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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