Algebraic cycles on Hilbert modular fourfolds and poles of L-functions

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.