Connecting Kani's Lemma and path-finding in the Bruhat-Tits tree to compute supersingular endomorphism rings

We give a deterministic polynomial time algorithm to compute the endomorphism ring of a supersingular elliptic curve in characteristic p, provided that we are given two noncommuting endomorphisms and the factorization of the discriminant of the ring $\mathcal{O}_0$ they generate. At each prime $q$ for which $\mathcal{O}_0$ is not maximal, we compute the endomorphism ring locally by computing a q-maximal order containing it and, when $q \neq p$, recovering a path to $\text{End}(E) \otimes \mathbb{Z}_q$ in the Bruhat-Tits tree. We use techniques of higher-dimensional isogenies to navigate towards the local endomorphism ring. Our algorithm improves on a previous algorithm which requires a restricted input and runs in subexponential time under certain heuristics. Page and Wesolowski give a probabilistic polynomial time algorithm to compute the endomorphism ring on input of a single non-scalar endomorphism. Beyond using techniques of higher-dimensional isogenies to divide endomorphisms by a scalar, our methods are completely different.