This is the discussion thread for the third session of season four of The Isogeny Club, where Gabrielle presents her talk titled: Connecting Kani’s Lemma and path-finding in the Bruhat-Tits tree to compute endomorphism rings of supersingular elliptic curves
Abstract
In this talk, we give a deterministic polynomial-time algorithm which computes the endomorphism ring of a supersingular elliptic curve, given a full-rank subring of the endomorphism ring. At each prime for which the input subring is not maximal, we compute the endomorphism ring locally. When the prime is not equal to the characteristic, we do this by using techniques of higher-dimensional isogenies to navigate towards the local endomorphism ring in the Bruhat-Tits tree. We review background on endomorphism rings and the Bruhat-Tits tree, briefly describe the “techniques of higher-dimensional isogenies”, and explain the algorithm in both the simplest case and the more general case. This is joint work with Kirsten Eisenträger.
Video
