Vol. 6, No. 3, 2012

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

Samit Dasgupta and Matthew Greenberg

Vol. 6 (2012), No. 3, 455–485
Abstract

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

Keywords
L-invariants, Shimura curves, Hida families, Stark–Heegner points
Mathematical Subject Classification 2000
Primary: 11F41
Secondary: 11G18, 11F67, 11F75