ℒ-invariants and Shimura curves

Dasgupta, Samit; Greenberg, Matthew

doi:10.2140/ant.2012.6.455

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

Samit Dasgupta and Matthew Greenberg

Vol. 6 (2012), No. 3, 455–485

DOI: 10.2140/ant.2012.6.455

Abstract

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.

Keywords

L-invariants, Shimura curves, Hida families, Stark–Heegner points

Mathematical Subject Classification 2000

Primary: 11F41

Secondary: 11G18, 11F67, 11F75

Milestones

Received: 15 July 2010

Revised: 8 April 2011

Accepted: 23 May 2011

Published: 5 July 2012

Authors

Samit Dasgupta
Department of Mathematics University of California, Santa Cruz 1156 High St Santa Cruz, CA 95064 United States
http://people.ucsc.edu/~sdasgup2/
Matthew Greenberg
Department of Mathematics and Statistics University of Calgary Calgary, AL T2N 1N4 Canada

Vol. 6, No. 3, 2012