2011/05/18 by Matteo Longo, Longo, M., Stefano Vigni +1 · 1 citation
Mathematics · #11G05 #14G05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1105.3721
openalex publication_date 2011/05/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.