Adjacency Matrices and the Spectrum of l‑Isogeny Graphs

Abstract

We study the symmetrised $\ell$‑isogeny graph attached to supersingular elliptic curves over $\mathbb F_{p^{2}}$.  Interpreting its adjacency matrix as the $\ell$‑th Brandt matrix, we establish a Ramanujan bound on all non‑trivial eigenvalues, derive exact trace identities, and obtain explicit mixing and resistance estimates for the associated random walk.  Numerical experiments up to $p<2000$ corroborate the Sato–Tate‑type distribution of the spectrum.