ALGEBRAIC CYCLES AND TATE CLASSES ON HILBERT MODULAR VARIETIES

Let E/Q be a totally real number field that is Galois over Q, and let p be a cuspidal, nondihedral automorphic representation of GL(2)(A(E)) that is in the lowest weight discrete series at every real place of E. The representation pi cuts out a "motive" M-et(pi(infinity)) from the l-adic middle degree intersection cohomology of an appropriate Hilbert modular variety. If l is sufficiently large in a sense that depends on pi we compute the dimension of the space of Tate classes in M-et(pi(infinity)). Moreover if the space of Tate classes on this motive over all finite abelian extensions k/E is at most of rank one as a Hecke module, we prove that the space of Tate classes in M-et(pi(infinity)) is spanned by algebraic cycles.