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 $\ell > 5$.