SUPERSINGULAR ABELIAN VARIETIES OVER FINITE FIELDS

HUI ZHU

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 krational points of A in terms of the characteristic polynomial f of the Frobenius endomorphism of A relative to k. Write f = ∏ g_{i}^{e_{i}} 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 size of the kernel and the cokernel of φ are bounded by an explicit function of dim A.