Étale Steenrod operations and the Artin–Tate pairing

Tony Feng

Type: Article

Publication Date: 2020-07-01

Citations: 3

DOI: https://doi.org/10.1112/s0010437x20007216

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.

Locations

arXiv (Cornell University) - View - PDF
DataCite API - View
Compositio Mathematica - View

Similar Works

Action	Title	Year	Authors
+	Steenrod Operations in Etale Cohomology and Applications to the Brauer Group of a Surface Over a Finite Field	2017	Tony Feng
+	An Alternating Property for Higher Brauer Groups	2017	Tony Feng
+	The geometric interpretation of the Tate pairing and its applications	2023	Damien Robert
+	The action of the Steenrod algebra on Tate cohomology	1993	David J. Benson J. P. C. Greenlees
+	The Cassels-Tate pairing on 2-Selmer groups of elliptic curves	2023	Himanshu Shukla Michael Stoll
+	p-adic Tate conjectures and abeloid varieties	2019	Oliver Gregory Christian Liedtke
+ PDF Chat	The Tate pairing for Abelian varieties over finite fields	2011	Peter Bruin
+	The Cassels-Tate pairing for finite Galois modules	2021	Adam Morgan Alexander Smith
+ PDF Chat	Visualizing elements of order in the Tate–Shafarevich group of an elliptic curve	2016	Tom Fisher
+ PDF Chat	Dyer–Lashof operations on Tate cohomology of finite groups	2012	Martin Langer
+	The Tate cohomology	2021	Ragnar-Olaf Buchweitz
+ PDF Chat	Recent progress on the Tate conjecture	2017	Burt Totaro
+ PDF Chat	Symmetric operations for all primes and Steenrod operations in algebraic cobordism	2015	Alexander Vishik
+	The Cassels-Tate pairing for finite Galois modules	2021	Adam Morgan Alexander Smith
+	The conjectures of Artin-Tate and Birch-Swinnerton-Dyer	2021	Stephen Lichtenbaum Niranjan Ramachandran Takashi Suzuki
+ PDF Chat	The conjectures of Artin-Tate and Birch-Swinnerton-Dyer	2022	Stephen Lichtenbaum Niranjan Ramachandran T. Suzuki
+ PDF Chat	Higher order Massey products and applications	2021	Ivan Limonchenko D. V. Millionshchikov
+	Stringy power operations in Tate K-theory	2007	Nora Ganter
+	A survey around the Hodge, Tate and Mumford-Tate conjectures for abelian varieties	2016	Victoria Cantoral‐Farfán
+	The Tate conjecture, abelian varieties and K3 surfaces	2021	Jean-Louis Colliot-Thélène Alexei N. Skorobogatov

Works That Cite This (3)

Action	Title	Year	Authors
+	Non-algebraic geometrically trivial cohomology classes over finite fields	2024	Federico Scavia Fumiaki Suzuki
+	An Alternating Property for Higher Brauer Groups	2017	Tony Feng
+	HASSE PRINCIPLES FOR ÉTALE MOTIVIC COHOMOLOGY	2018	Thomas Geisser

Works Cited by This (28)

Action	Title	Year	Authors
+	DUALITY THEOREMS IN GALOIS COHOMOLOGY OVER NUMBER FIELDS	2010	John Tate
+	Etale homotopy of simplicial schemes	1982	Eric M. Friedlander
+	Cubic Forms - Algebra, Geometry, Arithmetic	1986	Michiel Hazewinkel I︠u︡. I. Manin
+	Homotopy obstructions to rational points	2013	J. Harpaz Tomer M. Schlank
+	Etale Cohomology and the Weil Conjecture	1988	Eberhard Freitag Reinhardt Kiehl
+	Cohomology Operations and Applications in Homotopy Theory	2008	Robert E. Mosher Martin C. Tangora
+	Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer.	1965	J. W. S. Cassels
+	RATIONAL SURFACES OVER PERFECT FIELDS. II	1967	Ju I Manin
+	On the Brauer group of a surface	2004	Qing Liu Dino Lorenzini Michel Raynaud
+ PDF Chat	The Cassels-Tate Pairing on Polarized Abelian Varieties	1999	Bjorn Poonen Michael Stoll