$\mathscr{L}$-invariants and Shimura curves

Abstract

In earlier work, the second named author described how to extract Darmon-style $\mathcal{ℒ}$-invariants from modular forms on Shimura curves that are special at $p$. In this paper, we show that these $\mathcal{ℒ}$-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 $\mathbb{Q}$. 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.