Polarized superspecial abelian varieties over $\mathbb{F}_p$ via hermitian lattices

TL;DR

This paper classifies polarized superspecial abelian varieties over finite fields with specific Frobenius endomorphism properties by translating the problem into hermitian lattice classification, advancing understanding of supersingular loci.

Contribution

It establishes criteria for the nonemptiness and classifies the genera of these abelian varieties using hermitian lattice methods, extending previous theoretical frameworks.

Findings

Determined when the set of such abelian varieties is nonempty.

Classified the genera of these abelian varieties.

Reduced the problem to hermitian lattice classification using arithmetic methods.

Abstract

We study the set of isomorphism classes of polarized superspecial abelian varieties $(A,\lambda)$ of a fixed dimension over $\mathbb{F}_p$ with Frobenius endomorphism $\pi_A=\sqrt{-p}$ and $\ker \lambda =\ker \pi_A$. This set plays an important role in the geometry of the supersingular locus, and the generalizations of Deuring's $2T-H$ Theorem by Ibukiyama and Katsura. We determine when this set is nonempty and classify its genera. Our method reduces the problems of superspecial abelian varieties to those of certain hermitian lattices by the lattice description established by Jordan et. al and Ibukiyama--Karemaker--Yu, and we treat these problems on the lattices concerned by arithmetic methods.