Algebraic cycles and Tate classes on Hilbert modular varieties

Abstract

Let $E/\mathbb{Q}$ be a totally real number field that is Galois over $\mathbb{Q}$, and let $\pi$ be a cuspidal, nondihedral automorphic representation of $\mathrm{GL}_2(\mathbb{A}_E)$ that is in the lowest weight discrete series at every real place of $E$. The representation $\pi$ cuts out a "motive" $M_\mathrm{et}(\pi^{\infty})$ from the $\ell$-adic middle degree intersection cohomology of an appropriate Hilbert modular variety. If $\ell$ is sufficiently large in a sense that depends on $\pi$ we compute the dimension of the space of Tate classes in $M_\mathrm{et}(\pi^{\infty})$. 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 $M_\mathrm{et}(\pi^{\infty})$ is spanned by algebraic cycles.