Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
    
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. –  Kestutis Cesnavicius Aug 4 '12 at 7:09
    
You beat me to it by 6 minutes. –  Chandan Singh Dalawat Aug 4 '12 at 7:17
1  
The references above will (I think) give you the corresponding filtered $\phi$-module, but not the $(\phi,\Gamma)$-module. –  Laurent Berger Aug 4 '12 at 16:25
add comment

2 Answers

up vote 9 down vote accepted

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.

share|improve this answer
    
OK thanks I'll check them out! –  anon Aug 4 '12 at 20:49
    
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 $T_p(E)$? –  Tommaso Centeleghe Feb 27 '13 at 10:23
add comment

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).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.