In this talk I will describe joint work with Ted Chinburg on cup products in the etale cohomology of curves over finite fields.
The main result for elliptic curves is that computing cup products of so-called "normalized classes" is no more difficult than computing the Weil pairing of torsion points. For classes that are not normalized, a new pairing arises. We call this the Legendre derivative of Frobenius because of its formal similarity to the Legendre transform in real analysis.
For higher genus curves, we give a necessary and sufficient condition for the same statement about normalized classes to hold. Already in genus 2 there are examples for which this condition does not hold.