Cup products on curves over finite fields

TL;DR

This paper computes cup products in étale cohomology for curves over finite fields, revealing their relation to Weil pairings and the Picard group, with distinctions between algebraic closure and base field cases.

Contribution

It provides explicit calculations of cup products on curves over finite fields, connecting étale cohomology, Weil pairings, and Picard groups, highlighting differences over the base field.

Findings

Cup products relate to Weil pairings over algebraic closure.

Cup products over the base field take values in the Picard group modulo n.

Explicit formulas for cup products in étale cohomology of curves.

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 \'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 $\mathrm{H}^2(\overline{C},\mu_n) = \mathbb{Z}/n$.