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 be a quadratic extension of totally real number fields, and let be an elliptic curve over which is isogenous to its Galois conjugate over . A quadratic extension is said to be almost totally complex (ATC) if all archimedean places of but one extend to a complex place of . The main goal of this note is to provide a new construction for a supply of Darmon-like points on , which are conjecturally defined over certain ring class fields of . 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.