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 $A$.

arXiv comment9 pages