Paper 2025/042

Structural Results for Maximal Quaternion Orders and Connecting Ideals of Prime Power Norm in $B_{p,\mathrm{\infty }}$

Abstract

Fix odd primes $p,\ell$ with $p\equiv 3\mathrm{mod}4$ and $\ell \ne p$. Consider the rational quaternion algebra ramified at $p$ and $\mathrm{\infty }$ with a fixed maximal order ${\mathcal{O}}_{1728}$. We give explicit formulae for bases of all cyclic norm ${\ell }^n$ ideals of ${\mathcal{O}}_{1728}$ and their right orders, in Hermite Normal Form (HNF). Further, in the case $\ell \mid p+1$, or more generally, $-p$ is a square modulo $\ell$, we derive a parametrization of these bases along paths of the $\ell$-ideal graph, generalising the results of [1]. With such orders appearing as the endomorphism rings of supersingular elliptic curves defined over $\overline{{\mathbb{F}}_p}$, we note several potential applications to isogeny-based cryptography including fast ideal sampling algorithms. We also demonstrate how our findings may lead to further structural observations, by using them to prove a result on the successive minima of endomorphism rings of supersingular curves defined over $$. [1] = https://eprint.iacr.org/2025/033