Group Structures of Elementary Supersingular Abelian Varieties over Finite Fields

Abstract

Let A be a supersingular abelian variety over a finite field k which is k-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=ge 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/(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.