 For "k" of characteristic 0, showed that the Tate conjecture (as stated above) implies the semisimplicity of 

For "k" finite of order "q", Tate showed that the Tate conjecture plus the semisimplicity conjecture would imply the strong Tate conjecture, namely that the order of the pole of the zeta function "Z"("X", "t") at "t" = "q"−"j" is equal to the rank of the group of algebraic cycles of codimension "j" modulo numerical equivalence.Like the Hodge conjecture, the Tate conjecture would imply most of Grothendieck's standard conjectures on algebraic cycles.Namely, it would imply the Lefschetz standard conjecture (that the inverse of the Lefschetz isomorphism is defined by an algebraic correspondence); that the Künneth components of the diagonal are algebraic; and that numerical equivalence and homological equivalence of algebraic cycles are the same.