The Rationality of Quaternionic Darmon Points Over Genus Fields of Real Quadratic Fields
| Autor: | Matteo Longo, Stefano Vigni |
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| Jazyk: | angličtina |
| Rok vydání: | 2014 |
| Předmět: |
Pure mathematics
Mathematics - Number Theory General Mathematics Mathematics::Number Theory 14G05 11G05 Rationality Torus Type (model theory) Elliptic curve Mathematics - Algebraic Geometry Quadratic equation Genus (mathematics) FOS: Mathematics Quadratic field Number Theory (math.NT) Linear combination Algebraic Geometry (math.AG) Mathematics |
| Popis: | 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. 34 pages |
| Databáze: | OpenAIRE |
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