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Title:
Structural Results for Maximal Quaternion Orders and Connecting Ideals of Prime Power Norm in B_{p,\infty}
Authors: James Clements
Abstract:Fix odd primes p, \ell with p \equiv 3 \mod 4 and \ell \neq p. Consider the rational quaternion algebra ramified at p and \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 \mathbb{F}_p.
[1] = Parametrizing Maximal Orders Along Supersingular $\ell$-Isogeny Paths
ePrint: https://eprint.iacr.org/2025/042
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