Generalization of Kummer isomorphism?

Question

This is a question I asked on math.stackexchange without success.

Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ via the mod $p$ cyclotomic character.

We have a nice description of the vector space $H^1(G, \mathbb{F}_p(1))$ given by the Kummer isomorphism $H^1(G, \mathbb{F}_p(1)) \simeq \mathbb{Q}_p^* / \left(\mathbb{Q}_p^*\right)^p$.

Is there also a nice description of $H^1(G, \mathbb{F}_p(n))$ where $n$ is any (positive) integer? (here $\mathbb{F}_p(n)$ is just $\mathbb{F}_p$ with the action of the $n$-th power of the mod $p$ cyclotomic character)

Answer 1

It depends what you mean by "nice". There is, to the best of my knowledge, no such easy explicit description of $H^1(G,F_p(n))$ if $n \neq 0,1$. If $n=1$, it's Kummer theory as you have described. If $n=0$, the $H^1$ is the group of homomorphisms of $G$ into $F_p$, so it boils down to local class field theory.

In general, you can compute the dimension of $H^1(G,F_p(n))$ using the argument suggested by Timo Keller.

If you were looking at $H^1(G,Q_p(n))$ instead of $H^1(G,F_p(n))$, then you'd still have the "explicit" description as above, but in addition for $n \geq 2$, you'd have the Bloch-Kato exponential map, $exp_{Q_p(n)}$ which is (for $n \geq 2$) an isomorphism between a certain $1$-dimensional vector space $D_{cris}(Q_p(n))$ and $H^1(G,Q_p(n))$. This map is however not naturally "integral". This is typical of $p$-adic Hodge theory: as the "weight" increases, you have to replace simple integrality properties by more complicated $p$-adic analytical properties.

Another direction would be to use the theory of $(\phi,\Gamma)$-modules. The elements of $H^1(G,F_p(n))$ can then be seen as pairs of elements of $F_p[[X]][1/X]$ satisfying certain properties.

Answer 2

See [Neukirch-Schmidt-Wingberg], Cohomology of Number Fields, 2nd edition, p. 400, Proposition (7.3.10).

The idea is to directly calculate $H^0$, deduce from duality $H^2$ and then calculate the local Euler-Poincaré characterstics.