Isogeny complexes of superspecial abelian varieties

Abstract: We consider the structures formed by isogenies of abelian varieties with polarizations that are not necessarily principal, specifically with the $[\ell]$-polarizations we have previously defined. Our primary interest is in superspecial abelian varieties, where the isogenies are related to quaternionic hermitian forms. We first consider isogeny graphs. We show that these $[\ell]$-isogeny graphs are a generalized Brandt graph and construct them entirely in terms of definite quaternion algebras. We prove that they are connected and give examples to show that the regular graphs obtained are sometimes Ramanujan and sometimes not. Isogenies of $[\ell]$-polarized abelian varieties can be closed under composition, with the consequence that such isogenies naturally form semi-simplicial complexes as introduced by Eilenberg and Zilber in 1950 (later also called $\Delta$-complexes) -- the higher-dimensional analogues of multigraphs. We show that these isogeny complexes can be constructed from the arithmetic of hermitian forms over definite quaternion algebras and that they are quotients of the Bruhat-Tits building of the symplectic group by the action of a quaternionic unitary group. Working with quaternions these isogeny graphs and complexes are amenable to machine computation and we include many examples, concluding with a detailed examination of the $[2]$-isogeny complexes of superspecial abelian surfaces in characteristic $7$.