Isogeny graphs on superspecial abelian varieties: Eigenvalues and Connection to Bruhat-Tits buildings
We study for each fixed integer $g \ge 2$, for all primes $\ell$ and $p$ with $\ell \neq p$, finite regular directed graphs associated with the set of equivalence classes of $\ell$-marked principally polarized superspecial abelian varieties of dimension $g$ in characteristic $p$, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of $p$. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles-Goren-Lauter type cryptographic hash functions for abelian varieties. We give explicit lower bounds for the gaps in terms of the Kazhdan constant for the symplectic group when $g \ge 2$, and discuss optimal values in view of the theory of automorphic representations when $g=2$. As a by-product, we also show that the finite regular directed graphs constructed by Jordan-Zaytman also has the same property.