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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 $M_\\mathrm{et}(\\pi^{\\infty})$. 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Getz","submitted_at":"2015-07-16T00:34:51Z","abstract_excerpt":"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 $M_\\mathrm{et}(\\pi^{\\infty})$. 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