the Grössencharacter associated to a CM elliptic curve
Let $K$ be a quadratic imaginary field and let $A/F$ be an elliptic curve^{} defined over a number field^{} $F$ (such that $K\subset F$), with complex multiplication^{} by $K$. The socalled ‘Main Theorem of Complex Multiplication’ ([2], Thm. 8.2) implies the existence of a Grössencharacter of $F$, ${\psi}_{A/F}:{\mathcal{A}}_{F}^{\ast}\to {\u2102}^{\ast}$ associated to the curve $A/F$ satisfying several interesting properties which we collect in the following statement.
Theorem ([2], Thm. 9.1, Prop. 10.4, Cor. 10.4.1).
Let $\mathrm{\wp}$ be a prime of $F$ of good reduction for $A\mathrm{/}F$, i.e. the reduction^{} $\stackrel{\mathrm{~}}{A}\mathrm{/}F$ of $A$ modulo $\mathrm{\wp}$ is smooth. There exists a Grössencharacter of $F$, ${\psi}_{A\mathrm{/}F}\mathrm{:}{\mathrm{A}}_{F}^{\mathrm{\ast}}\mathrm{\to}{\mathrm{C}}^{\mathrm{\ast}}$, such that:

1.
${\psi}_{A/F}$ is unramified at a prime $\U0001d514$ of $F$ if and only if $A/F$ has good reduction at $\U0001d514$;

2.
${\psi}_{A/F}(\mathrm{\wp})$ belongs to ${\mathcal{O}}_{K}$, thus multiplication by $[{\psi}_{A/F}(\mathrm{\wp})]$ is a well defined endomorphism^{} of $A/F$. Moreover ${N}_{\mathbb{Q}}^{F}(\mathrm{\wp})={N}_{\mathbb{Q}}^{K}({\psi}_{A/F}(\mathrm{\wp}))$;

3.
the following diagram is commutative^{}
$\text{xymatrix}A\text{ar}\mathrm{@}>[d]\text{ar}\mathrm{@}>{[r]}^{[{\psi}_{A/F}(\mathrm{\wp})]}\mathrm{\&}A\text{ar}\mathrm{@}>[d]\mathrm{\&}\stackrel{~}{A}\text{ar}\mathrm{@}>{[r]}^{{\varphi}_{\mathrm{\wp}}}\mathrm{\&}\stackrel{~}{A}\mathrm{\&}$
where ${\varphi}_{\mathrm{\wp}}:\stackrel{~}{A}\to \stackrel{~}{A}$ be the ${N}_{\mathbb{Q}}^{F}(\mathrm{\wp})$power Frobenius map^{} and the vertical maps are reduction mod $\mathrm{\wp}$;

4.
let $\stackrel{~}{A}({\mathcal{O}}_{F}/\mathrm{\wp})$ be the number of points in $\stackrel{~}{A}$ over the finite field^{} ${\mathcal{O}}_{F}/\mathrm{\wp}$ and put ${a}_{\mathrm{\wp}}={N}_{\mathbb{Q}}^{F}(\mathrm{\wp})+1\stackrel{~}{A}({\mathcal{O}}_{F}/\mathrm{\wp})$. Then
$${a}_{\mathrm{\wp}}={\psi}_{A/F}(\mathrm{\wp})+\overline{{\psi}_{A/F}(\mathrm{\wp})}=2\cdot \mathrm{\Re}({\psi}_{A/F}(\mathrm{\wp})).$$ 
5.
(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 $K\u2288F$, and ${F}^{\prime}=FK$, then $L(E/F,s)=L({\psi}_{A/{F}^{\prime}},s)$.
In particular, if ${h}_{K}=1$ then $A$ is defined over $K$ (actually, it may be defined over $\mathbb{Q}$), ${\psi}_{A/K}(\mathrm{\wp})$ is a generator^{} of $\mathrm{\wp}$ (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 ${\psi}_{A/K}^{k}(\mathrm{\wp})={\alpha}^{k}$ where $\alpha $ is any generator of $\mathrm{\wp}$. Also, by part (5), $L(A/\mathbb{Q},s)=L({\psi}_{A/K},s)$.
References
 1 J. H. Silverman, The Arithmetic of Elliptic Curves, SpringerVerlag, New York.
 2 J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. SpringerVerlag, New York, 1994.
Title  the Grössencharacter associated to a CM elliptic curve 

Canonical name  TheGrossencharacterAssociatedToACMEllipticCurve 
Date of creation  20130322 15:45:29 
Last modified on  20130322 15:45:29 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  4 
Author  alozano (2414) 
Entry type  Definition 
Classification  msc 11G05 
Related topic  Grossencharacter 
Related topic  EllipticCurve 
Defines  grossencharacter associated to an elliptic curve 