On fields of definition of components of the Siegel supersingular locus

Abstract

Recently Ibukiyama proves an explicit formula for the number of certain non-principal polarizations on a superspecial abelian surface, extending his earlier work with Katsura for principal polarizations [Compos. Math. 1994]. As a consequence of Ibukiyama's formula, there exists a geometrically irreducible component of the Siegel supersingular locus which is defined over the prime finite field. In this note we give a direct proof of this result.