Document type
ArticleVersion
Accepted versionPublication date
Publication license
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/195115
Almost totally complex points on elliptic curves
Journal Title
Director/Tutor
Journal ISSN
Volume Title
Related resource
Abstract
Let $F / F_0$ be a quadratic extension of totally real number fields, and let $E$ be an elliptic curve over $F$ which is isogenous to its Galois conjugate over $F_0$. A quadratic extension $M / F$ is said to be almost totally complex (ATC) if all archimedean places of $F$ but one extend to a complex place of $M$. The main goal of this note is to provide a new construction for a supply of Darmon-like points on $E$, which are conjecturally defined over certain ring class fields of $M$. These points are constructed by means of an extension of Darmon's ATR method to higher-dimensional modular abelian varieties, from which they inherit the following features: they are algebraic provided Darmon's conjectures on ATR points hold true, and they are explicitly computable, as we illustrate with a detailed example that provides numerical evidence for the validity of our conjectures.
Subject (English)
Citation
Citation
GUITART MORALES, Xavier, ROTGER, Victor and ZHAO, Yu. Almost totally complex points on elliptic curves. Transactions of the American Mathematical Society. 2014. Vol. 366, num. 5, pags. 2773-2802. ISSN 0002-9947. [consulted: 8 of June of 2026]. Available at: https://hdl.handle.net/2445/195115