Group structures of elementary supersingular abelian varieties over finite fields
Hui Zhu
Let $A$ be a supersingular abelian variety over a finite field ${\bf k}$ which is isogenous to a power of a simple abelian variety over ${\bf k}$. Write the characteristic polynomial of the Frobenius endomorphism of $A$ relative to ${\bf k}$ as $f=g^e$ for a monic irreducible polynomial $g$ and a positive integer $e$, we show that the group of ${\bf k}$-rational points $A({\bf k})$ on $A$ is isomorphic to $({\bf Z}/g(1){\bf Z})^e$ unless $A$'s simple component is of dimension $1$ or $2$, in which case we prove that $A({\bf k})$ is isomorphic to $({\bf Z}/g(1){\bf Z})^a\times({\bf Z}/{\frac{g(1)}{2}}{\bf Z}\times{\bf Z}/2{\bf Z})^b$ for some non-negative integers $a,b$ with $a+b=e$. In particular, if the characteristic of ${\bf k}$ is $2$ or $A$ is simple of dimension greater than $2$, then $A({\bf k})\cong ({\bf Z}/g(1){\bf Z})^e$.