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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/195115
Almost totally complex points on elliptic curves
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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.
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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: 30 of June of 2026]. Available at: https://hdl.handle.net/2445/195115