Mathematics – Number Theory
Scientific paper
2011-05-18
Mathematics
Number Theory
34 pages
Scientific paper
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.
Longo Manfredi
Vigni Stefano
No associations
LandOfFree
The rationality of quaternionic Darmon points over genus fields of real quadratic fields does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The rationality of quaternionic Darmon points over genus fields of real quadratic fields, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The rationality of quaternionic Darmon points over genus fields of real quadratic fields will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-54657