My question today concerns Hecke characters or Größencharaktere. I'm doing a project on complex multiplication of elliptic curves and need some help understanding Hecke characters. My main references are Silverman's advanced topics on elliptic curves and Ireland-Rosen Classical Intro to Number Theory.

In Silverman, [II.9.2](see below for print of the theorem) gives us a Hecke character for the base field [see (a)]. (b) mentions how the character is related to the curve. But the rest of the way, Silverman did not make use of the property that the character is unramified at $\mathfrak{P}$ [see (b)]. So it seems that characters are defined only on the base field and has nothing to do with the curve, which would be silly, wouldn't it? Although $\alpha_{E/L}$ may yet have a hand in bringing the curve in.

So these are my questions:

- How are Hecke characters defined for each curve wrt to a certain ground field?
- How does (b) contribute to how we find the character?
- What are the invariants on the Hecke character that makes it unique to a certain $E/K$?

Hope these questions make sense and a BIG thank you!

**Theorem 9.2.** Let $E/L$ be an elliptic curve with complex multiplication by the ring of integers $R_K$ of $K$, assume that $K\subset L$, and let $\alpha_{E/L}: \mathbf{A}^{*}_{L}\rightarrow K^*$ be the map such that

(i) $\alpha R_K=(s)$, where $(s)\subset K$ is the ideal of $s$.

(ii) For any fractional ideal $\mathfrak{a}\subset K$ and any analytic isomorphism
$$f:\mathbb{C}/\mathfrak{a}\rightarrow E(\mathbb{C})$$
the following diagram commutes:

$$\begin{array}[c]{ccc} K/\mathfrak{a}&\stackrel{\alpha s^{-1}}{\longrightarrow}&K/\mathfrak{a}\\ \downarrow\scriptstyle{f}&&\downarrow\scriptstyle{f}\\ E(L^{\text{ab}})&\stackrel{[x,L]}{\longrightarrow}&E(L^{\text{ab}}) \end{array}.$$

For any idele $s\in\mathbf{A}^{*}_{L}$, let $s_{\infty}\in \mathbb{C}^*$ be the component of $s$ correspoinding to the unique archimedean absolute value on $K$. Define the map
$$\psi_{E/L}: \mathbf{A}^{*}_{L}\rightarrow \mathbb{C}^*,\quad\psi_{E/L}(x)=\alpha_{E/L}(x)N^L_K(x^{-1})_{\infty}$$
(a) $\psi_{E/L}$ is a Hecke character of L.

(b) Let $\mathfrak{P}$ be a prime of $L$. Then $\psi_{E/L}$ is unramified at $\mathfrak{P}$ iff $E$ has good reduction at $\mathfrak{P}$.