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Étale Steenrod operations and the Artin–Tate pairing

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  13 July 2020

Tony Feng
Tony Feng*
Affiliation:
Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA02139, USA email [email protected]
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Abstract

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 analog 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.

Keywords

arithmetic geometryBrauer groupsSteenrod operationsArtin–Tate conjecture

MSC classification

Primary: 11G25: Varieties over finite and local fields
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 7 , July 2020 , pp. 1476 - 1515
Copyright
© The Author 2020

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Footnotes

The author was supported by an NSF Graduate Fellowship, a Stanford ARCS Fellowship, and an NSF Postdoctoral Fellowship during the completion of this paper.

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