Cup products on curves over finite fields

Suppose $k$ is a finite field, that $C$ is a smooth projective geometrically irreducible curve over $k$, and that $n$ is a positive integer not divisible by the characteristic of $k$. In this paper we compute cup products of elements of the 'etale cohomology groups ${\mathrm{H}}^1(C,\mathbb{Z}\mathrm{/}n)$ and ${\mathrm{H}}^1(C,{\mu }_n)$. Over the algebraic closure $\bar{k}$ of $k$, such cup products are connected to values of the Weil pairing on the $n$-torsion of the Jacobian of $\bar{C}=\bar{k}{\otimes }_{k}C$ by using a fixed isomorphism between $\mathbb{Z}\mathrm{/}n$ and ${\mu }_n$ over $\bar{C}$. Over $k$, such cup products are more subtle due to the fact that they take values in the group ${\mathrm{H}}^2(C,{\mu }_n)=\mathrm{P}\mathrm{i}\mathrm{c}(C)\mathrm{/}n\cdot \mathrm{P}\mathrm{i}\mathrm{c}(C)$ rather than in the group ${\mathrm{H}}^2(\bar{C},{\mu }_n)=\mathbb{Z}\mathrm{/}n$.