Decomposition group of $G_{K,S}$ Let $K$ be a number field and fix a prime $p$. Let S be a finite set of places of $K$ containing the places above $p$ and the infinite places. Let $G_{K,S}$ be the absolute Galois group of the maximal extension of $K$ in a fixed algebraic closure of $K$ unramified outside $S$.
1) Let $\mathfrak{p}$ be a prime of $K$ above $p$. Then what is meant by the decomposition group of $G_{K,S}$ at $\mathfrak{p}$, and is it equal to the decomposition group of $G_K$ at $\mathfrak{p}$? $\mathfrak{p}$ is not a place of the maximal extension unramified outside $S$, so I do not know how to make sense out of this.
2) Let further $\rho: G_{K,S}\to GL_2(\mathbb{F}_p)$ be a continuous absolutely irreducible representation. For $v$ a place not in $S$, how is $\rho(Frob_v)$ (or rather its trace) well-defined, where $Frob_v\in G_{K,S}$ is a Frobenius element? I am not assuming any unramifiedness, and again the question is how to make sense of $v$ as a place of the maximal extension unramified outside $S$.
 A: (1) "The decomposition group at $\mathfrak{p}$" is not well defined on the nose as a subgroup of $G_{K, S}$. It depends on a choice of a prime above $\mathfrak{p}$ in $K^S$ (the maximal extension unramified outside $S$). However, any two such primes are in the same $G_{K, S}$-orbit, so their decomposition groups are conjugate in $G_{K, S}$. So if you're intending to (for instance) analyse the restriction of a $G_{K, S}$-representation to a decomposition group of a prime above $\mathfrak{p}$, then it doesn't matter which prime above $\mathfrak{p}$ you pick.
(2) You write "I am not assuming any unramifiedness..." -- but you are, because you're assuming $v$ is not in $S$. If $w$ is a choice of prime of $K^S$ above $v$, then $(K^S)_w / K_v$ is unramified and the decomposition group of $w$ in $K^S / K$ is (topologically) generated by a uniquely determined Frobenius element. This element depends on $w$, but its conjugacy class does not. So $\rho(Frob_v)$ is well-defined up to conjugation in $GL_2$, and hence has a uniquely defined trace.
(As an aside: among these two questions, which fairly elementary, you have smuggled in a seriously deep question! You ask if the decomposition groups of a prime above $\mathfrak{p}$ in $G_K$ and $G_{K, S}$ are the same. If $K = \mathbf{Q}$ and $|S| \ge 2$ this is indeed true, but this is a rather recent theorem, proved in 2009 by Chenevier and Clozel using sophisticated techniques from the Langlands program.)
