Expander Properties of Superspecial Isogeny Digraphs with Level Structure

Charles, Goren and Lauter proved that the supersingular $\ell$-isogeny graph is a Ramanujan graph, which is an optimal expander. Jordan and Zaytman argued that this is no longer true in dimension two, but Florit and Smith showed that those graphs exhibit good expansion properties nonetheless. Castryck, Decru and Smith however have pointed out that the higher-dimensional analogue setting should only consider a subset of all edges, namely the paths corresponding to $(\ell^k,\ell^k)$-isogenies, so-called good extensions, instead of all $(\ell^a,\ell^b,\ell^c,\ell^d)$-isogenies in general, which contain bad extensions too. Such bad extensions lead to many small cycles in the graph, which are a cryptographic problem due to collisions and a graph-theoretic nuisance as these superfluous edges counteract part of the expansion properties. Restricting to good extensions makes the resulting graph directed, as outgoing edges now depend on the incoming edge.

We study abelian surfaces with $(\ell,\ell)$-level structure and $(\ell)^g$-isogeny digraphs restricted to good extensions for concrete small dimensions and degrees $\ell$. These graphs exhibit excellent expander properties: by our heuristic evidence, they converge to weakly Ramanujan graphs for all primes $\ell$ in dimension 1, and for $\ell = 2$ in dimension 2. Our main conjecture implies that this would still be the case for $\ell=3$ in dimension 2, but not for any larger $\ell$ in dimension 2, or any $\ell$ in dimension 3 and up. Furthermore, we generalize the work of Florit and Smith from $\ell = 2$ to general primes $\ell$, by classifying all abelian surfaces with nontrivial automorphism groups and their actions on their maximal isotropic $(\ell,\ell)$-subgroups.