On the supersingular loci of quaternionic Siegel space

TL;DR

This paper investigates the structure of the supersingular locus in a specific moduli space of abelian varieties with quaternionic multiplication, revealing that its irreducible components are Fermat surfaces of degree p+1.

Contribution

It establishes that the irreducible components of the supersingular locus are Fermat surfaces of degree p+1, providing a geometric description of this locus in quaternionic Siegel space.

Findings

Irreducible components are Fermat surfaces of degree p+1.

Provides geometric characterization of the supersingular locus.

Advances understanding of quaternionic moduli spaces.

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.