PlanetMath: the Grössencharacter associated to a CM elliptic curve

Theorem 1 ([2], Thm. 9.1, Prop. 10.4, Cor. 10.4.1) Let $\wp$ be a prime of $F$ of good reduction for $A/F$ , i.e. the reduction $\widetilde{A}/F$ of $A$ modulo $\wp$ is smooth. There exists a Grössencharacter of $F$ , $\psi_{A/F}:\mathcal{A}_F^\ast \to \Complex^\ast$ , such that:

$\psi_{A/F}$ is unramified at a prime $\mathfrak{Q}$ of $F$ if and only if $A/F$ has good reduction at $\mathfrak{Q}$ ;
$\psi_{A/F}(\wp)$ belongs to $\RInts$ , thus multiplication by $[\psi_{A/F}(\wp)]$ is a well defined endomorphism of $A/F$ . Moreover $N_\Rats^F(\wp)=N_\Rats^K(\psi_{A/F}(\wp))$ ;
the following diagram is commutative
$\xymatrix{ A \ar@{->}[d] \ar@{->}[r]^{[\psi_{A/F}(\wp)]} & A \ar@{->}[d] &\ \widetilde{A} \ar@{->}[r]^{\phi_\wp} & \widetilde{A} & }$
where $\phi_\wp:\widetilde{A}\to\widetilde{A}$ be the $N_\Rats^F(\wp)$ -power Frobenius map and the vertical maps are reduction mod $\wp$ ;
let $|\widetilde{A}(\mathcal{O}_F/\wp)|$ be the number of points in $\widetilde{A}$ over the finite field $\mathcal{O}_F/\wp$ and put $a_\wp=N_{\Rats}^F(\wp) +1 - |\widetilde{A}(\mathcal{O}_F/\wp)|$ . Then $$a_{\wp}=\psi_{A/F}(\wp)+\overline{\psi_{A/F}(\wp)}=2\cdot \Re (\psi_{A/F}(\wp)).$$
(due to Deuring) let $L(A/F,s)$ be the $L$ -function associated to the elliptic curve $A/F$ . If $K\subset F$ then $L(A/F,s)=L(\psi_{A/F},s)L(\overline{\psi_{A/F}},s)$ . If $K\nsubseteq F$ , and $F'=FK$ , then $L(E/F,s)=L(\psi_{A/F'},s)$ .

In particular, if $h_K=1$ then $A$ is defined over $K$ (actually, it may be defined over $\Rats$ ), $\psi_{A/K}(\wp)$ is a generator of $\wp$ (by part (2), and the explicit generator can be pinned down using part (4)). Thus, if $e$ is the number of roots of unity in $K$ , then $\psi_{A/K}^k(\wp)=\alpha^k$ where $\alpha$ is any generator of $\wp$ . Also, by part (5), $L(A/\Rats,s)=L(\psi_{A/K},s)$ .

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