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?



(phi, Gamma) module of ordinary elliptic curve
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.


