Neighborhood of the supersingular elliptic curve isogeny graph at $j=0$ and $1728$

TL;DR

This paper analyzes the local structure of the supersingular elliptic curve isogeny graph at special j-invariants 0 and 1728 over finite fields, providing detailed descriptions under certain prime size conditions.

Contribution

It characterizes the neighborhood of specific supersingular elliptic curves in the isogeny graph for primes larger than certain bounds, extending understanding of the graph's local structure.

Findings

Describes the neighborhood of [E_0] in the isogeny graph for p > 3ℓ^2.

Describes the neighborhood of [E_{1728}] in the isogeny graph for p > 4ℓ^2.

Provides explicit descriptions of the local graph structure at these special vertices.

Abstract

We describe the neighborhood of the vertex $[E_0]$ (resp. $[E_{1728}]$) in the $\ell$-isogeny graph $\mathcal{G}_\ell(\mathbb{F}_{p^2}, -2p)$ of supersingular elliptic curves over the finite field $\mathbb{F}_{p^2}$ when $p>3\ell^2$ (resp. $p>4\ell^2$) with $E_0: y^2=x^3+1$ (resp. $E_{1728}: y^2=x^3+x$) supersingular.