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Orientations and cycles in supersingular isogeny graphs. (arXiv:2205.03976v2 [math.NT] UPDATED)
Dec. 6, 2022, 2:10 a.m. | Sarah Arpin, Mingjie Chen, Kristin E. Lauter, Renate Scheidler, Katherine E. Stange, Ha T. N. Tran
cs.CR updates on arXiv.org arxiv.org
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 …
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