= Hecke character of a CM elliptic curve
{c}
{title2=$\psi_E:\mathbb A_F^\times/F^\times\to\mathbb C^\times$}
Suppose $F$ contains the CM field $K$ and all CM endomorphisms are defined over $F$. <CM reciprocity> compares the torus with its idele-transformed lattice and gives $a(s)\in K^\times$ for finite ideles. Then $\psi_E(s)=a(s_f)/N_{F/K}(s)_\infty$. On principal ideles, $a(b_f)=N_{F/K}(b)$, so the character descends to the <idèle class group>. At a good prime its ideal value reduces to the <Frobenius isogeny>. If the original field omits $K$, construct the character over $FK$ and induce the corresponding representation down to the original field.
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