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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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Why is every elliptic curve over a proper base over $\mathbf{F}_q$ isotrivial? 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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Why is every elliptic curve over a proper base over $\mathbf{F}q$ \mathbf{F}_q$ isotrivial? If the moduli space of elliptic curves $\mathbf{A}^1\mathbf{Z}$ \mathbf{A}^1_\mathbf{Z}$ were fine, it would be clear to me.
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