# 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}
}
• Published 26 October 2013
• Mathematics
• Inventiones mathematicae
We prove that all elliptic curves defined over real quadratic fields are modular.
On the modularity of elliptic curves over a composite field of some real quadratic fields
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
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
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
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
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
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

## References

SHOWING 1-10 OF 84 REFERENCES
Modularity of some elliptic curves over totally real fields
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
A First Course in Modular Forms
• Mathematics
• 2008
Modular Forms, Elliptic Curves, and Modular Curves.- Modular Curves as Riemann Surfaces.- Dimension Formulas.- Eisenstein Series.- Hecke Operators.- Jacobians and Abelian Varieties.- Modular Curves
Abelian Varieties over Q and Modular Forms
Let C be an elliptic curve over Q. Let N be the conductor of C. The Taniyama conjecture asserts that there is a non-constant map of algebraic curves X 0 (N) — C which is defined over Q. Here, X o (N)
Serre’s Conjecture for mod 7 Galois Representations
We give an account of Serre’s conjecture for Galois representations with values in GL2(IF7). For this, we construct elliptic curves over totally real soluble extensions with given mod 7
Congruences between Hilbert modular forms: Constructing ordinary lifts, II
• Mathematics
• 2013
In this paper, we improve on the results of our earlier paper [BLGG12], proving a near-optimal theorem on the existence of ordinary lifts of a mod l Hilbert modular form for any odd prime l.
Implementing 2-descent for Jacobians of hyperelliptic curves
• M. Stoll
• Mathematics, Computer Science
• 2001
This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2-Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus
The invariants of a genus one curve
It was first pointed out by Weil that we can use classical invariant theory to compute the Jacobian of a genus one curve. The invariants required for curves of degree n=2, 3, 4 were already known to
On embeddings of modular curves in projective spaces
We use explicit results on modular forms (Muić, Ramanujan J 27:188–208, 2012) via uniformization theory to obtain embeddings of modular curves and more generally of compact Riemann surfaces attached
Bielliptic curves and symmetric products
• Mathematics
• 1991
We show that the twofold symmetric product of a nonhyperelliptic, nonbielliptic curve does not contain any elliptic curves. Applying a theorem of Faltings, we conclude that such a curve defined over