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 they generate. At each prime for which is not maximal, we compute the endomorphism ring locally by computing a q-maximal order containing it and, when , recovering a path to 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.
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