  In this paper we compute cup products of elements of the first \'etale
cohomology group of $\mu_\ell$ over a smooth projective geometrically
irreducible curve $C$ over a finite field $k$ when $\ell$ is a prime and $\#k
\equiv 1$ mod $\ell$. Over the algebraic closure of $k$, such products are
values of the Weil pairing on the $\ell$-torsion of the Jacobian of $C$. Over
$k$, such cup products are more subtle due to the fact that they naturally take
values in $\mathrm{Pic}(C) \otimes_{\mathbb{Z}} \tilde{\mu}_\ell$ rather than
in the group $\tilde{\mu}_\ell$ of $\ell^{\mathrm{th}}$ roots of unity in
$k^*$.
