Hodge classes and the Jacquet-Langlands correspondence

We prove that the Jacquet-Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in $\mathrm{\ell }$-adic cohomology is Galois invariant for all $\mathrm{\ell }$.