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?



Does the p-adic Tate module of an elliptic curve with ordinary reduction decompose?
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.


