# 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 so-called ‘Main Theorem of Complex Multiplication’ ([2], Thm. 8.2) implies the existence of a Grössencharacter of $F$, $\psi_{A/F}:\mathcal{A}_{F}^{\ast}\to\mathbb{C}^{\ast}$ 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 $\wp$ 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 $F$, $\psi_{A/F}:\mathcal{A}_{F}^{\ast}\to\mathbb{C}^{\ast}$, such that:

1. 1.

$\psi_{A/F}$ is unramified at a prime $\mathfrak{Q}$ of $F$ if and only if $A/F$ has good reduction at $\mathfrak{Q}$;

2. 2.

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

3. 3.

the following diagram is commutative

$\xymatrix{A\ar@{->}[d]\ar@{->}[r]^{[\psi_{A/F}(\wp)]}&A\ar@{->}[d]&\inner@par% \widetilde{A}\ar@{->}[r]^{\phi_{\wp}}&\widetilde{A}&}$

where $\phi_{\wp}:\widetilde{A}\to\widetilde{A}$ be the $N_{\mathbb{Q}}^{F}(\wp)$-power Frobenius map and the vertical maps are reduction mod $\wp$;

4. 4.

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 $a_{\wp}=N_{\mathbb{Q}}^{F}(\wp)+1-|\widetilde{A}(\mathcal{O}_{F}/\wp)|$. Then

 $a_{\wp}=\psi_{A/F}(\wp)+\overline{\psi_{A/F}(\wp)}=2\cdot\Re(\psi_{A/F}(\wp)).$
5. 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 $K\nsubseteq F$, 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 $\mathbb{Q}$), $\psi_{A/K}(\wp)$ 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), $L(A/\mathbb{Q},s)=L(\psi_{A/K},s)$.

## References

• 1 J. H. Silverman, , Springer-Verlag, New York.
• 2 J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
Title the Grössencharacter associated to a CM elliptic curve TheGrossencharacterAssociatedToACMEllipticCurve 2013-03-22 15:45:29 2013-03-22 15:45:29 alozano (2414) alozano (2414) 4 alozano (2414) Definition msc 11G05 Grossencharacter EllipticCurve grossencharacter associated to an elliptic curve