Abstract
We reinterpret Kim's nonabelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of etale 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 BrauerManin obstruction, allowing us to determine when Kim's maps recover the BrauerManin 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 adelic elliptic curve with global Tate module underlying a global elliptic curve.
Original language  English 

Pages (fromto)  699756 
Journal  Algebraic and Geometric Topology 
Volume  20 
Issue number  2 
DOIs  
Publication status  Published  23 Apr 2020 
Keywords
 math.AT
 math.AG
 math.NT
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Jon Pridham
 School of Mathematics  Personal Chair of Derived Algebraic Geometry
Person: Academic: Research Active