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



Examples of (Phi,Gamma)-modules
(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. 


