  Let $\ell$ be a prime, $k$ a finitely generated field of characteristic
different from $\ell$, and $X$ a smooth geometrically connected curve over $k$.
Say a semisimple representation of $\pi_1^{\mathrm{et}}(X_{\bar k})$ is
arithmetic if it extends to a finite index subgroup of
$\pi_1^{\mathrm{et}}(X)$. We show that there exists an effective constant
$N=N(X,\ell)$ such that any semisimple arithmetic representation of
$\pi_1^{\mathrm{et}}(X_{\bar k})$ into $\mathrm{GL}_n(\bar{\mathbb{Z}_\ell})$,
which is trivial mod $\ell^N$, is in fact trivial. This extends a previous
result of the second author from characteristic zero to all characteristics.
The proof relies on a new noncommutative version of Siegel's linearization
theorem and the $\ell$-adic form of Baker's theorem on linear forms in
logarithms.
