Isogeny graphs on superspecial abelian varieties: Eigenvalues and Connection to Bruhat-Tits buildings

Abstract

We show that for each fixed integer g ≥ 2, for all primes ℓ and p with ℓ = p, finite regular undirected graphs associated to (ℓ) g -isogenies of principally polarized superspecial abelian varieties of dimension g in characteristic p form a family of expanders as 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 second smallest eigenvalues of corresponding Laplacians in terms of the Kazhdan constant for the symplectic group when g ≥ 2, and discuss optimal values in view of the theory of automorphic representations when g = 2.2.1. Superspecial abelian varieties 2.2. Principal polarizations 2.3. Class number of the principal genus for quaternion Hermitian lattices 2.4. Ibukiyama-Katsura-Oort-Serre's result in terms of adelic language