Supersingular Abelian Varieties over Finite Fields Original Research Article

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=∏ geii for distinct monic irreducible polynomials gi and positive integers ei. We show that there is a group homomorphism phi: A(k)→∏ (Z/gi(1) Z)ei that is “almost” an isomorphism in the sense that the sizes of the kernel and the cokernel of phi are bounded by an explicit function of dim A.