Soulé vanishing for $H^2$ of $\mathbb{Q}_p(n)$ In his paper "The motivic fundamental group of the projective line minus three points and the theorem of Siegel", M. Kim uses the following result:
let $p$ be an odd prime, $T$ be a finite set of primes of $\mathbb{Z}$ that contains $p$, and let $\mathbb{Q}_T$ be the maximal extension of $\mathbb{Q}$ unramified outside $T$; finally, let $G_T$ be the absolute Galois group of $\mathbb{Q}_T$. Then
$$
H^1(G_T, \mathbb{Q}_p(2n))=0 \quad \forall n \geq 1 \quad \text{and}\quad \dim H^1(G_T, \mathbb{Q}_p(2n+1)) =1 \quad \forall n \geq 1.
$$
In Kim's lectures at the IHES (see http://www.ihes.fr/~abbes/CAGA/kim.html, lecture 3, around minute 55), he explains that the equalities above are proven by studying the Euler characteristic of the Galois module $\mathbb{Q}_p(n)$ and using the fact that $H^2(G_T, \mathbb{Q}_p(n))=0$ by "Soulé vanishing". The reference for this result given in Kim's paper is to Soulé's "K-théorie des anneaux d'entiers de corps de nombres et cohomologie étale" (Inventiones mathematicae (1979), volume 55, pp. 251-296); I've tried reading part of this paper, but unfortunately I don't know the first thing about K-theory, and I don't see how to extract the statement $H^2(G_T, \mathbb{Q}_p(n))=0$ from Soulé's results; more precisely, one of the main sources of my confusion is the fact that most of the theorems in Soulé's paper seem to only be stated for even $n$ (see for example his Théorème 6, part (iii), which gives a surjective morphism from $K_{2i-2}(A) \otimes \mathbb{Z}_\ell$ onto $H^2(\operatorname{Spec}A', \mathbb{Z}_\ell(i))$ only for $i$ even), while - if I understand correctly - Kim's proof needs the result for all values of $n$.
It is more than likely that I've misunderstood something either in Kim's explanation or in Soulé's paper, and I would be more than grateful for any help with the matter.
 A: The results that you cite can be generalized to an arbitrary number field $F$ . Keeping your notations (over $F$ instead of $\mathbf Q$), it is classically known that for $p$ odd, $cd_p (G_T) \le 2$. Recall also that $H^i(G_T, \mathbf Q_p (m))= H^i(G_T, \mathbf Z_p (m))\otimes \mathbf Q_p$, so that the problem amounts to computing the $\mathbf Z_p$-ranks of $H^i(G_T, \mathbf Z_p (m))$ for $i=1,2$. Now, the starting point is a result of Tate (1976) that in the cohomology of the exact sequence $0\to \mathbf Z_p (m)\to \mathbf Q_p (m)\to \mathbf Q_p /\mathbf Z_p (m)\to 0$, the coboundary map $ H^i(G_T, \mathbf Q_p/\mathbf Z_p (m))\to H^{i+1}(G_T, \mathbf Z_p (m))$ has kernel Div $H^i(G_T, \mathbf Q_p/\mathbf Z_p (m))$ (where Div = maximal divisible subgroup) and image $H^{i+1}(G_T, \mathbf Z_p (m))_{tor}$ (where $(.)_{tor}$ = torsion subgroup). It follows that rank $H^i(G_T, \mathbf Z_p (m))=$ corank $H^i(G_T, \mathbf Q_p/\mathbf Z_p (m))$ (where rank means $\mathbf Z_p$-rank and corank = rank of the Pontryagin dual).
Let us first compute for $i=1$. The $G_T$-cohomology of the "$p$-th power" exact sequence  $0\to \mathbf Z/p\mathbf Z (m) \to \mathbf Q_p/\mathbf Z_p(m)\to \mathbf Q_p/\mathbf Z_p(m)\to 0$ easily gives that the corank of $H^1(G_T, \mathbf Q_p/\mathbf Z_p (m))= \chi (G_T,\mathbf Z/p\mathbf Z(m))  + \epsilon + \delta_m (F)$, where $\epsilon =1$ for $m=0$, $\epsilon=1$  otherwise, $\chi$ is the Euler-Poincaré characteristic, and $\delta_m (F)$ = corank $H^2(G_T, \mathbf Q_p/\mathbf Z_p (m))$. It is known that $\chi (G_T,\mathbf Z/p\mathbf Z(m))= -r_2 (F)$ for $m$ even, $-r_2 (F) - r_1 (F)$ for $m$ odd, but the proof is not short (see e.g. Tate, Stockholm ICM 1962).
It remains to compute $\delta_m (F)$ = corank $H^2(G_T, \mathbf Q_p/\mathbf Z_p (m))$. It is a conjecture that for $m\neq 1$, this $H^2$ is null. For $m=0$, this is the famous Leopoldt conjecture, proved by transcendental methods by A. Brumer if $F$ is abelian. For $m\ge 2$, the conjecture holds true for any $F$ by Soulé's theorem: the $p$-adic Chern maps $K_{2m-i} (O_F)\otimes \mathbf Z_p \to H^i(G_T, \mathbf Z_p (m)),m\ge 2, i=1,2$ are surjective with finite kernels. We conclude by using a result of A. Borel in K-theory (1977): for $m\ge 2$, $K_{2m-2} (O_F)$ is finite. Note that the Borel regulator map also gives the $\mathbf Z$- rank of $K_{2m-1} (O_F)$, hence, combined with Soulé's theorem, it yields a direct proof for the rank of $H^1(G_T, \mathbf Z_p (m))$ for $m\ge2$. But no proof not appealing to Borel is known. 
Final remark: the celebrated Quillen-Lichtenbaum conjecture asserts that the Chern class maps above should be isomorphisms for $m\ge 2$. This is now a theorem of Voevodsky, Rost et. al ./.
