# Elliptic curves over real quadratic fields are modular

@article{Freitas2013EllipticCO, title={Elliptic curves over real quadratic fields are modular}, author={Nuno Freitas and Bao V. Le Hung and Samir Siksek}, journal={Inventiones mathematicae}, year={2013}, volume={201}, pages={159-206} }

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

## 83 Citations

On the modularity of elliptic curves over a composite field of some real quadratic fields

- Mathematics
- 2016

Let K be a composite field of some real quadratic fields. We give a sufficient condition on K such that all elliptic curves over K are modular.

Elliptic curves over Q$_{∞}$ are modular

- Mathematics
- 2015

We show that if p is a prime, then all elliptic curves defined over the cyclotomic Zp-extension of Q are modular.

Elliptic curves over totally real cubic fields are modular

- Mathematics
- 2019

We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real…

Modularity of some elliptic curves over totally real fields

- Mathematics
- 2014

We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some…

Torsion of Q-curves over quadratic fields

- Mathematics
- 2019

We find all the possible torsion groups of $\Q$-curves over quadratic fields and determine which groups appear finitely and which appear infinitely often.

Darmon points on elliptic curves over number fields of arbitrary signature

- Mathematics, Computer Science
- 2015

New constructions of complex and p ‐adic Darmon points on elliptic curves over base fields of arbitrary signature are presented and it is conjecture that these points are global.

Elliptic curves over totally real quartic fields not containing $\sqrt{5}$ are modular

- Mathematics
- 2021

We prove that every elliptic curve defined over a totally real number field of degree 4 not containing √ 5 is modular. To this end, we study the quartic points on four modular curves.

On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences

- Mathematics
- 2021

We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the from y = x(x + b), where b is any positive integer.…

Modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7

- Mathematics
- 2016

This version improves the old version entitled "On the modularity of elliptic curves with a residually irreducible representation".
Let $E$ be an elliptic curve over an abelian totally real field…

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