GROUP STRUCTURES OF ELEMENTARY SUPERSINGULAR ABELIAN VARIETIES OVER FINITE FIELDS

HUI ZHU

Abstract. Let A be a supersingular abelian variety over a finite field k which is isogenous to a power of a simple abelian variety over k. Write the characteristic polynomial of the Frobenius endomorphism of A relative to k as f = g^{e} for a monic irreducible polynomial g and a positive integer e, we show that the group of k-rational points A(k) on A is isomorphic to (Z/g(1)Z)^{e} unless A’s simple component is of dimension 1 or 2, in which case we prove that A(k) is isomorphic to (Z/g(1)Z)^{a} × (Z/\frac{g(1)}{2}Z × Z/2Z)^{b} for some non-negative integers a, b with a + b = e. In particular, if the characteristic of k is 2 or A is simple of dimension greater than 2, then A(k) ≅ (Z/g(1)Z)^{e}.