Group structures of elementary supersingular abelian varieties over finite fields

Abstract

Let A be a supersingular abelian variety over a finite field k. We give an approximate description of the structure of the group A(k) of rational points of A over k in terms of the characteristic polynomial f of the Frobenius endomorphism of A relative to k. If f=g^e for a monic irreducible polynomial g and a positive integer e, we show that there is a group homomorphism A(k) --> (Z/g(1)Z)^e whose kernel and cokernel are elementary abelian 2-groups. In particular, this map is an isomorphism if the characteristic of k is 2 or A is simple of dimension greater than 2; in the last case one has e=1 or 2, and A(k) is isomorphic to (Z/g(1)Z)^e.