  Darmon points on p-adic tori and Jacobians of Shimura curves over Q were
introduced in previous joint works 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. 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 M. 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.
