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?

$\begingroup$ I'm not sure whether this answers your question completely, but you may find Volkov, Maja, Les représentations ladiques associées aux courbes elliptiques sur Qp. J. Reine Angew. Math. 535 (2001), 65–101. and Volkov, Maja, A class of padic Galois representations arising from abelian varieties over Qp. Math. Ann. 331 (2005), no. 4, 889–923. relevant. $\endgroup$– Kestutis CesnaviciusAug 4, 2012 at 7:09

$\begingroup$ You beat me to it by 6 minutes. $\endgroup$– Chandan Singh DalawatAug 4, 2012 at 7:17

1$\begingroup$ The references above will (I think) give you the corresponding filtered $\phi$module, but not the $(\phi,\Gamma)$module. $\endgroup$– Laurent BergerAug 4, 2012 at 16:25
2 Answers
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 CherbonnierColmez (JAMS) and Colmez (eg his paper on trianguline representations), you can say a number of interesting things.

$\begingroup$ What if we give ourself the knowledge of the jinvariant of E? Do you think that using j_E one can say more about the structure of $T_p(E)$? $\endgroup$ Feb 27, 2013 at 10:23
As Laurent has already pointed out, the representation is reducible and hence so is the phiGamma 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 onedimensional, so up to isomophism there is only one nonsplit extension, and you just need to write down any old nonsplit extension and you're done. Sarah Zerbes and I describe a way of doing this in our paper "Wach modules and critical slope padic Lfunctions"; 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).