Orientations and Cycles in Supersingular Isogeny Graphs

The chapter concerns several theoretical aspects of oriented supersingular \(\ell \) -isogeny volcanoes and their relationship to closed walks in the supersingular \(\ell \) -isogeny graph. Our main result is a bijection between the rims of the union of all oriented supersingular \(\ell \) -isogeny volcanoes over \(\overline {\mathbb {F}}_p\) (up to conjugation of the orientations) and isogeny cycles (non-backtracking closed walks that are not powers of smaller walks) of the supersingular \(\ell \) -isogeny graph over \(\overline {\mathbb {F}}_p\) . The exact proof and the statement of this bijection are made more intricate by special behaviours arising from extra automorphisms and the ramification of p in certain quadratic orders. We use the bijection to count isogeny cycles of given length in the supersingular \(\ell \) -isogeny graph exactly as a sum of class numbers of these orders and also give an explicit upper bound by estimating the class numbers.