Hecke Characters

Question

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:

1. How are Hecke characters defined for each curve wrt to a certain ground field?
   
2. How does (b) contribute to how we find the character?
3. 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!

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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 $\mathfrak{P}$.

Answer 1

This is my answer for your question 2.

Let $\upsilon$ be a prime ideal of $L$. We denote the completion of $L$ at $\upsilon$ with $L_\upsilon$ and the integer ring with $R_\upsilon$. Recall $A^\ast_L = \{\alpha = (\alpha_\upsilon) \in \Pi_\upsilon L^\ast_\upsilon \shortmid \alpha_\upsilon \in R^\ast_\upsilon,$ almost for all $\upsilon \}$. We define as follows;

$\psi_{E/L}$ is unramified at $\upsilon$ $=_{def}$ $\psi_{E/L}(R^\ast_\upsilon) = 1$.

Thus, (b) means

$\psi_{E/L}(R^\ast_\upsilon) = 1$ iff $E$ has good reduction at $\upsilon$.

I dare to state that (b) doesn't contribute to find the character.

I guess you confuse the unramification of a character with the unramification of a prime ideal. I recommend you to read Tate’s paper Fourier analysis in number fields and Hecke's zeta-functions. You can find it on the net.