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Title: Almost totally complex points on elliptic curves
Author: Guitart Morales, Xavier
Rotger, Victor
Zhao, Yu
Keywords: Funcions L
Geometria algebraica aritmètica
Teoria de nombres
Corbes el·líptiques
Arithmetical algebraic geometry
Number theory
Elliptic curves
Issue Date: May-2014
Publisher: American Mathematical Society (AMS)
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.
Note: Versió postprint del document publicat a:
It is part of: Transactions of the American Mathematical Society, 2014, vol. 366, num. 5, p. 2773-2802
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ISSN: 0002-9947
Appears in Collections:Articles publicats en revistes (Matemàtiques i Informàtica)

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