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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 $$ V_{\ell}(\psi):=H^{1,\text{ét}}(E(\mathbb{C})\otimes_L\overline{\mathbb{Q}},\mathbb{Q}_\ell)^{\otimes r}\otimes_{K\otimes\mathbb{Q}_\ell}L_{\tilde{\ell}} $$ where $\otimes r$ is taken over $K\otimes\mathbb{Q}_\ell$.

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).

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).

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 $$ V_{\ell}(\psi):=H^{1,\text{ét}}(E(\mathbb{C})\otimes_L\overline{\mathbb{Q}},\mathbb{Q}_\ell)^{\otimes r}\otimes_{K\otimes\mathbb{Q}_\ell}L_{\tilde{\ell}} $$ where $\otimes r$ is taken over $K\otimes\mathbb{Q}_\ell$.

My question is: How to prove that the $\ell$-adic Tate module $V_\ell(E)$ is isomorphic to $V_{\ell}(\psi)$ as representations of $G_K$. (References would be appreciated).

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 $$ V_{\ell}(\psi):=H^{1,\text{ét}}(E(\mathbb{C})\otimes_L\overline{\mathbb{Q}},\mathbb{Q}_\ell)^{\otimes r}\otimes_{K\otimes\mathbb{Q}_\ell}L_{\tilde{\ell}} $$ where $\otimes r$ is taken over $K\otimes\mathbb{Q}_\ell$.

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).

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 $$ V_{\tilde{\ell}}(\psi):=H^{1,\text{ét}}(E(\mathbb{C})\otimes_L\overline{\mathbb{Q}},\mathbb{Q}_\ell)^{\otimes r}\otimes_{K\otimes\mathbb{Q}_\ell}L_{\tilde{\ell}} $$ where $\otimes r$ is taken over $K\otimes\mathbb{Q}_\ell$.

My question is: How to prove that the $\ell$-adic Tate module $V_\ell(E)$ is isomorphic to $V_{\ell}(\psi)$ as representations of $G_K$. (References would be appreciated).

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 $$ V_{\ell}(\psi):=H^{1,\text{ét}}(E(\mathbb{C})\otimes_L\overline{\mathbb{Q}},\mathbb{Q}_\ell)^{\otimes r}\otimes_{K\otimes\mathbb{Q}_\ell}L_{\tilde{\ell}} $$ where $\otimes r$ is taken over $K\otimes\mathbb{Q}_\ell$.

My question is: How to prove that the $\ell$-adic Tate module $V_\ell(E)$ is isomorphic to $V_{\ell}(\psi)$ as representations of $G_K$. (References would be appreciated).