On superspecial abelian varieties over finite fields

TL;DR

This paper introduces a new lattice-based framework for understanding superspecial abelian varieties over finite fields, providing explicit formulas for their enumeration in certain cases.

Contribution

It offers a novel lattice description of superspecial abelian varieties over finite fields, with explicit counting formulas when the field size is an odd power of the prime.

Findings

New lattice description for superspecial abelian varieties

Explicit formula for counting superspecial abelian surfaces over finite fields of odd prime power order

Dependence of description on the parity of the field extension degree

Abstract

In this paper we establish a new lattice description for superspecial abelian varieties over a finite field $\mathbb {F}_q$ of $q=p^a$ elements. Our description depends on the parity of the exponent $a$ of $q$. When $q$ is an odd power of the prime $p$, we give an explicit formula for the number of superspecial abelian surfaces over $\mathbb F_q$.