  We study a natural birational invariant for varieties over finite fields and
show that its vanishing on projective space is equivalent to the Tate
conjecture, the Beilinson conjecture, and the Grothendieck--Serre
semi-simplicity conjecture for all smooth projective varieties over finite
fields. We further show that the Tate, Beilinson, and 1-semi-simplicity
conjecture in half of the degrees implies those conjectures in all degrees.
