We sketch a strategy to prove the Tate conjecture on algebraic cycles for a good amount of quaternionic Shimura varieties.
A key point is a twisted adjoint L-value formula relative to each quaternion algebra D/F for a totally real field F and its scalar extension B to a totally real quadratic extension E/F. The theta base-change lift f of a Hilbert modular form f to the multiplicative group of B has period integral over the Shimura subvariety Sh(D) inside Sh(B) given by the adjoint L-value for f twisted by the quadratic character of E/F at 1. Since the adjoint L-value at 1 does not vanish (its abscissa of convergence), Sh(D) gives rise to a non-trivial Tate cycle in the degree 2r cohomology group of Sh(D) for the dimension r of Sh(D).