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In paragraph 3 of chapter 7 of Mumford's Geometric Invariant Theory, the author proves that the coarse moduli space $A_{g,d,n}$ of abelian varieties of dimension $g$, with a degree $d^2$ polarization and a level $n$ structure, exists over $\text{Spec } \mathbb{Z}$ and is quasi-projective over $\text{Spec }\mathbb{Z}[1/p]$ for any prime number $p$.

He next claims that $A_{g,d,n}$ is in fact quasi-projective over $\text{Spec }\mathbb{Z}$ but does not provide a proof. How is this done nowadays?

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    $\begingroup$ If memory serves, this is proved in Faltings-Chai. $\endgroup$
    – Jason Starr
    Commented Sep 27, 2016 at 10:21
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    $\begingroup$ And in my thesis, Astérisque 129 (1985), VII, Th. 4.2. (Of course the question arises only for $n=1$). $\endgroup$
    – Laurent Moret-Bailly
    Commented Sep 27, 2016 at 15:30
  • $\begingroup$ @Moret-Bailly Thank you for this reference, it is really well explained. Would you like to post this as an answer so I can accept it and close the topic? $\endgroup$
    – user85435
    Commented Oct 1, 2016 at 11:16

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