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

TL;DR

This paper analyzes the local structure of vertices in the isogeny graph of superspecial abelian surfaces, focusing on cases where one elliptic curve is defined over a smaller field, and offers a new proof of a key theorem.

Contribution

It determines the local structure of vertices in the isogeny graph for superspecial abelian surfaces with specific field conditions and provides a simplified proof of a main existing theorem.

Findings

Characterization of local vertex structure in the isogeny graph.

Identification of cases where E or E' is over _p.

A new, simplified proof of the main theorem in prior work.

Abstract

For two supersingular elliptic curves $E$ and $E'$ defined over $\mathbb{F}_{p^2}$, let $[E \times E']$ be the superspecial abelian surface with the principal polarization $\{0\} \times E' + E \times \{0\}$. We determine local structure of the vertices $[E \times E']$ in the $(\ell, \ell)$-isogeny graph of principally polarized superspecial abelian surfaces where either $E$ or $E'$ is defined over $\mathbb{F}_p$. We also present a simple new proof of the main theorem in \cite{LOX20}.