L-invariants and Shimura curves

In earlier work, the second named author described how to extract Darmon-style L-invariants from modular forms on Shimura curves that are special at p. In this paper, we show that these L-invariants are preserved by the Jacquet-Langlands correspondence. As a consequence, we prove the second named author's period conjecture in the case where the base field is Q. As a further application of our methods, we use integrals of Hida families to describe Stark-Heegner points in terms of a certain Abel-Jacobi map.