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.



supersingular Abelian scheme
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](https://mathoverflow.net/questions/265505/extending-homomorphisms-of-abelian-schemes).


