I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G\_{Q\_p}$ over $F\_p$ which are in correspondence with 1-dimensional etale (phi,Gamma)-modules over $F\_p((T))$. 


There are finitely many such Galois representations. Moreover, their associated (phi,Gamma)-modules are very simple -- the action of phi and Gamma can be described (on some basis element) as scaling by an element of $F\_p$ (as opposed to $F\_p((T))$).


My question: can one see this directly on the (phi,Gamma)-module side? That is, given a 1-dimensional etale (phi,Gamma)-module $D$ over $F\_p((T))$, find a $D'$ isomorphic $D$ such that $D'$ has a basis in which the matrices for phi and elements of Gamma are in $F\_p^\times$.



One dimensional (phi,Gamma)-modules in char p
OK...I think I see how to do this now. In the end, I am seeing $(p-1)^2$ distinct $(\phi,\Gamma)$-modules which matches well with the Galois side.


To do this, let $D$ be any 1-dimensional etale $(\phi,\Gamma)$-module. Let $e$ be a basis, and set $\phi(e)=h(T)e$ with $h(T) \in F\_p((T))^\times$. Write $h(T) = h\_0 T^a f(T)$ with $h\_0 \in F\_p^\times$ and $f(T) \in F\_p[[T]]$ with $f(0)=1$. 


Changing basis from $e$ to $u(T)e$ with $u(T) \in F\_p((T))^\times$ gives
$$
\phi(u(T)e) = u(T^p)h(T)e = \frac{u(T^p)}{u(T)} h(T) (u(T)e).
$$
I claim one can find $u(T)$ such that $u(T)/u(T^p)$ equals any element of $1+TF\_p[[T]]$. Indeed, for such an element $g(T)$, the infinite product $\prod\_{j=1}^\infty \phi^j(g(T))$ (which hopefully converges since $g(0)=1$) works. 


Thus, we can change basis so that $\phi$ has the form $\phi(e) = h\_0 T^a e$ -- i.e. we can kill off the $f(T)$ term. Further, by making a change of basis of the form $e$ goes to $T^b e$, we may assume that $0 \leq a < p-1$. 


Now, we use the fact that the $\phi$ and $\Gamma$ actions commute (which is a strong condition even in dimension 1). Namely, let $\gamma$ be a generator of $\Gamma$, and set $\gamma e = g(T) e$. Then $\gamma \phi e = \phi \gamma e$ implies
$$
((1+T)^{\chi(\gamma)}-1)^a g(T) = g(T^p) T^a.
$$
Comparing leading coefficients, we see this is only possible if $a=0$ and $g(T)$ is a constant.


Thus, $D$ has a basis $e$ so that $\phi(e) = h\_0 e$ and $\gamma(e) = g\_0 e$ with $h\_0,g\_0 \in F\_p^\times$ as desired.


Does this look okay? Any takers for the 2-dimensional case?


