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

TL;DR

This paper investigates the spectral properties of isogeny graphs on superspecial abelian varieties, demonstrating rapid mixing and potential cryptographic applications, with explicit bounds related to automorphic representations.

Contribution

It establishes that adjacency matrices of these graphs have real eigenvalues with spectral gaps independent of the characteristic p, extending known results beyond elliptic curves.

Findings

Eigenvalues are real with p-independent spectral gaps

Rapid mixing of random walks on isogeny graphs

Potential for cryptographic hash functions using abelian varieties

Abstract

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.