Almost totally complex points on elliptic curves
Authors:
Xavier Guitart, Victor Rotger and Yu Zhao
Journal:
Trans. Amer. Math. Soc. 366 (2014), 27732802
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 Darmonlike 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 higherdimensional 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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 C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over : wild 3adic exercises, J. Amer. Math. Soc. 14 (2001), no.4, 843939. MR 1839918 (2002d:11058)
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 B.H. Gross and D.B. Zagier, Heegner points and derivatives of series. Invent. Math. 84 (1986), no.2, 225320. MR 833192 (87j:11057)
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 X. Guitart, M. Masdeu, Elementary matrix decomposition and the computation of Darmon points with higher conductor, to appear in Mathematics of Computation.
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 M. Longo, V. Rotger, S. Vigni, On rigid analytic uniformizations of Jacobians of Shimura curves, to appear in American J. Math.
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 M. Longo, S. Vigni, The rationality of quaternionic Darmon points over genus fields of real quadratic fields, preprint 2011.
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 J. S. Milne, Introduction to Shimura varieties, available at http://www.jmilne.org/math.
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 J. S. Milne, On the arithmetic of abelian varieties, Invent. Math. 178 (2009), no.3, 485504. MR 0330174 (48:8512)
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 Y. Matsushima, G. Shimura, On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes, Ann. Math. (2) 78 (1963), 417449. MR 0155340 (27:5274)
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 J. Quer, Fields of definition of building blocks, Math. Comp. 78 (2009), 537554. Appendix available at http://arxiv.org/abs/1202.3061 MR 2448720 (2010a:11109)
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Additional Information
Xavier Guitart
Affiliation:
MaxPlanckInstitute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany – and – Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, C. Jordi Girona 13, 08034 Barcelona, Spain
Address at time of publication:
Institut für Experimentelle Mathematik, Universität DuisburgEssen, 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 13, 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/S000299472013059818
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.
