## Experimental Mathematics

- Experiment. Math.
- Volume 13, Issue 3 (2004), 327-329.

### Existence of Nonelliptic mod ${\ell}$ Galois Representations for Every $\ell > 5$

#### Abstract

For $\ell =$ 3 and 5 it is known that every odd, irreducible, two-dimensional representation of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ and determinant equal to the cyclotomic character must "come from'' the $\ell$-torsion points of an elliptic curve defined over $\Q$. We prove, by giving concrete counter-examples, that this result is false for every prime $\ell > 5$.

#### Article information

**Source**

Experiment. Math., Volume 13, Issue 3 (2004), 327-329.

**Dates**

First available in Project Euclid: 22 December 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1103749840

**Mathematical Reviews number (MathSciNet)**

MR2103330

**Zentralblatt MATH identifier**

1091.11018

**Subjects**

Primary: 11G05: Elliptic curves over global fields [See also 14H52]

Secondary: 11F80: Galois representations

**Keywords**

Elliptic curves Galois representations

#### Citation

Dieulefait, Luis. Existence of Nonelliptic mod ${\ell}$ Galois Representations for Every $\ell > 5$. Experiment. Math. 13 (2004), no. 3, 327--329. https://projecteuclid.org/euclid.em/1103749840