Isogeny graphs of superspecial abelian varieties and Brandt matrices

Abstract

Fix primes $p$ and $\ell$ with $\ell\neq p$. If $(A,\lambda)$ is a $g$-dimensional principally polarized abelian variety, an $(\ell)^g$-isogeny of $(A,\lambda)$ has kernel a maximal isotropic subgroup of the $\ell$-torsion of $A$; the image has a natural principal polarization. We define three isogeny graphs associated to such $(\ell)^g$-isogenies -- the big isogeny graph $\mathit{Gr}_{\!g}(\ell,p)$, the little isogeny graph $\mathit{gr}_{\!g}(\ell,p)$, and the enhanced isogeny graph $\widetilde{\mathit{gr}}_{\!g}(\ell, p)$. We prove that all three isogeny graphs are connected. One ingredient of the proof is strong approximation for the quaternionic unitary group, which has previously been applied to moduli of abelian varieties in charateristic $p$ by Chai, Ekedahl/Oort, and Chai/Oort. The adjacency matrices of the three isogeny graphs are given in terms of the Brandt matrices defined by Hashimoto, Ibukiyama, Ihara, and Shimizu. We study some basic properties of these Brandt matrices and recast the theory using the notion of Brandt graphs. We show that the isogeny graphs $\mathit{Gr}_{\!g}(\ell, p)$ and $\mathit{gr}_{\!g}(\ell, p)$ are in fact our Brandt graphs. We give the $\ell$-adic uniformization of $\mathit{gr}_{\!g}(\ell,p)$ and $\widetilde{\mathit{gr}}_{\!g}(\ell,p)$. The $(\ell+1)$-regular isogeny graph $\mathit{Gr}_1(\ell,p)$ for supersingular elliptic curves is well known to be Ramanujan. We calculate the Brandt matrices for a range of $g>1$, $\ell$, and $p$. These calculations give four examples with $g>1$ where the regular graph $\mathit{Gr}_{\!g}(\ell,p)$ has two vertices and is Ramanujan, and all other examples we computed with $g>1$ and two or more vertices were not Ramanujan. In particular, the $(\ell)^g$-isogeny graph is not in general Ramanujan for $g>1$.