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Abstract: I will discuss some joint work (in progress) with Ichino. In
previous work, we showed that the Jacquet-Langlands correspondence for
cohomological Hilbert modular forms preserves rational Hodge structures,
as predicted by the Tate conjecture. In this talk, I will discuss a related
result in the case of weight one forms, which are non-cohomological. In
this case, the Tate conjecture does not apply and thus it is not obvious
what the content of such a result should be. I will motivate and explain
the statement, which is suggested by another recent development, namely
the conjectural connection between motivic cohomology and the cohomology of
locally symmetric spaces.
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