  In this paper we give some evidence for the Tate (and Hodge) conjecture(s)
for a class of Hilbert modular fourfolds X, whose connected components arise as
arithmetic quotients of the fourfold product of the upper half plane by
congruence subgroups \Gamma of SL(2, O_K), where O_K denotes the ring of
integers of a quartic, Galois, totally real number field K. The expected
relationship to the orders of poles of the associated L-functions is verified
for abelian extensions of \Q. Also shown is the existence of homologically
non-trivial cycles of codimension two which are not intersections of divisors.
