
If $Y$ is an elliptic curve, then this is false. Let $f:Xto Y$ be a morphism. Now, any isogeny $phi:Yto Y$ defines a new morphism $phi circ f:Xto Y$ 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 $Xto Y$ 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 $X_Kto Y_K$ with $X$ and $Y$ smooth proper curves of genus $>0$ over $O_{K,T}$ extends (uniquely) to a morphism $Xto Y$. 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