Steenrod operations and symplectic arithmetic duality

TL;DR

The paper addresses Tate's Symplecticity Conjecture, which predicts a symplectic structure on the Brauer group of smooth projective surfaces over finite fields, realized through the Milne--Artin--Tate pairing. It develops a unifying dictionary across topological, $oldsymbol{ ext{ell}}$-adic, and $p$-adic cohomology, then connects the pairing to Steenrod operations via a key Bockstein identity, reducing the alternation problem to lifting a Wu-class $v_{2d}$ in cohomology. For $oldsymbol{ extell} eq p$, the arithmetic Wu formula and lifting arguments prove symplecticity, while for $oldsymbol{ extell}=p$ the paper CF introduces two incarnations of syntomic Steenrod operations, perfectoid nearby cycles, and spectral prismatization to establish the defining symplecticity in characteristic $p=2$. Collectively, these developments complete Tate's conjecture in all characteristics and generalize the underlying duality structures through modern homotopical and prismatic methods, with broader implications for Artin--Tate type conjectures and BSD analogies in positive characteristic.

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 Séminaire 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}.

Figures (3)

• Figure 1: This roadmap (animated by Nano Banana Pro) depicts the long, winding journey to Theorem \ref{['thm:symplecticityatp']}.
• Figure 2: $\ell k(L, K)=2$
• Figure 3: Organization of the remaining sections of this paper.

Theorems & Definitions (36)

• Conjecture 1.2.1: Tate's symplecticity conjecture
• Theorem 1.4.1
• Theorem 1.4.2
• Remark 1.4.3: Higher dimensional generalizations
• Remark 2.1.1
• Lemma 2.3.1
• proof
• Lemma 2.3.2
• proof
• Definition 2.4.1: Milne--Artin--Tate pairing