Supersingular Abelian Varieties over Finite Fields

Let A be a supersingular abelian variety defined over a finite field k. We give an approximate description of the structure of the group A(k) of k-rational points of A in terms of the characteristic polynomial f of the Frobenius endomorphism of A relative to k. Write f=∏g^e_ii for distinct monic irreducible polynomials g_i and positive integers e_i. We show that there is a group homomorphism φ:A(k)→∏(Z/g_i(1)Z)^e_i that is "almost" an isomorphism in the sense that the sizes of the kernel and the cokernel of φ are bounded by an explicit function of dimA.