# On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises

@article{Breuil2001OnTM, title={On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises}, author={Christophe Breuil and Brian Conrad and Fred Diamond and Richard Taylor}, journal={Journal of the American Mathematical Society}, year={2001}, volume={14}, pages={843-939} }

We complete the proof that every elliptic curve over the rational numbers is modular.

## 236 Citations

### Elliptic curves over real quadratic fields are modular

- Mathematics
- 2013

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

### The Joint Universality for L-Functions of Elliptic Curves

- Mathematics
- 2004

A joint universality theorem in the Voronin sense for L-functions of elliptic curves over the field of rational numbers is proved.

### Discrete value: Distribution of L-functions of elliptic curves

- Mathematics
- 2004

A discrete universality theorem in the Voronin sense for L-functions of elliptic curves is proved.

### 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.

### 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…

### 1 8 M ay 2 01 5 Elliptic curves over Q ∞ are modular

- Mathematics, Computer Science
- 2015

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

### 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…

### On the topological aspects of arithmetic elliptic curves

- Mathematics
- 2011

In this short note, we shall construct a certain topological family which contains all elliptic curves over Q and, as an application, show that this family provides some geometric interpretations of…

### MODULAR FORMS AND ELLIPTIC CURVES OVER

- Mathematics
- 2020

. We survey our joint work with Paul Gunnells and Farshid Hajir on a computational investigation of the modularity of elliptic curves over the cyclotomic ﬁeld $ \mathbb{Q}(\zeta_{5})$ , including the…

### Potential modularity for elliptic curves and some applications

- Mathematics, Computer Science
- 2009

## References

SHOWING 1-10 OF 59 REFERENCES

### On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields

- MathematicsNagoya Mathematical Journal
- 1971

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z).…

### Introduction to the arithmetic theory of automorphic functions

- Mathematics
- 1971

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary…

### Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?)

- Mathematics
- 1995

When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This…

### BASE CHANGE FOR GL(2)

- Mathematics
- 1980

R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a cuspdidal automorphic…

### Ramified deformation problems

- Mathematics
- 1999

is absolutely irreducible. His proof relies on extending the scope of the deformation-theoretic tools. The remaining obstacle to having an unconditional proof of the Taniyama-Shimura Conjecture is to…

### Artin’s conjecture for representations of octahedral type

- Mathematics
- 1981

Let L/F be a finite Galois extension of number fields. E. Artin conjectured that the Z-series of a nontrivial irreducible complex representation of Gal(L/F) is entire, and proved this for monomial…

### Groupes $p$-divisibles, groupes finis et modules filtrés

- Mathematics
- 2000

Let k be a perfect field of characteristic p>0. When p>2, Fontaine and Laffaille have classified p-divisibles groups and finite flat p-groups over the Witt vectors W(k) in terms of filtered modules.…

### Modularity of Certain Potentially Barsotti-Tate Galois Representations

- Mathematics
- 1999

where is the reduction of p. These results were subject to hypotheses on the local behavior of p at X, i.e., the restriction of p to a decomposition group at X, and to irreducibility hypotheses on…

### Ring-Theoretic Properties of Certain Hecke Algebras

- Mathematics
- 1995

The purpose of this article is to provide a key ingredient of [W2] by establishing that certain minimal Hecke algebras considered there are complete intersections. As is recorded in [W2], a method…