E Explicit Class‑Field Generation via Chains of Modular Polynomials

Abstract

We introduce an 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
$\ell\in\{2,3,5\}$.