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 $\mathrm{H}^1(C,\mathbb{Z}/n)$ and $\mathrm{H}^1(C,\mu_n)$. Over the algebraic closure $\overline{k}$ of $k$, such cup products are connected to values of the Weil pairing on the $n$-torsion of the Jacobian of $\overline{C} = \overline{k} \otimes_k C$ by using a fixed isomorphism between $\mathbb{Z}/n$ and $\mu_n$ over $\overline{C}$. Over $k$, such cup products are more subtle due to the fact that they take values in the group $\mathrm{H}^2(C,\mu_n)=\mathrm{Pic}(C)/n\cdot \mathrm{Pic}(C)$ rather than in the group $\mathrm{H}^2(\overline{C},\mu_n) = \mathbb{Z}/n$.