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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?

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up vote 14 down vote accepted

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.

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David, what do you mean by "Otherwise the image of Galois i open..."? I don't see what this has to do with the conclusion. But I can certainly miss something here. In any case, please enlighten me! – Daniel Larsson Jun 2 '10 at 8:50
If there was a decomposition $V_\ell(E) = V_1 \oplus V_2$ with $V_i$ stable under (some open subgroup of) the Galois group, then the image of Galois would have to be a p-adic Lie group of dimension at most 2. Since the Borel is 3-dimensional, this isn't the case (the image of Galois is "as large as possible" given that it has to preserve the kernel of reduction). – David Loeffler Jun 2 '10 at 8:55
Brilliant! Thanks. – Daniel Larsson Jun 2 '10 at 8:57
Sorry, typo: I meant $V_p(E)$ (the Tate module tensored up to $\mathbb{Q}_p$), not $V_\ell(E)$. – David Loeffler Jun 2 '10 at 8:59
It's thm. A.2.4 in Serre's book. – fherzig Jun 2 '10 at 19:27

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.

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