(phi, Gamma) module of ordinary elliptic curve

Question

Suppose $E$ is an elliptic curve over $\mathbb{Q}_p$ with good ordinary reduction. Can someone please tell me how to compute the associated $(\phi,\Gamma)$-module of the Tate module of $E$, or give me a reference to where it is computed?

Answer 1

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.

Answer 2

As Laurent has already pointed out, the representation is reducible and hence so is the phi-Gamma module, and writing down the two composition factors is easy; describing the extension class is harder. But it can be done: the relevant ext group is one-dimensional, so up to isomophism there is only one non-split extension, and you just need to write down any old non-split extension and you're done. Sarah Zerbes and I describe a way of doing this in our paper "Wach modules and critical slope p-adic L-functions"; see section 4 of http://arxiv.org/abs/1012.0175. There is another approach in one of Pierre Colmez's papers ("La serie principale unitaire" if I remember rightly).