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 l-adiques associées aux courbes elliptiques sur Qp. J. Reine Angew. Math. 535 (2001), 65–101. and Volkov, Maja, A class of p-adic Galois representations arising from abelian varieties over Qp. Math. Ann. 331 (2005), no. 4, 889–923. relevant. $\endgroup$ Aug 4, 2012 at 7:09
  • $\begingroup$ You beat me to it by 6 minutes. $\endgroup$ Aug 4, 2012 at 7:17
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    $\begingroup$ The references above will (I think) give you the corresponding filtered $\phi$-module, but not the $(\phi,\Gamma)$-module. $\endgroup$ Aug 4, 2012 at 16:25

2 Answers 2


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.

  • $\begingroup$ What if we give ourself the knowledge of the j-invariant 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 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).


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