论文标题
超级等值型图中的方向和周期
Orientations and cycles in supersingular isogeny graphs
论文作者
论文摘要
该论文涉及定向超级$ \ ell $ - 发育的火山的几个理论方面及其与超级$ \ ell $ isegem-isegeny图中的封闭步行的关系。我们的主要结果是在$ \ overline {\ Mathbb {f}} _ p $上方的所有定向超级$ \ ell $ $ $ $ $ $ $ $ $ $ $ \ ell $ - 发育的火山之间进行两次培养超过$ \ overline {\ mathbb {f}} _ p $。通过额外的自动形态和某些二次订单中$ p $的后果引起的特殊行为，使这一培养的确切证明和陈述更加复杂。我们使用两次射击来计算超级$ \ ell $ - 发育图中给定长度的同一循环，将其完全作为这些顺序的类数量的总和，并且还通过估计班级数量给出了明确的上限。
The paper 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 which are not powers of smaller walks) of the supersingular $\ell$-isogeny graph over $\overline{\mathbb{F}}_p$. 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 $\ell$-isogeny graph exactly as a sum of class numbers of these orders, and also give an explicit upper bound by estimating the class numbers.
