Talk 2: Modular forms of weight one, motivic cohomology and the Jacquet-Langlands correspondence

I will discuss some joint work (in progress) with Ichino. In a previous paper, 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. Since weight one forms are not cohomological, the Tate conjecture does not apply and thus it is not at all 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.