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

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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 Γof SL(2, Oₖ), where Oₖ 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. Also shown is the existence of homologically non-trivial cycles of codimension two which are not intersections of divisors.