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By a supersingular Abelian scheme, I mean an Abelian scheme which is fibrewise a supersingular Abelian variety, i.e. isogenous to a product of supersingular elliptic curves (F. Oort, Subvarieties of moduli spaces. Invent. Math. 24 (1974), 95–119).

Let $\mathscr{A}$ be a supersingular Abelian scheme over an integral variety over $\mathbf{F}_p$. Is it true that $\mathscr{A}$ is isogenous to a product of relative elliptic curves over $X$? I am looking for a reference. Equations (0.4.1), (0.4.2) in [Li-Oort, Moduli of Supersingular Abelian Varieties, LNM 1680] seem to imply this.

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    $\begingroup$ You want this to be true without a base change of your integral variety $S$? For a finite subgroup $\Gamma$ of $\textbf{GL}_n(\mathbf{Z})$ and for a $\Gamma$-torsor $S'\to S$, for every elliptic curve $E$ over $\mathbf{F}_p$, for the Abelian variety $A=E^n = E\times_{\text{Spec}(\mathbf{F}_p)}\dots \times_{\text{Spec}(\mathbf{F}_p)} E$ with its natural action of $\textbf{GL}_n(\mathbf{Z})$, there is an Abelian scheme over $S$, $\mathcal{A} = (S' \times_{\text{Spec}(\mathbf{F}_p)} A)/\Gamma$. Do you expect $\mathcal{A}$ to be isogenous to a product over $S$? $\endgroup$
    – Jason Starr
    Commented Mar 1, 2017 at 16:43
  • $\begingroup$ @Jason Starr: I think a finite étale base change would be OK. $\endgroup$
    – user19475
    Commented Mar 1, 2017 at 17:04
  • $\begingroup$ You can also assume $\mathscr{A}$ to be principally polarised. $\endgroup$
    – user19475
    Commented Mar 1, 2017 at 17:07
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    $\begingroup$ Here's a naive remark which I hope might be useful. Let $x$ be an $\mathbb F_p$-point of the base $S$ (assuming it exists) and let $E$ be the fibre of $\mathcal A$ over $x$. Then, do you expect $\mathcal A$ to be isogenous to $E^g\times S$, up to finite etale base change? If so, then you could try to show that the scheme of isogenies $Isog_S(\mathcal A, E^g\times S)$ is finite etale over $S$. It is at least surjective by the results in Li-Oort...(You might want to consider a subscheme parametrizing isogenies of some bounded degree, and maybe also of degree coprime to $p$.) $\endgroup$
    – Ariyan Javanpeykar
    Commented Mar 2, 2017 at 8:34

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Let $K = K(X)$ be the function field of $X$. By Frans Oort, Subvarieties of moduli spaces. In: Invent. Math., 24 (1974), 95–119., p. 113, Theorem 4.2, $\mathscr{A}_{\bar{K}}$ is isogenous to $E_{\bar{K}}^g$ with $E_{\bar{K}}/\bar{K}$ any (!) supersingular elliptic curve (any two supersingular elliptic curves over an algebraically closed field are isogenous, see loc. cit., p. 113). Note that for any prime $p$, there exists a supersingular elliptic curve over $\mathbf{F}_p$, see Silverman, The Arithmetic of Elliptic Curves, p. 148f., Theorem V.4.1(c) for $p > 2$ and the text before this theorem for $p = 2$. By Milne, http://jmilne.org/math/articles/1986b.pdf, p. 146, Corollary 20.4(b) applied to the primary field extension $\bar{K}/K^{\mathrm{sep}}$, there is a separable field extension $L/K$ and an isogeny $E_L^g \to \mathscr{A}_L$. Since $E/\mathbf{F}_p$ extends to $E \times_{\mathbf{F}_p} X$ over $X$, the claim follows from extending homomorphisms of Abelian schemes.

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