If $A/k$ is a supersingular abelian variety, then it is $k$-isogenous to $A'/k$ a superspecial abelian variety, see "A Note on supersingular abelian varieties" by Chia-Fu Yu. And if $A'/k$ is superspecial (of dimension $g>1$), it is isomorphic over some finite extension $k'/k$ to $E_0^g$, $E_0$ any supersingular elliptic curve over $k$.
My question is what happens when we keep track of a principal polarisation? Namely:
- If $(A, \Theta)/k$ is a principally polarised supersingular abelian variety, is it isogenous over $k$ (or an extension) to a principally polarised superspecial abelian variety $(A', \Theta')$? Here by an isogeny $f: (A, \Theta) \to (A', \Theta')$ between principally polarised abelian varieties, I mean that $f^{\ast} \Theta' \simeq n \Theta$ for some integer $n>0$. (Here since I am speaking about polarisations, $\simeq$ means algebraic equivalence, and I only ask the algebraic equivalence class of $\Theta'$ to be defined over $k$ rather than $\Theta'$ itself.)
- If $(A', \Theta')/k$ is a principally polarised superspecial abelian variety, is it isogenous over a finite extension of $k$ to a product of supersingular elliptic curves $E_1 \times \dots \times E_g$ with the product principal polarisation? Can we take $E_0=E_1=\cdots=E_g$ for any supersingular curve $E_0$?
