The Tate conjecture asserts that, for a smooth projective algebraic variety 
over a finitely generated field and a prime 
distinct from the field
characteristic, the classes invariant under the Galois
group in its 
-adic
cohomology are generated by classes of algebraic
cycles. In codimension 
, it predicts the surjectivity of the cycle-class map after
tensoring with 
onto the invariant part of 
.
Over a finite field, the conjecture is related to the order of the pole of the zeta
function of 
at the point corresponding to codimension 
. It is an arithmetic analogue of the Hodge
conjecture and is known in several important special cases, but remains open
in general.
