Let $E$ be an elliptic curve over $K$ (totally real number field) with complex multiplication by the field $L$. Let $\psi$ be the Grössencharacter associated to $E$, assume that $\psi$ of type $(-r,0)$ (i.e., the restriction of $\psi$ to the archimedian part $\mathbb{C}^\times$ of the idele group of $K$ has the form $z\mapsto z^{-r}$). Set 
$$
V\_{\ell}(\psi):= (H^{1}(E(\mathbb{C}),\mathbb{Q})^{\otimes r}\otimes\_K L)\otimes\_{K\otimes\mathbb{Q}\_\ell}L\_{\tilde{\ell}}
$$
where $\otimes r$ is taken over $K$.


My question is: How to prove that the $\ell$-adic Tate module $V\_\ell(E)$ is isomorphic to $V\_{\ell}(\psi)\oplus\imath V\_{\ell}(\psi)$ as representations of $G\_K$, where $\imath\in G\_K$ is the complex conjugation. (References would be appreciated).



Isomorphism of the $\ell$-adic Tate module of an elliptic curve with CM
**Edit:** Following Will Sawin's comments, I realized that my first answer contained many mistakes. I try to emend my answer, writing $G\_\mathbb{Q}=\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ and $G\_K=\operatorname{Gal}(\overline{\mathbb{Q}}/K)$ throughout.


First of all, your $r$ is actually equal to $1$: indeed, if you look at Theorem II.9.1 in Silverman's *Advanced Topics in the Arithmetic of Elliptic Curves*, you see that what he calls $\alpha\_{E/L}$ is a character which is trivial when restricted to infinite idèles (use property (i) there), and therefore the property in Theorem II.9.2 defining the Grössencharakter, namely $\psi\_{E/L}(x)=\alpha\_{E/L}(x)N^L\_K(x^{-1})\_\infty$, shows that $\psi\_{E/L}$ is of infinite type $(-1,0)$. Secondly, I think that you have a sign-problem: you should have the same sign in the infinity-type (negative) and in the tensor power showing up in your equation.


Secondly, you say in the comments that you took the above equation from Kato's 2004 *Astérisque* paper *$p$-adic Hodge theory and values of zeta functions of modular forms* but beware that in his setting $K$ is imaginary quadratic (see the beginning of §15) and that he is working with a classical modular form (of some weight $k$): it is certainly a good idea to try to grasp what he's doing in case $k=2$, but this corresponds to an elliptic curve $E$ precisely when the curve is defined over $\mathbb{Q}$, so that $K$ is an imaginary quadratic field of class number $1$ (assuming that $E$ has complex multiplication by the full ring of integers). A final remark about Kato's paper: although on page 160 he says that $``$as representation of $G\_\mathbb{Q}$ there is an isomorphism
$$
V\_\ell(E)\cong V\_{\ell}(\psi)\oplus \sigma V\_\ell(\psi)"
$$
he **does not mean** that the above is a direct sum of representations. Indeed, in his notation three lines below his statement, if you pick $(x,0)\in V\_\ell(E)$ and $\gamma=\iota\tau\notin G\_K$, you get $\gamma(x,0)=(0,\gamma x)$, showing that the direct sum is not a sum of representation. Indeed, as Will Sawin observed, the Galois representation $V\_\ell(E)$ is *irreducible*.


I consider then an elliptic curve $E/\mathbb{Q}$ with CM by $\mathcal{O}\_K$. I write $E\_{/K}$ for its base-change to $K$ and I write $V\_\ell(E),V\_\ell(E\_{/K})$ for the respective Tate modules. They are somehow different creatures: the first is endowed with an action of $G\_\mathbb{Q}$ while the second has an action of the subgroup $G\_K$ and is, moreover, a free $\mathbb{Q}\_\ell\otimes K=K\_\ell$-module of rank $1$, as discussed for instance in [Tate module of CM elliptic curves](https://mathoverflow.net/questions/13349/tate-module-of-cm-elliptic-curves): this is not true for $V\_\ell(E)$, because the complex multiplication is not defined over $\mathbb{Q}$ and, in particular, it does not commute with the Galois action of the full group $G\_\mathbb{Q}$. Let's analyze these two representations separately:


