Does the p-adic Tate module of an elliptic curve with ordinary reduction decompose? Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E$ an elliptic curve over $K$ with good ordinary reduction.
The p-adic Tate module $T_p(E)$ is (after tensoring with $\mathbb{Q}_p$) a 2-dimensional $\mathbb{Q}_p$-representation of $\mathop{\mathrm{Gal}}(\bar{K}/K)$.
It is reducible: the kernel of reduction to the residue field is an invariant line.
Does $T_p(E) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$ contain another invariant line?
 A: 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.
A: If you go to the maximal unramified extension of $K$ (so the residue field is algebraically closed) then you can write $T_p(E)$ as an extension of $\mathbb{Z}_p$ by $T_p(\mu)$. The class of this extension is the Serre-Tate parameter and can be viewed as a one-unit in the ground field. The Serre-Tate parameter parametrizes the set of elliptic curves with a fixed ordinary reduction.
To answer your question, you get another invariant line if and only if the Serre-Tate parameter is a root of unity (since the extension of groups splits up to isogeny). As in David's answer, this only happens if the elliptic curve has CM. The curve with Serre-Tate parameter equal to one is the canonical lift of the reduction ($T_p(E)$ is a direct sum) and the other CM curves are called quasi-canonical lifts.
