Etale Steenrod operations and the Artin-Tate pairing

We prove a 1966 conjecture of Tate concerning the Artin-Tate pairing on the Brauer group of a surface over a finite field, which is the analogue of the Cassels-Tate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of Poonen-Stoll on the Cassels-Tate pairing. Our method is based on studying a connection between the Artin-Tate pairing and (generalizations of) Steenrod operations in étale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate the relevant étale Steenrod operations in terms of characteristic classes.