Endomorphisms of abelian varieties and points of finite order in characteristic $p$

An analog of the Tate hypothesis on homomorphisms of Abelian varieties is proved, in which points of sufficiently large prime order figure in place of the Tate modules. As is the case with the Tate hypothesis, this assertion follows formally from a finiteness hypothesis for isogenies of Abelian varieties, which is proved in characteristic $p>2$ and for finite fields. The same methods are used to prove the finiteness of the set of Abelian varieties of a given dimension over a finite field.