On fields of definition of components of the Siegel supersingular locus

TL;DR

This paper provides a direct proof that a geometrically irreducible component of the Siegel supersingular locus is defined over the prime finite field, building on Ibukiyama's explicit formulas for polarizations.

Contribution

It offers a new, direct proof of the existence of a geometrically irreducible component over the prime field, complementing previous results based on explicit formulas.

Findings

Existence of a geometrically irreducible component over the prime finite field

Direct proof of the component's field of definition

Extension of Ibukiyama's formulas to geometric properties

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.