Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 

 

Almost totally complex points on elliptic curves


Authors: Xavier Guitart, Victor Rotger and Yu Zhao
Journal: Trans. Amer. Math. Soc. 366 (2014), 2773-2802
MSC (2010): Primary 11G05, 11G40
Published electronically: September 19, 2013
MathSciNet review: 3165655
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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.


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Additional Information

Xavier Guitart
Affiliation: Max-Planck-Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany – and – Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain
Address at time of publication: Institut für Experimentelle Mathematik, Universität Duisburg-Essen, Ellernstr. 29, 45326, Essen, Germany
Email: xevi.guitart@gmail.com

Victor Rotger
Affiliation: Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain
Email: victor.rotger@upc.edu

Yu Zhao
Affiliation: Department of Mathematics, John Abbott College, Montreal, Quebec, Canada H9X 3L9
Email: yu.zhao@johnabbott.qc.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05981-8
Received by editor(s): April 16, 2012
Received by editor(s) in revised form: May 28, 2012, October 2, 2012, and October 4, 2012
Published electronically: September 19, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.