On superspecial abelian surfaces over finite fields

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 ${\mathbb{F}}_q$.