MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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)

share|cite|improve this question
The correct generalization of the Kummer isomorphism may be the one given by the Bloch-Kato conjecture (aka Theorem of Rost-Voevodsky - Merkurjev-Suslin) which relates Galois cohomology and Milnor K-theory: for a field $F$, $H^n(G_F,\mathbf Z/\ell\mathbf Z(n))\simeq K_n(F)\otimes \mathbf Z/\ell\mathbf Z$. See the book by Gille and Szamuely for a first introduction. – ACL Jul 31 '13 at 18:21
up vote 6 down vote accepted

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.

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.