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Elliptic curves over real quadratic fields are modular

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Abstract

We prove that all elliptic curves defined over real quadratic fields are modular.

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Notes

  1. In \({{\mathrm{GL}}}_2({\mathbb {F}}_3)\) the normalizer of a split Cartan subgroup is contained as an index \(2\) subgroup in the normalizer of a non-split Cartan subgroup, as the latter is a 2-Sylow subgroup.

  2. Indeed, the absence of (c) is the reason why we do not yet have a parametrization of quadratic points on the family \(X(\mathrm {b}N)\), even though (a) and (b) are known [37] for prime \(N>71\), where \(\mathcal {A}\) is the Eisenstein quotient of \(J_0(N)\), and \(p\) is any prime \(\ne 2\), \(3\), \(5\).

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Acknowledgments

We would like to thank the referees for useful comments. It is a pleasure to express our sincere gratitude to a large number of colleagues for their help and advice during the course of writing this paper: Samuele Anni, Alex Bartel, Peter Bruin, Frank Calegari, Tommaso Centeleghe, John Cremona, Lassina Dembélé, Fred Diamond, Luis Dieulefait, Toby Gee, Ariel Pacetti, Richard Taylor and Damiano Testa. We would also like to thank Rajender Adibhatla, Shuvra Gupta, Derek Holt, David Loeffler and Panagiotis Tsaknias for useful discussions.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Bayreuth, 95440, Bayreuth, Germany

    Nuno Freitas

  2. Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA, 02138, USA

    Bao V. Le Hung

  3. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Samir Siksek

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  1. Nuno Freitas
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  2. Bao V. Le Hung
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  3. Samir Siksek
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Correspondence to Samir Siksek.

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The first-named author is supported through a Grant within the framework of the DFG Priority Programme 1489 Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory. The third-named author is supported by an EPSRC Leadership Fellowship EP/G007268/1, and EPSRC LMF: L-Functions and Modular Forms Programme Grant EP/K034383/1.

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Freitas, N., Le Hung, B.V. & Siksek, S. Elliptic curves over real quadratic fields are modular. Invent. math. 201, 159–206 (2015). https://doi.org/10.1007/s00222-014-0550-z

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