Neighborhood of vertices in the isogeny graph of principally polarized superspecial abelian surfaces

Abstract

For two supersingular elliptic curves E and $E^{\prime }$ defined over $F_{p^2}$, let $[E\times E^{\prime }]$ be the superspecial abelian surface with the principal polarization $\left\{0\right\}\times E^{\prime }+E\times \left\{0\right\}$. We determine local structure of the vertices $[E\times E^{\prime }]$ in the $(\ell ,\ell )$-isogeny graph of principally polarized superspecial abelian surfaces where either E or $E^{\prime }$ is defined over $F_p$. We also present a simple new proof of the main theorem in [26].