Endomorphisms of Abelian Varieties over Finite Fields

Almost all of the general facts about abelian varieties which we use without comment or refer to as "well known" are due to WEIL, and the references for them are [12] and [3]. Let k be a field, k its algebraic closure, and A an abelian variety defined over k, of dimension g. For each integer m > 1, let A m denote the group of elements aeA(k) such that ma=O. Let l be a prime number different from the characteristic of k, and let T~(A) denote the projective limit of the groups A~ with respect to the maps A~n.l~Av, which are induced by multiplication by l. It is well known that Tt(A) is a free module of rank 2g over the ring Z l of l-adic integers. The group G=Gal(k./k) operates on Tt(A). Let A' and A" be abelian varieties defined over k. The group HOmk(A', A") of homomorphisms of A' into A" defined over k is Z-free, and the canonical map