Title: Non-abelian reciprocity laws and higher Brauer-Manin obstructions,
Abstract: We reinterpret Kim's non-abelian 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
Brauer--Manin obstruction, allowing us to determine when Kim's maps recover the
Brauer-Manin locus. A tower based on relative completions yields non-trivial
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.