What is the (Phi,Gamma)-module of an elliptic curve over Z_p, expressed by a direct construction ?
[Math] Examples of (Phi,Gamma)-modules
arithmetic-geometrynt.number-theory
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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?
If your elliptic curve has good ordinary reduction, then the attached Galois representation is reducible : it is an extension of $\eta_2$ by $\eta_1 \chi$ where $\eta_{1,2}$ are unramified characters and $\chi$ is the cyclotomic character. The $(\phi,\Gamma)$-modules of $\eta_2$ and $\eta_1 \chi$ are easy to compute, so it remains to say something about the extension, ie the upper right star in the matrices of $\phi$ and $\gamma \in \Gamma$. This is less easy; one can't simply write general formulas, but by using the results of Cherbonnier-Colmez (JAMS) and Colmez (eg his paper on trianguline representations), you can say a number of interesting things.
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(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.