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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 $F_p^\times$.

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I added the etale condition per Brian's correction. –  Robert Pollack Feb 20 '10 at 20:33
    
Hey Rob---I once failed to do that exercise too :-) I could see all the phi-Gamma modules associated to the Galois reps but couldn't prove from first principles that there weren't any more. –  Kevin Buzzard Feb 20 '10 at 20:45

3 Answers 3

up vote 11 down vote accepted

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 $\phi(e) = g_0 e$ with $h_0,g_0 \in F_p^\times$ as desired.

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

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Yes, it looks good. This same style of calculation comes up when using Dieudonne theory to work out p-torsion commutative groups of order p^2, as well as Fontaine-Laffaille modules (as in the original paper of Fontaine-Laffaille). The fun part is to dig out the Galois action in the rings to identify exactly which character corresponds to the parameters (p_0, g_0) (which one can guess in advance, up to some inversions perhaps). I should revise the CMI lecture notes to address the general p-torsion rank-1 case (in the form of an exercise...)! As for the irreducible 2-dimensional case...oy. –  BCnrd Feb 21 '10 at 1:36

The correspondence requires the $(\phi,\Gamma)$-module to have the \'etale property for its underlying $\phi$-module, and this plays an essential role in the proof of the correspondence (see Fontaine's original articles, or the CMI summer school lecture notes on $p$-adic Hodge theory, for example). So I assume you mean to impose the \'etale property. But even for the trivial representation (of dimension 1) the actions of $\phi$ and $\Gamma$ are more complicated than $\mathbf{F}_p^{\times}$-scalings relative to a suitable basis. Do you mean to ask why in a suitable basis the $(\phi,\Gamma)$-modules associated to powers of the mod $p$ cyclotomic character are related to the case of the trivial character by a "Tate twist" on the $(\phi, \Gamma)$-module side? If so, see Example 13.6.6 in the CMI summer school lecture notes (which handles $\mathbf{Z} _p(r)$ more generally, for any $p$-adic field $K$, and can pass to mod $p$ version there in the same way to get to the case you ask about).

Looking back at the current draft of the notes, I now see a typo in the notes there (I think $\chi^t$ at the end should be $\chi^r$). I'd better fix it for the final draft, so thanks for asking the question, Rob!

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Yes. I certainly meant to include etale. In this case (i.e. 1-dimensional in char p), this is simply asserting that $\phi$ is non-zero, yes? Regarding the trivial representation, its associate $(\phi,\Gamma)$-module must just be $F_p((T))$ with the standard $\phi$ and $\Gamma$ actions. Since $\phi(1)=1$ and $\gamma(1)=1$ for all $\gamma \in \Gamma$, with respect to the basis ``1" all of these operators have matrix 1 (certainly in $F_p^\times$). What am I missing? –  Robert Pollack Feb 20 '10 at 20:29
    
The etale property in this rank-1 case mod p is indeed the same as phi being nonzero on some basis vector. Concerning the 2nd part of your comment, nothing is being missed; I was just expressing (perhaps in an unclear way) that there's a lot of nontrivial action on the coefficients. When reading the question I misread it as not accounting for that (but I see it was implicit when you wrote "(on some basis element)"). Anyway, see the Example 13.6.6, which I hope will clear things up. –  BCnrd Feb 20 '10 at 20:42

Hey Rob! You may want to take a look at Colmez's "Représentations triangulines de dimension 2" (here). He's working with (φ, Γ)-modules over Qp (by which I mean the Robba ring of Qp) not Fp((T)) though. But he shows that every one-dimensional (φ, Γ)-module over Qp is given by a p-adic character δ of $\mathbf{Q}_p^\times$ with the action of φ on a basis vector given by δ(p) and the action of $\gamma\in\Gamma$ given by $\delta(\chi(\gamma))$ (with $\chi$ the $p$-adic cyclotomic character). See section 2 and proposition 3.1 in particular of Colmez's paper (in which he also describes the $H^1$'s of these one-dimensional $(\phi,\Gamma)$-modules).

EDIT: looking at Colmez's paper, it looks like the way that he knows that this is all one-dimensional $(\phi,\Gamma)$-modules is by using the equivalence of catgeories with Galois representations, so I guess this doesn't really answer your question as it doesn't address the $(\phi,\Gamma)$-modules intrinsically.

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