Nonabelian reciprocity laws and higher Brauer–Manin obstructions

Pridham, Jonathan

doi:10.2140/agt.2020.20.699

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Nonabelian reciprocity laws and higher Brauer–Manin obstructions

Jonathan P Pridham

Algebraic & Geometric Topology 20 (2020) 699–756

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.

Keywords

Diophantine obstructions, étale homotopy

Mathematical Subject Classification 2010

Primary: 55Q05

Secondary: 11D99, 14F35, 55S35

References

Publication

Received: 27 January 2018

Revised: 17 May 2019

Accepted: 10 June 2019

Published: 23 April 2020

Authors

Jonathan P Pridham
School of Mathematics and Maxwell Institute The University of Edinburgh Edinburgh United Kingdom

Volume 20, issue 2 (2020)