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?
[Math] (phi, Gamma) module of ordinary elliptic curve
galois-representationsnt.number-theory
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OK...I think I see how to do this now. In the end, I am seeing $(p-1)^2$ distinct $(\phi,\Gamma)$-modules which matches well with the Galois side.
To do this, let $D$ be any 1-dimensional etale $(\phi,\Gamma)$-module. Let $e$ be a basis, and set $\phi(e)=h(T)e$ with $h(T) \in F_p((T))^\times$. Write $h(T) = h_0 T^a f(T)$ with $h_0 \in F_p^\times$ and $f(T) \in F_p[[T]]$ with $f(0)=1$.
Changing basis from $e$ to $u(T)e$ with $u(T) \in F_p((T))^\times$ gives $$ \phi(u(T)e) = u(T^p)h(T)e = \frac{u(T^p)}{u(T)} h(T) (u(T)e). $$ I claim one can find $u(T)$ such that $u(T)/u(T^p)$ equals any element of $1+TF_p[[T]]$. Indeed, for such an element $g(T)$, the infinite product $\prod_{j=1}^\infty \phi^j(g(T))$ (which hopefully converges since $g(0)=1$) works.
Thus, we can change basis so that $\phi$ has the form $\phi(e) = h_0 T^a e$ -- i.e. we can kill off the $f(T)$ term. Further, by making a change of basis of the form $e$ goes to $T^b e$, we may assume that $0 \leq a < p-1$.
Now, we use the fact that the $\phi$ and $\Gamma$ actions commute (which is a strong condition even in dimension 1). Namely, let $\gamma$ be a generator of $\Gamma$, and set $\gamma e = g(T) e$. Then $\gamma \phi e = \phi \gamma e$ implies $$ ((1+T)^{\chi(\gamma)}-1)^a g(T) = g(T^p) T^a. $$ Comparing leading coefficients, we see this is only possible if $a=0$ and $g(T)$ is a constant.
Thus, $D$ has a basis $e$ so that $\phi(e) = h_0 e$ and $\gamma(e) = g_0 e$ with $h_0,g_0 \in F_p^\times$ as desired.
Does this look okay? Any takers for the 2-dimensional case?
Serre has shown that there exists a complementary subspace invariant under the Lie algebra $\mathfrak{g}$ if and only if E has complex multiplication. Otherwise the image of Galois is open in the Borel subgroup of $\operatorname{GL}_2(\mathbb{Q}_p)$. I learnt this from the paper by Coates and Howson ("Euler characteristics and elliptic curves II", beginning of section 5); they reference Serre's book "Abelian l-adic representations and elliptic curves", but I don't have a copy of that to hand right now to check.
Best Answer
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.