Hecke character

A Hecke character of a number field is a continuous character of its idèle class group. Its finite conductor records the local unit subgroups on which it is trivial. As an ideal character it is defined on fractional ideals prime to that conductor, with a prescribed rule on principal ideals coming from its archimedean character. A CM elliptic curve yields an algebraic Hecke character with one CM embedding in its infinity type of a Hecke character.

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.

For an algebraic Hecke character, the character on archimedean ideles is a product of integral powers of embeddings and their complex conjugates. In the convention $\psi ((a))=a$ for primary $a$ over an imaginary quadratic field, the ideal character has type $(1,0)$ and the idele character has archimedean component $z^{-1}$. Distinguishing these conventions avoids inverse errors in CM Galois representations.