Hecke character of a CM elliptic curve (${\psi }_E:{\mathbb{A}}_F^{\times }/F^{\times }\to {\mathbb{C}}^{\times }$)

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.

For $E:y^2=x^3-x$ over $\mathbb{Q}(i)$, the Hecke character has conductor $(1+i{)}^3$ and ideal infinity type $(1,0)$. Its value on an ideal prime to $2$ is its unique primary Gaussian integer generator. Rational $(1+i{)}^3$-torsion fixes the unit ambiguity of the Frobenius value. At a split prime $\mathfrak{p}$, its value reduces to the degree-$p$ Frobenius isogeny; at an inert rational prime $p$, its value is $-p$, reducing to the $p^2$-power Frobenius isogeny.