On the supersingular loci of quaternionic Siegel space

Abstract

The paper studies the supersingular locus of the characteristic p moduli space of principally polarized abelian 8-folds that are equipped with an action of a maximal order in a quaternion algebra, that is non-split at the infinite place, but split at p. The main result is that its irreducible components are Fermat surfaces of degree p+1.

The work is partially based on results proved in the author's 1997 Oxford thesis entitled "On the mod-p reduction of ordinary CM points"