Bad places for the approximation property for finite groups

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 $A$.