The rationality of quaternionic Darmon points over genus fields of real quadratic fields

Abstract

Darmon points on p-adic tori and Jacobians of Shimura curves over Q were introduced in joint articles with Rotger as generalizations of Darmon's Stark-Heegner points. In this article, we study the algebraicity over extensions of a real quadratic field K of the projections of Darmon points to elliptic curves, which coincide with the points on elliptic curves previously defined by M. Greenberg. More precisely, we prove that linear combinations of Darmon points on elliptic curves weighted by certain genus characters of K are rational over the predicted genus fields of K. This extends to an arbitrary quaternionic setting the main theorem on the rationality of Stark-Heegner points obtained by Bertolini and Darmon, and at the same time gives evidence for the rationality conjectures formulated in a joint paper with Rotger and by Greenberg in his article on Stark-Heegner points. In light of this result, quaternionic Darmon points represent the first instance of a systematic supply of points of Stark-Heegner type other than Darmon's original ones for which explicit rationality results are known. © 2013 The Author(s) 2013. Published by Oxford University Press. All rights reserved.