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Why is every elliptic curve over a proper (edit: smooth and geometrically connected) base over $\mathbf{F}_q$ isotrivial, i.e. is constant after base changing with $\bar{\mathbf{F}}_q$? If the moduli space of elliptic curves $\mathbf{A}^1_\mathbf{Z}$ were fine, it would be clear to me.

It probably follows by considering the functor $\mathcal{M}_{1,1} \to \mathbf{A}^1_\mathbf{Z}$ in some way.

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It's false when the base is non-reduced (e.g., dual numbers over a field). You want the base to be smooth and geometrically connected, not just proper. It would be good if you indicated in the question precisely how you are defining isotriviality (in case your problem is caused by using a definition which is too restrictive). –  Boyarsky Jul 1 '10 at 19:22
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The coarse moduli space is a canonical map from $\mathcal{M}_{Ell}$ to the $j$-line, and the properties of your base imply its image in the $j$-line is a point. Since the $j$-invariant classifies elliptic curves up to isomorphism, all $\overline{\mathbb{F}_q}$-fibers of the family are isomorphic. –  S. Carnahan Jul 1 '10 at 19:57
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The fiberwise definition of isotrivial in Scott's comment is not the one in the question, and is too weak; the definition in the revised question is too strong. For a smooth proper geometrically connected curve $X$ over a field $k$ so that there is a geometrically connected etale double cover $X' \rightarrow X$, the corresponding quadratic twist $\mathcal{E}$ of a constant elliptic curve $E \times X$ is a new elliptic curve over $X$ which is not isotrivial in the sense of the OP, but it is isotrivial in the right sense: becomes constant over an etale cover of the base (namely $X'$!). –  Boyarsky Jul 1 '10 at 20:21

2 Answers 2

I give an argument using the correct definition of isotriviality (relating it to the weak definition), and the reader who is familiar with Deligne-Mumford stacks can see that the method is actually applicable quite generally to separated Deligne-Mumford stacks locally of finite presentation over an algebraically closed field (these hypotheses ensure the existence of a coarse moduli space; perhaps it can all be done more generally).

Let $k$ be an algebraically closed field and $E \rightarrow S$ an elliptic curve over a reduced $k$-scheme $S$ such that $j:S \rightarrow \mathbb{A}^1_k$ is a constant map to some $j_0 \in k$. Let $E_0$ be an elliptic curve over $k$ with $j$-invariant $j_0$. We claim there is an etale cover $S' \rightarrow S$ such that $E_{S'} \simeq (E_0)_{S'} := (E_0) \times_{{\rm{Spec}}(k)} S'$. In other words, the weak notion of isotriviality implies the right notion when $S$ is reduced. This will proceed by "standard" EGA-style reduction steps (and the proof will show by conceptual reasons that the reducedness hypothesis on $S$ cannot be dropped when $j_0 = 0, 1728$).

Consider the functor on $S$-schemes ${\rm{Isom}}(E,(E_0)_S)$ whose $T$-points are ${\rm{Isom}}_T(E_T, (E_0)_T)$ for an $S$-scheme $T$. This is represented by a finite type unramified $S$-scheme, and we just need to check that it is etale surjective over $S$ (so then we can take it to be $S'$, over which there is a "universal isomorphism"). We can assume $S$ is affine, then by limit arguments we can assume it is finite type over $k$, and then local (and essentially finite type over $k$) with residue field $k$ provided we prove the etale property for the Isom-scheme over the closed point. We can pass to the completion, so now the problem is rather concrete: if $R$ is a reduced complete local noetherian $k$-algebra with residue field $k$ and if $E$ is an elliptic curve over $R$ with special fiber $E_0$ such that $j(E) \in R$ is equal to $j_0 := j(E_0)$ then we claim that $E \simeq (E_0)_R$ (so the Isom-scheme of interest is the base change to $R$ of the Aut-scheme of $E_0$, so it is etale over $R$ as desired).

Let $A$ denote the universal deformation ring of $E_0$, so $E$ corresponds to a local $k$-algebra map $A \rightarrow R$ which we want to show kills the maximal ideal of $A$. Since $R$ is reduced, it suffices to show that some power of the maximal ideal of $A$ dies in $R$. Let $G_0 = {\rm{Aut}}_k(E_0)$ be the finite automorphism group of $E_0$, so the completed local ring $O$ at $j_0$ on the coarse moduli space $\mathbb{A}^1_k$ is the ring $A^{G_0}$. (I am omitting the rigorous proof of the "well-known" fact that $\mathbb{A}^1_k$ is the coarse moduli space for the moduli stack of elliptic curves in a good sense which includes this description of the completed local ring at geometric points. If such a proof is unclear then someone can always post an MO query about it. This link with deformation rings is the heart of the argument, as will soon become clear.) The map $A^{G_0} \rightarrow A$ is finite and local (since $G_0$ is a finite group), so the maximal ideal of $A^{G_0}$ generates a max-primary ideal of $A$. By hypothesis $A^{G_0} \rightarrow R$ kills the maximal ideal, whence $A \rightarrow R$ kills a power of the maximal ideal. We have seen that this suffices for our needs (due to the reducedness of $R$). QED

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Maybe one could also give an argument along the following lines:

Given an elliptic curve $E$ over some base scheme $S$ (edit: proper reduced and connected) over some field $k$. Choose an integer $N$ such that $N \ge 3$ and $N$ is prime to $p$.. The group scheme $E[N]$ is finite and etale over $S$, so we find an connected finite etale covering $S' \to S$ such that there exits an isomorphism $\alpha \colon E[N] \times_S S' \simeq (Z/n)^2$. We say $\alpha$ is a level-$N$ structure on $E' = E \times_S S'$. By basic results form Katz and Mazur, there exists a fine moduli space of elliptic curves with level-$N$ structures: $M[\Gamma(N)]$ with a universal family $(E^{univ} \to M[\Gamma(N)], \alpha^{univ})$.

However, $M[\Gamma(N)]$ is affine, so the map $S' \to M[\Gamma(N)]$ given by the pair $(E, \alpha)$ factors over the affine hull of $S'$, i.e. $Spec(k')$ where $k'$ is a finite extension of $k$.

It follows $E' \simeq E^{univ} \times_{M[\Gamma(N)]} Spec(k')$, thus we get a product family after etale basechange.

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Holger, you want to replace $S'$ with $S$ without losing initial hypotheses, so you must assume $S$ is proper, reduced and connected: then by taking $S'$ to be a connected component of the finite etale $S$-scheme of full level-$N$ structures you can rename $S'$ as $S$. Then you can conclude as you do. This argument is certainly more elementary than the one I gave using deformation theory. To allow more general $S$ by imposing fibral hypotheses on $E \rightarrow S$, or to handle other moduli problems, the deformation method works well. –  Boyarsky Jul 2 '10 at 15:10
    
Thank you. I hope I edited the answer accordingly. –  Holger Partsch Jul 4 '10 at 9:54

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