Examples of (Phi,Gamma)-modules

Question

What is the (Phi,Gamma)-module of an elliptic curve over Z_p, expressed by a direct construction ?

Answer 1

(By an elliptic curve over $\mathbb Z_p$, I assume you mean an elliptic scheme over spec $\mathbb Z_p$, or which amount to the same, an elliptic curve over $\mathbb Q_p$ which has good reduction.) The Galois representation on $V_p(E)$ is then crystalline, so its $(\phi,\Gamma)$-module $D$ will be triangulate -- in other words, an extension of two $(\phi,\Gamma)$-modules of dimension $1$, a sub-object $D_1$ and a quotient $D_2=D/D_1$.

As you probably know (since otherwise you would have begun by this case), a $(\phi,\Gamma)$-module of dimension $1$ is described by a continuous character $\delta: \mathbb Q_p^\ast \rightarrow \mathbb Q_p^\ast$. More precisely, it is always isomorphic to the Robba ring as a module over the Robba ring, with actions that are twisted from the original one, which by $\delta_{|\mathbb Z_p^\ast}$ for the $\Gamma$-action and by $\delta(p)$ for the $\phi$-action. Now let us describe the characters $\delta_1$ and $\delta_2$ corresponding to $D_1$ and $D_2$. Let $\alpha$ and $\beta$ be the two roots of the polynomial $X^2-a_pX + p =0$, ordered so that $v_p(\alpha) < v_p (\beta)$. Here $a_p$ is as usual $E(\mathbb F_p) -1 - p$, and $v_p$ is the $p$-valuation. So let $\delta_1$ be the character which sends $z \in \mathbb Z^\ast_p$ to $1$ and $p$ to $\alpha$ (so the character $z \mapsto \alpha^{v_p(z)}$ in short), and let $\delta_2$ be the character which sends $z \in \mathbb Z^\ast_p$ to $1$ and $p$ to $\beta$.

This describes $D_1$ and $D_2$. Am I done? not quite. I need to say which extension it is. But I have to run, so I will finish later... (later...) So if you go look to the paper of Colmez on triangulline representations, you will see that there is a result computing the Ext group $Ext^1(D_1,D_2)$ in the category of $(\phi,Gamma)$-modules, and that in our case the dimension is $1$. So the extension $D$ has two possibility: either it is trivial, or it is non-trivial (and then the proof of that Colmez's theorem give you and explicit description of what it is).

When your elliptic curve $E$ is super singular, $D$ is always the non-trivial extension. When it is ordinary, it is more complicated. Let $\rho_{E,p}$ be the Galois representation of $G_{\mathbb Q_p}$ on the Tate module of $E$. It is always reducible, but it may be decomposable or not (with a conjecture saying that it is decomposable if and only if E has CM). Well the extension $D$ is split if and only if $\rho_{E,p}$ is decomposable.

Okay, this describes explicitly $D$. I practice, it is not fundamental to know when $D$ is a split extension or not, the fact that it is an extension of the very concrete $D_1$ and $D_2$ is enough. If you want to understand all this better, I think that COlmez' paper on triangulline representation is the best place to start.

Answer 2

This is a remark (I am not an expert). But in his course notes, http://perso.ens-lyon.fr/laurent.berger/ihp2010.php, Laurent Berger explains how $(\Phi,\Gamma)$-modules can be given concretely by giving two matrices $P=Mat(\phi)$ and $G=Mat(\gamma)$ satisfying some semilinear commutation relation etc. This is chapter $10$. In $10.2$ are concrete examples.

And here is an answer in this direction by Berger himself: (phi, Gamma) module of ordinary elliptic curve