Joint PU/IAS Number Theory

The Tate conjecture predicts that in many instances, Langlands functoriality should be given by algebraic cycle classes. In previous joint work with Ichino, we showed that the Jacquet-Langlands correspondence for cohomological modular forms on GL(2) is given by a Hodge class, which is also Galois invariant. In this talk, I will discuss an analogous result in the case of weight one forms. Since weight one forms are not cohomological, it is not even clear how to formulate the statement of such a result. I will motivate the statement, which connects to another theme, namely motivic actions on the cohomology of locally symmetric spaces, and sketch the idea of the proof. (Joint work in progress with Ichino.)