Let $V$ be p-torsion representation. Then, $H^n(K,V)＝0$ for all $n≧3$ in the context of class field theory

Question

Let $V$ be p-torsion representation. Then, $H^n(K,V)＝0$ for all $n≧3$, where $K$ is absolute Galois group of $\Bbb {Q_p}$.

I heard this is classical result of class field theory, but also proved without using class field theory via theory of $(φ,Γ)$modules.

But I have never met this statement in the context of class field theory. Could you give me reference or self-contained explanation of the proof of the titled statement in the context of class field theory ?

Answer 1

See Serre's Galois cohomology, proposition II.2.4 for the following statement:

Proposition Let $k$ be a field of characteristic $\neq p$, and let $n$ be an integer $\geq 1$. Then the following conditions are equivalent:

• $\operatorname{cd}_p(G_k) \leq n$, where $\operatorname{cd}_p$ denotes the $p$-cohomological dimension.
• For any separable finite extension $K/k$, we have $H^{n+1}(K, \Bbb G_m)=0$ and $H^n(K,\Bbb G_m)$ is $p$-divisible.

Now one important theorem of local class field theory is that $H^2(K,\Bbb G_m)\cong \Bbb Q/\Bbb Z$ for a local field $K$, which is divisible.
Another important theorem for class field theory is Tate's theorem in group cohomology which implies that $$H^3(K,\Bbb G_m) \cong H^1(K,\Bbb Z)=\mathrm{Hom}_{cont}(G_K,\Bbb Z)=0$$