MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

What is the (Phi,Gamma)-module of an elliptic curve over Z_p, expressed by a direct construction ?

share|cite|improve this question
This doesn't really answer your question, but this paper of Laurent Berger is in the right direction: – Jeff H May 29 '13 at 16:31 – user5831 Oct 14 '13 at 14:12
up vote 9 down vote accepted

(By an elliptic curve over $\mathbb Z_p$, I assume you mean an elliptic scheme over spec $\mathbb Z_p$, or which amount to the same, an elliptic curve over $\mathbb Q_p$ which has good reduction.) The Galois representation on $V_p(E)$ is then crystalline, so its $(\phi,\Gamma)$-module $D$ will be triangulate -- in other words, an extension of two $(\phi,\Gamma)$-modules of dimension $1$, a sub-object $D_1$ and a quotient $D_2=D/D_1$.

As you probably know (since otherwise you would have begun by this case), a $(\phi,\Gamma)$-module of dimension $1$ is described by a continuous character $\delta: \mathbb Q_p^\ast \rightarrow \mathbb Q_p^\ast$. More precisely, it is always isomorphic to the Robba ring as a module over the Robba ring, with actions that are twisted from the original one, which by $\delta_{|\mathbb Z_p^\ast}$ for the $\Gamma$-action and by $\delta(p)$ for the $\phi$-action. Now let us describe the characters $\delta_1$ and $\delta_2$ corresponding to $D_1$ and $D_2$. Let $\alpha$ and $\beta$ be the two roots of the polynomial $X^2-a_pX + p =0$, ordered so that $v_p(\alpha) < v_p (\beta)$. Here $a_p$ is as usual $E(\mathbb F_p) -1 - p$, and $v_p$ is the $p$-valuation. So let $\delta_1$ be the character which sends $z \in \mathbb Z^\ast_p$ to $1$ and $p$ to $\alpha$ (so the character $z \mapsto \alpha^{v_p(z)}$ in short), and let $\delta_2$ be the character which sends $z \in \mathbb Z^\ast_p$ to $1$ and $p$ to $\beta$.

This describes $D_1$ and $D_2$. Am I done? not quite. I need to say which extension it is. But I have to run, so I will finish later... (later...) So if you go look to the paper of Colmez on triangulline representations, you will see that there is a result computing the Ext group $Ext^1(D_1,D_2)$ in the category of $(\phi,Gamma)$-modules, and that in our case the dimension is $1$. So the extension $D$ has two possibility: either it is trivial, or it is non-trivial (and then the proof of that Colmez's theorem give you and explicit description of what it is).

When your elliptic curve $E$ is super singular, $D$ is always the non-trivial extension. When it is ordinary, it is more complicated. Let $\rho_{E,p}$ be the Galois representation of $G_{\mathbb Q_p}$ on the Tate module of $E$. It is always reducible, but it may be decomposable or not (with a conjecture saying that it is decomposable if and only if E has CM). Well the extension $D$ is split if and only if $\rho_{E,p}$ is decomposable.

Okay, this describes explicitly $D$. I practice, it is not fundamental to know when $D$ is a split extension or not, the fact that it is an extension of the very concrete $D_1$ and $D_2$ is enough. If you want to understand all this better, I think that COlmez' paper on triangulline representation is the best place to start.

share|cite|improve this answer

This is a remark (I am not an expert). But in his course notes,, Laurent Berger explains how $(\Phi,\Gamma)$-modules can be given concretely by giving two matrices $P=Mat(\phi)$ and $G=Mat(\gamma)$ satisfying some semilinear commutation relation etc. This is chapter $10$. In $10.2$ are concrete examples.

And here is an answer in this direction by Berger himself: (phi, Gamma) module of ordinary elliptic curve

share|cite|improve this answer

Your Answer


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.