# Non-abelian reciprocity laws and higher Brauer-Manin obstructions

## TL;DR

This paper reinterprets Kim's non-abelian reciprocity maps as obstruction towers in etale homotopy types, extending the theory to include new obstructions like the Brauer--Manin, with applications to Shimura varieties and modular curves.

## Contribution

It introduces a new perspective on Kim's reciprocity maps as obstruction towers, removing previous technical constraints and extending the framework to broader classes of varieties.

## Key findings

- Obstruction towers can recover the Brauer--Manin locus.
- Non-trivial reciprocity maps exist for Shimura varieties.
- Obstructions relate to Galois cohomology of modular forms.

## 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.