Algebraic cycles and tate classes on hilbert modular varieties
Let E/ be a totally real number field that is Galois over , and let be a cuspidal, nondihedral automorphic representation of GL2(E) that is in the lowest weight discrete series at every real place of E. The representation cuts out a motive Mét(π∞) 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 π we compute the dimension of the space of Tate classes in M ét(π∞). 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 ét(π∞) is spanned by algebraic cycles. © 2014 World Scientific Publishing Company.
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