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2020 Nonabelian reciprocity laws and higher Brauer–Manin obstructions
Jonathan P Pridham
Algebr. Geom. Topol. 20(2): 699-756 (2020). DOI: 10.2140/agt.2020.20.699

Abstract

We reinterpret Kim’s nonabelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of étale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer–Manin obstruction, allowing us to determine when Kim’s maps recover the Brauer–Manin locus. A tower based on relative completions yields nontrivial reciprocity maps even for Shimura varieties; for the stacky modular curve, these take values in Galois cohomology of modular forms, and give obstructions to an adèlic elliptic curve with global Tate module underlying a global elliptic curve.

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Jonathan P Pridham. "Nonabelian reciprocity laws and higher Brauer–Manin obstructions." Algebr. Geom. Topol. 20 (2) 699 - 756, 2020. https://doi.org/10.2140/agt.2020.20.699

Information

Received: 27 January 2018; Revised: 17 May 2019; Accepted: 10 June 2019; Published: 2020
First available in Project Euclid: 30 April 2020

zbMATH: 07195375
MathSciNet: MR4092310
Digital Object Identifier: 10.2140/agt.2020.20.699

Subjects:
Primary: 55Q05
Secondary: 11D99, 14F35, 55S35

Rights: Copyright © 2020 Mathematical Sciences Publishers

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