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Proving modularity for a given elliptic curve over an imaginary quadratic field


Authors: Luis Dieulefait, Lucio Guerberoff and Ariel Pacetti
Journal: Math. Comp. 79 (2010), 1145-1170
MSC (2000): Primary 11G05; Secondary 11F80
DOI: https://doi.org/10.1090/S0025-5718-09-02291-1
Published electronically: August 4, 2009
MathSciNet review: 2600560
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Abstract: We present an algorithm to determine if the $ L$-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree. By the work of Harris-Soudry-Taylor, Taylor, and Berger-Harcos, we can associate to an automorphic representation a family of compatible $ \ell$-adic representations. Our algorithm is based on Faltings-Serre's method to prove that $ \ell$-adic Galois representations are isomorphic. Using the algorithm we provide the first examples of modular elliptic curves over imaginary quadratic fields with residual $ 2$-adic image isomorphic to $ S_3$ and $ C_3$.


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Additional Information

Luis Dieulefait
Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585. 08007 Barcelona, Spain
Email: ldieulefait@ub.edu

Lucio Guerberoff
Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria. C.P:1428, Buenos Aires, Argentina - Institut de Mathématiques de Jussieu, Université Paris 7, Denis Diderot, 2, place Jussieu, F-75251 Paris Cedex 05, France
Email: lguerb@dm.uba.ar

Ariel Pacetti
Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria. C.P:1428, Buenos Aires, Argentina
Email: apacetti@dm.uba.ar

DOI: https://doi.org/10.1090/S0025-5718-09-02291-1
Keywords: Elliptic Curves Modularity
Received by editor(s): November 5, 2008
Received by editor(s) in revised form: April 7, 2009
Published electronically: August 4, 2009
Additional Notes: The second author was supported by a CONICET fellowship
The third author was partially supported by PICT 2006-00312 and UBACyT X867
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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