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.
"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