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 etale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate the relevant etale Steenrod operations in terms of characteristic classes.

中文翻译：

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Étale Steenrod 操作和 Artin-Tate 配对

我们证明了 1966 年 Tate 的猜想，涉及有限域上表面的 Brauer 群上的 Artin-Tate 配对，这是 Cassels-Tate 配对的类似物。Tate 询问这种配对是否总是交替的，我们找到了肯定的答案，鉴于 Poonen-Stoll 在 Cassels-Tate 配对上的工作，这有点令人惊讶。我们的方法基于研究 Artin-Tate 配对和 etale 上同调中的 Steenrod 操作（的推广）之间的联系。受流形代数拓扑类比的启发，我们开发了工具，允许我们根据特征类计算相关的 etale Steenrod 操作。