1. As said, $V\_\ell(E\_{/K})$ is of rank $1$ over $K\_\ell$ and I claim that it is isomorphic to $K\_\ell(\psi)$, which is the rank-$1$-module $K\_\ell$ endowed with an action of $G\_K$ given by $g\cdot v=\psi(g)v$, seeing $\psi(g)\in K^\times\subseteq K\_\ell^\times$. This is the commutativity of the diagram in Theorem 9.1 (ii) of Silverman's *Advanced topics...*, together with the definition of $\psi$ in Theorem 9.2 *ibid.*. As a by-product, you get an isomorphism
$$
K\_\ell(\psi)\cong H^1(E(\mathbb{C}),\mathbb{Q}\_\ell)^{-1}
$$
because on the one hand
$$
H^1(E(\mathbb{C}),\mathbb{Q}\_\ell)^{-1}\cong H^1\_\mathrm{et}(E\_{/K},\mathbb{Q}\_\ell)^{-1}\cong V\_\ell(E\_{/K})
$$
(this is well-explained here [Etale cohomology and l-adic Tate modules](https://mathoverflow.net/questions/7318/etale-cohomology-and-l-adic-tate-modules), upon realizing that $(\ast\otimes -1)$ means ``taking the dual''); and, on the other, because
$$
H^1(E(\mathbb{C}),\mathbb{Q}\_\ell)^{-1}=\mathbb{Q}\_\ell\otimes\_{\mathbb{Q}}H^1(E(\mathbb{C}),\mathbb{Q})^{-1}=\mathbb{Q}\_\ell\otimes\_{\mathbb{Z}}H\_1(E(\mathbb{C}),\mathbb{Z})=\mathbb{Q}\_\ell\otimes \mathcal{O}\_K
$$
where the last isomorphism is the map $\gamma\mapsto \int{\_\gamma} {\omega\_E}$ (in general, the image of this map is a lattice $\Lambda$ such that $E(\mathbb{C})=\mathbb{C}/\Lambda$, so in this case $\Lambda=\mathcal{O}\_K$).
2. The tensor product $V\_\ell(E)\otimes\_{\mathbb{Q}\_\ell}K\_\ell$ has rank $2$ over $K\_\ell$ and $G\_\mathbb{Q}$ acts on it on the first factor. It turns out that there is an isomorphism of $G\_\mathbb{Q}$-representations
$$
V\_\ell(E)\otimes\_{\mathbb{Q}\_\ell}K\_\ell\cong \operatorname{Ind}^K\_\mathbb{Q}\big(K\_\ell(\psi)\big)
$$
as proven (in the general setting of CM cuspforms) in Ribet's 1977 paper *Galois representations attached to eigenforms with Nebentypus*, Theorem 4.5. You can retrace the line of Ribet's argument (who was inspired by the classical proof for elliptic curves) by doing exercices 2.29-2.32 in Silverman's *Advanced topics...*: the main point is to compare the traces of both representations and then to invoke a general result saying that two $2$-dimensional irreducible representations sharing the same trace are isomorphic.


Now, by restricting the $G\_\mathbb{Q}$-representation constructed above to the subgroup $G\_K$ you **don't find** $V\_\ell(E\_{/K})$ (which is of rank $1$ over $K\_\ell$), but you rather find
$$
\operatorname{Res}\_{G\_K}\big(V\_\ell(E)\otimes K\_\ell\big)\cong\operatorname{Res}\_{G\_K}\operatorname{Ind}^K\_\mathbb{Q}\big(K\_\ell(\psi)\big)
$$
and you can apply Proposition 22 of Serre's *Représentations linéaires des groupes finis* to deduce that this latter object is $K\_\ell(\psi)\oplus \sigma K\_\ell(\psi)$, in turn isomorphic to $$
\Big(H^1(E(\mathbb{C}),\mathbb{Q}\_\ell)^{-1}\Big)\oplus \sigma \Big(H^1(E(\mathbb{C}),\mathbb{Q}\_\ell)^{-1}\Big)
$$
by point 1. above.


