Supersingular loci of low dimensions and parahoric subgroups

Abstract

The theory of polarized supersingular abelian varieties $(A,\lambda)$ is often essentially related to the theory of quaternion hermitian lattices. In this paper, we add one more such relation by giving an adelic description of supersingular loci of low dimensions. Katsura, Li and Oort have shown that the supersingular locus in the moduli of principally polarized abelian varieties is not irreducible and that the number of its irreducible components is equal to the class number of certain maximal quaternion hermitian lattices. Now the locus has certain natural algebraic subsets consisting of $A$ with fixed \textit{$a$-numbers} that are defined as the dimensions of embeddings from $\alpha_p$ to $A$. For low dimensional cases when $\dim A\leq 3$, we describe configuration of these subsets in the locus by intersection properties of some adelic subgroups of quaternion hermitian groups corresponding to parahoric subgroups locally at characteristic $p$. In particular, we describe which components each superspecial point lies on. This is proved by using certain liftability property of isogenies of abelian varieties, where the isogenies are interpreted to cosets of $GL_n$ and of parahoric subgroups of the quaternion hermitian groups acting on quaternion hermitian matrices.