The Realizability Problem with Inertia Conditions

In this paper, we consider the inverse Galois problem with described inertia behavior. For a finite group $G$, one of its subgroups $I$ and a prime integer $p$, we ask whether or not $G$ and $I$ can be realized as the Galois group and the inertia subgroup at $p$ of an extension of $\mathbb{Q}$. We first discuss the result when $G$ is an abelian group. Then in the case that $G$ is of odd order, Neukirch showed that there exists such an extension if and only if the given inertia condition is realizable over $\mathbb{Q}_p$, from which we obtain the answer for this case by studying the structure of extensions of $\mathbb{Q}_p$ and applying techniques from embedding problems. As a corollary, we give an explicit presentation of the Galois group of the maximal pro-odd extension of $\mathbb{Q}_p$. When $G=\operatorname{GL}_2(\mathbb{F}_p)$ for an odd prime $p$, we relate our realizability problem to modular Galois representations and use elliptic curves to give answers for those subgroups $I$ corresponding to weight 2 modular forms. Finally, we provide an example arising from Grunwald-Wang's counterexample for which the local-global principle of our realizability problem fails.