Supersingular Abelian Varieties over Finite Fields

Abstract

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 ϕ: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 ϕ are bounded by an explicit function of dimA.