Inventiones mathematicae

, Volume 201, Issue 1, pp 159–206 | Cite as

Elliptic curves over real quadratic fields are modular

  • Nuno Freitas
  • Bao V. Le Hung
  • Samir Siksek


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

Mathematics Subject Classification

Primary 11F80 Secondary 11G05 



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.

Supplementary material

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA
  3. 3.Mathematics InstituteUniversity of WarwickCoventryUK

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