the Grössencharacter associated to a CM elliptic curve

Let $K$ be a quadratic imaginary field and let $A/F$ be anelliptic curve defined over a number field $F$ (such that$K\subset F$), with complex multiplication by $K$. The so-called‘Main Theorem of Complex Multiplication’ ([2], Thm. 8.2)implies the existence of a Grössencharacter of $F$,${\psi }_{A/F}:{𝒜}_F^{\ast }\to {ℂ}^{\ast }$ associated to thecurve $A/F$ satisfying several interesting properties which wecollect 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 $\mathrm{\wp }$ issmooth. 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 $𝔔$ of $F$if and only if $A/F$ has good reduction at $𝔔$;

2. ${\psi }_{A/F}(\mathrm{\wp })$ belongs to ${𝒪}_K$, thus multiplicationby $[{\psi }_{A/F}(\mathrm{\wp })]$ 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

   where ${\varphi }_{\mathrm{\wp }}:\tilde{A}\to \tilde{A}$ be the$N_{\mathbb{Q}}^F(\mathrm{\wp })$-power Frobenius map and the vertical maps arereduction mod $\mathrm{\wp }$;

4. let $|\tilde{A}({𝒪}_F/\mathrm{\wp })|$ be the number ofpoints in $\tilde{A}$ over the finite field${𝒪}_F/\mathrm{\wp }$ and put $a_{\mathrm{\wp }}=N_{\mathbb{Q}}^F(\mathrm{\wp })+1-|\tilde{A}({𝒪}_F/\mathrm{\wp })|$. 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 $L$-functionassociated 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$K⊈F$, 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 generatorof $\mathrm{\wp }$ (by part (2), and the explicit generator can be pinneddown using part (4)). Thus, if $e$ is the number of roots of unityin $K$, then ${\psi }_{A/K}^k(\mathrm{\wp })={\alpha }^k$ where $\alpha$ is any generator of $\mathrm{\wp }$. Also, by part (5),$L(A/\mathbb{Q},s)=L({\psi }_{A/K},s)$.