Paper 2025/033

Parametrizing Maximal Orders Along Supersingular $\ell$-Isogeny Paths

Abstract

Suppose you have a supersingular $\ell$-isogeny graph with vertices given by $j$-invariants defined over ${\mathbb{F}}_{p^2}$, where $p=4\cdot f\cdot {\ell }^e-1$ and $\ell \equiv 3(\mathrm{mod}4)$. We give an explicit parametrization of the maximal orders in $B_{p,\mathrm{\infty }}$ appearing as endomorphism rings of the elliptic curves in this graph that are $\le e$ steps away from a root vertex with $j$-invariant 1728. This is the first explicit parametrization of this kind and we believe it will be an aid in better understanding the structure of supersingular $\ell$-isogeny graphs that are widely used in cryptography. Our method makes use of the inherent directions in the supersingular isogeny graph induced via Bruhat-Tits trees, as studied in [1]. We also discuss how in future work other interesting use cases, such as $$, could benefit from the same methodology.