Steenrod operations and symplectic arithmetic duality

TL;DR

This paper introduces a conjecture about symplectic forms on Brauer groups of surfaces over finite fields, discusses related work, and provides an overview of the proof, connecting algebraic geometry and arithmetic duality.

Contribution

It offers an accessible exposition of a conjecture linking symplectic structures to Brauer groups and summarizes recent proof developments.

Findings

Confirmation of the conjecture's plausibility

Outline of the proof strategy from recent literature

Connections between Steenrod operations and arithmetic duality

Abstract

This expository article elaborates upon my talk at the 2025 AMS Summer Institute on Algebraic Geometry. It gives an introduction to a conjecture from Tate's 1966 S\'eminaire Bourbaki report, predicting the existence of a symplectic form on Brauer groups of surfaces over finite fields, and then an informal tour of the proof in \cite{Feng20} and \cite{CF}.