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Let $k$ be a field of characteristic not 2 or 3. Then the set of elliptic curves over $k$ can be parametrized by the affine variety $S=D(4a^3+27b^2)\subset\mathbb{A}^2_k$ via the family $E\to S$ where $E\subset\mathbb{P}^2_k\times S$ given is by the equation $y^2=x^3+ax+b$.

  1. Is this in some mild sense universal? For example is every family of elliptic curves $E'\to S'$ over some smooth $k$-variety $S'$ the pull-back of $E\to S$ via a (necessarily non-unique) morphism $S'\to S$? If not, at least locally?

  2. Are there analogous constructions for abelian varieties of higher dimension? Is there a family of abelian varieties of dimension $g$ over some irreducible (affine? smooth?) variety $S$ over $k$ such that every abelian variety of dimension $g$ over $k$ is a fiber over a rational point of $S$?

In the literature I have only found (stronger) results over algebraically closed fields or when we additionally require something like a $\Gamma_1(N)$ structure or something.

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    $\begingroup$ The term you are looking for is "coarse moduli space". $\endgroup$
    – Wojowu
    Jul 27 at 15:44
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    $\begingroup$ @Wojowu Really? It seems closer to "versal deformation spaces" to me. $\endgroup$
    – Will Sawin
    Jul 27 at 15:46
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    $\begingroup$ @WillSawin You are right, this is probably closer to what OP meant. $\endgroup$
    – Wojowu
    Jul 27 at 18:42

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  1. This is true in a Zariski-local sense. Any elliptic curve together with a nowhere vanishing differential can be put in this form. For an elliptic curve over $S$, the differentials form a line bundle over $\operatorname{Spec} S$, so we can get a nowhere vanishing differential over a Zariski open cover where this line bundle is trivial.

  2. For principally polarized abelian varieties, you can choose $n$ such that $GL_n$ admits $Sp_{2g}( \mathbb Z/3)$ as a subgroup and consider the moduli space parameterizing pairs of a principally polarized abelian variety of genus $g$ together with a function from the set of full level 3-structures on the abelian variety compatible with the polarization to $GL_n$ which is $Sp_{2g}(\mathbb Z/3)$-equivariant. This moduli space will be a scheme because it is a bundle over $GL_n/Sp_{2g}(\mathbb Z/3)$ with fiber the moduli space of abelian varieties with full level $3$ structure, and both are schemes, and every family of abelian varieties factors through this Zariski-locally since the obstruction to doing that is a $GL_n$-torsor which splits locally.

But there is almost no reason to do this in practice, and you are better off working with the moduli stack of abelian varieties for most purposes.

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