Bad places for the approximation property for finite groups

TL;DR

This paper investigates the limitations of the Tame Approximation Problem for finite groups over number fields, demonstrating that certain obstructions are sharp by constructing explicit counterexamples involving abelian groups.

Contribution

It proves that the set of bad places for the approximation property is precisely characterized, providing explicit counterexamples with abelian groups where surjectivity fails.

Findings

The set Bad_G is sharp for the Tame Approximation Problem.

Explicit counterexamples are constructed for abelian groups.

Surjectivity of the cohomology restriction map can fail at specific places.

Abstract

Given a number field $k$ and a finite $k$-group $G$, the Tame Approximation Problem for $G$ asks whether the restriction map $H^1(k,G)\to\prod_{v\in\Sigma}H^1(k_v,G)$ is surjective for every finite set of places $\Sigma\subseteq\Omega_k$ disjoint from $\text{Bad}_G$, where $\text{Bad}_G$ is the finite set of places that either divides the order of $G$ or ramifies in the minimal extension splitting $G$. In this paper we prove that the set $\text{Bad}_G$ is "sharp". To achieve this we prove that there are finite abelian $k$-groups $A$ where the map $H^1(k,A)\to\prod_{v\in\Sigma_0}H^1(k_v,A)$ is not surjective in a set $\Sigma_0\subseteq\text{Bad}_A$ with particular properties, namely $\Sigma_0$ is the set of places that do not divide the order of $A$ and ramify in the minimal extension splitting $A$.