Paper 2022/562

Orientations and cycles in supersingular isogeny graphs

Sarah Arpin and Mingjie Chen and Kristin E. Lauter and Renate Scheidler and Katherine E. Stange and Ha T. N. Tran

Abstract

The paper concerns several theoretical aspects of oriented supersingular l-isogeny volcanoes and their relationship to closed walks in the supersingular l-isogeny graph. Our main result is a bijection between the rims of the union of all oriented supersingular l-isogeny volcanoes over $\overline{\mathbb{F}}_p$ (up to conjugation of the orientations), and isogeny cycles (non-backtracking closed walks which are not powers of smaller walks) of the supersingular l-isogeny graph modulo p. The exact proof and 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 l-isogeny graph exactly as a sum of class numbers, and also give an explicit upper bound by estimating the class numbers.