There is a well-known conjecture of Serre that any continuous, irreducible representation $\overline{\rho}:G_\mathbf{Q}\rightarrow {\rm GL}_2(\overline{\mathbf{F}}_\ell)$ which is odd arises from a newform. Here $G_\mathbf{Q}$ is the absolute Galois group of $\mathbf{Q}$ , and $\overline{\mathbf{F}}_\ell$ is an algebraic closure of the finite field $\mathbf{F}_\ell$ of $\ell$ of ℓ elements. We formulate such a conjecture for $n$ -dimensional mod ℓ representations of $\pi_1(X)$ for any positive integer $n$ and where $X$ is a geometrically irreducible, smooth curve over a finite field $k$ of characteristic $p$ ( $p \neq \ell$ ), and we prove this conjecture in a large number of cases. In fact, a proof of all cases of the conjecture for $\ell>2$ follows from a result announced by Gaitsgory in [G]. The methods are different.