The Neukirch-Uchida theorem with restricted ramification

Abstract: Let $K$ be a number field and $S$ a set of primes of $K$. We write $K_S/K$ for the maximal extension of $K$ unramified outside $S$ and $G_{K,S}$ for its Galois group. In this paper, we prove the following generalization of the Neukirch-Uchida theorem under some assumptions: "For $i=1,2$, let $K_i$ be a number field and $S_i$ a set of primes of $K_i$. If $G_{K_1,S_1}$ and $G_{K_2,S_2}$ are isomorphic, then $K_1$ and $K_2$ are isomorphic." Here the main assumption is that the Dirichlet density of $S_i$ is not zero for at least one $i$. A key step of the proof is to recover group-theoretically the $l$-adic cyclotomic character of an open subgroup of $G_{K,S}$ for some prime number $l$.