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Title:
Exploring Kaneko’s bound: On multi-edges, loops and the diameter of the supersingular \ell-isogeny graph
Authors: Sebastiano Boscardin, Sebastian A. Spindler
Abstract:We analyze Kaneko’s bound to prove that, away from the j-invariant 0, edges of multiplicity at least three can occur in the supersingular \ell-isogeny graph \mathcal{G}_\ell(p) only if the base field’s characteristic satisfies p < 4\ell^3. Further we prove a diameter bound for \mathcal{G}_\ell(p), while also showing that most vertex pairs have a substantially smaller distance, in the directed case; this bound is then used in conjunction with Kaneko’s bound to deduce that the distance of 0 and 1728 in \mathcal{G}_\ell(p) is at least one fourth of the graph’s diameter if p \equiv 11 \mathrel{\operatorname{mod}} 12. We also study other phenomena in \mathcal{G}_\ell(p) with Kaneko’s bound and provide data to demonstrate that the resulting bounds are optimal; for one of these bounds we investigate the connection between loop multiplicities in isogeny graphs and the factorization of the `diagonal’ classical modular polynomial \Phi_\ell(X,X) in positive characteristic.
ePrint: https://eprint.iacr.org/2025/1361
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