Cup products on curves over finite fields

Abstract

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 étale 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 ${\mathrm{H}}^2(\overline{C},{\mu }_n)=\mathbb{Z}/n$.

30 pages; the paper has been rewritten to deal with the general case when the prime $\ell$ is replaced by an arbitrary positive integer $n$ that is relatively prime to the characteristic of the base field $k$ and $k$ does not necessarily contain a primitive $n$-th root of unity

References in corpus (1)

• Cup products in the etale cohomology of number fields