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 analog of the Hodge
conjecture and is known in several important special cases, but remains open
in general.
Brantner (2026) proved the integral geometric Tate conjecture for ordinary Calabi-Yau varieties of dimension three over finite
fields. Here ordinary means that its Hodge and Newton polygons agree. The result
also holds for 1-ordinary threefolds with 
. Here "integral" means using
the l-adic integers 
as coefficients rather than 
. The word "geometric" means that algebraic
cycles are taken over the algebraic closure 
of the base field 
. The cycle-class map must be a surjection
onto the classes in 
fixed by the Galois group 
for some finite extension
field 
,
for every prime 
distinct from the field
characteristic (Brantner 2026). It is not a proof of the general conjecture.
