Computation of ATR Darmon points on nongeometrically modular elliptic curves
ATR points were introduced by Darmon as a conjectural construction of algebraic points on certain elliptic curves for which in general the Heegner point method is not available. So far the only numerical evidence, provided by Darmon-Logan and Gärtner, concerned curves arising as quotients of Shimura...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2013 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/193372 |
| Acceso en línea: | https://hdl.handle.net/2445/193372 |
| Access Level: | acceso abierto |
| Palabra clave: | Teoria de nombres Geometria algebraica aritmètica Funcions L Grups discontinus Number theory Arithmetical algebraic geometry L-functions Discontinuous groups |
| Sumario: | ATR points were introduced by Darmon as a conjectural construction of algebraic points on certain elliptic curves for which in general the Heegner point method is not available. So far the only numerical evidence, provided by Darmon-Logan and Gärtner, concerned curves arising as quotients of Shimura curves. In those special cases the ATR points can be obtained from the already existing Heegner points, thanks to results of Zhang and Darmon-Rotger-Zhao. In this paper we compute for the first time an algebraic ATR point on a curve which is not uniformizable by any Shimura curve, thus providing the first piece of numerical evidence that Darmon's construction works beyond geometric modularity. To this purpose we improve the method proposed by Darmon and Logan by removing the requirement that the real quadratic base field be norm-euclidean, and accelerating the numerical integration of Hilbert modular forms. |
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