Zeta functions of isogeny graphs and spectral properties of adjacency operators

Resumen

We introduce an \emph{augmented Ihara zeta function} for supersingular $\ell$‑isogeny graphs that records both the degree label and the orientation determined by dual isogenies. A Bass–Hashimoto style determinant formula is proved, and we show that the resulting zeta function factors as the characteristic polynomial of the Hecke operator $T_{\ell}$ acting on weight‑$2$ cusp forms of level~$p$. Deligne’s bound on Hecke eigenvalues then yields a \emph{uniform Ramanujan property} for supersingular isogeny graphs with any prime $\ell<p/4$. We extend the zeta formalism to non‑regular ordinary \emph{isogeny volcanoes}, derive a rationality result, and relate the dominant pole to the volcano height. Finally, explicit cycle‑counting formulas lead to an equidistribution theorem for cyclic isogeny chains, confirmed by numerical experiments for primes $p\le 1000$ and $\ell\in\{2,3,5\}$