Group Structures of Elementary Supersingular Abelian Varieties over Nite Elds

Let A be a supersingular abelian variety over a nite eld k which is isogenous to a power of a simple abelian variety over k. Write the characteristic polynomial of the Frobenius endomorphism of A relative to k as f = g e for a monic irreducible polynomial g and a positive integer e, we show that the group of k-rational points A(k) on A is isomorphic to (Z=g(1)Z) e unless A's simple component is of dimension 1 or 2, in which case we prove that A(k) is isomorphic to (Z=g(1)Z) a (Z= g(1) 2 ZZ=2Z) b for some non-negative integers a; b with a + b = e. In particular, if the characteristic of k is 2 or A is simple of dimension greater than 2, then A(k) = (Z=g(1)Z) e .