Question about Größencharaktere in imaginary quadratic number fields

Question

Presumably, one could ask this question for a Größencharakter in an arbitrary number field, but I'll restrict my attention to the case I'm interested in. Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$ and let $\mathfrak{f}$ be a nonzero ideal of $\mathcal{O}_K$. Let $\chi$ be a character on the multiplicative group $(\mathcal{O}_K/\mathfrak{f})^\times$ and let $\lambda$ be a generator for the infinite-order Größencharaktere of $K$ mod $\mathfrak{f}$. On principal ideals $(\alpha)$, $\lambda$ has the property that $$ \lambda^m((\alpha)) = \left(\frac{\alpha}{|\alpha|}\right)^{mg}, $$ where $g$ is the number of units $\varepsilon \in \mathcal{O}_K$ with $\varepsilon \equiv 1\ (\text{mod}\ \mathfrak{f})$. In Coleman - The Distribution of Points at which Binary Quadratic Forms are Prime, the author is interested, for a fixed $m$, in the "collection of characters $\chi$ such that $\lambda^m\chi$ is a function of ideals only." Thus my question,

Question: What conditions ensure that $\lambda^m\chi$ is a function of ideals only? Further, is it enough to ensure that these conditions hold on principal ideals (in which case this seems like a question about behavior on units)?

Größencharaktere are a subject that I can never seem to fully wrap my head around, so any help is most appreciated.

Answer 1

Let $P$ be the group of principal fractional ideals $(\alpha)$ where $\alpha \in K^\times$ and let $I$ be the group of all fractional ideals in $K$. I am not going to look at the paper you cite, but will assume "function of ideals only" is a clumsy way to say "multiplicative function on $I$". And I assume all these multiplicative functions are taking values in the unit circle.

In order that a multiplicative function that starts out on $K^\times$ be a well-defined multiplicative function on the group $P$, we need it to be trivial on the units in $\mathcal O_K$ since we need its value on each $\alpha$ and $\alpha{u}$ to be equal for all units $u$, which is the same as saying its values on all units has to be $1$. When $K$ is imaginary quadratic, the only units in $\mathcal O_K$ are $\pm 1$ unless $K = \mathbf Q(i)$ or $K = \mathbf Q(\zeta_3)$.

Once you have a multiplicative function on $P$, how can it be extended to a multiplicative function on $I$? Thanks to finiteness of ideal class groups, $P$ has finite index in $I$: $[I:P] = |I/P|$ is the size of the ideal class group of $K$, which is usually denoted $h(K)$. Every multiplicative function on $P$ extends in $[I:P]$ ways to a multiplicative function on $I$: see Theorem 3.4 and Remark 3.8 here.