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In Elliptic Modular Forms and Their Applications, p.89, Zagier defines a "größencharakter" $\psi_N$ on the field $K = \mathbb{Q}(i)$. This is a character on ideals. Since $\mathcal{O}_K = \mathbb{Z}[i]$ is a PID we can write every integral ideal $I\subseteq \mathcal{O}_K$ as $I = (\lambda)$ for a choice of $4$ different generators $\lambda\in\mathcal{O}_K$ (since $w_K = \lvert\mathcal{O}_K^\times\rvert=4$). Specifically, define

$$\psi_N (I) = \overline{\lambda}^N, \quad \text{ if } w_K\vert N \text{ and } I = (\lambda),$$

and set it to be $0$ for $N$ not divisible by $w_K$.

More generally, assume that $K$ is an imaginary quadratic number field with class number $h_K = 1$, and let $w_K$ denote the size of the unit group $\mathcal{O}_K^\times$. Define

$$\psi_N(I)=\overline{\lambda}^N, \quad \text{ if } w_K\vert N \text{ and } I = (\lambda).$$

Question: How can $\psi_N$ be interpreted as a größencharakter on $K$?

A größencharakter $\chi$ has 2 equivalent formulations (I believe): either as a continuous character on the idele group $\chi: \mathbb{A}_K^\times\to\mathbb{C}^\times$ with $K^\times$ in its kernel, or (modulo some ideal $\mathfrak{m}\subseteq \mathcal{O}_K$) as a character defined on the group $J^\mathfrak{m}$ of ideals coprime to $\mathfrak{m}$ such that there exist characters $$\chi_f: (\mathcal{O}_K/\mathfrak{m})^\times\to\mathbb{C}^\times, \quad \chi_\infty:\mathbf{R}^\times\to\mathbb{C}^\times$$ which determine $\chi$ on principal integral ideals: $\chi((a)) = \chi_f(a)\chi_\infty(a)$ when $a\in\mathcal{O}_K$ is coprime to $\mathfrak{m}$. Here, $\mathbf{R}$ is the $\mathbb{R}$-algebra $$\mathbf{R} = \left\{(z,\overline{z}): z\in\mathbb{C}\right\}$$ and $K\hookrightarrow\mathbf{R}$ via $a\mapsto (a, \overline{a})$. (This definition of a größencharakter can be found, for example, in Neukirch's Algebraic Number Theory, p.470.)

Ideally I would like to see how either of these could be written down explicitly for this "größencharakter" $\psi_N$, and to understand how to extend this ad-hoc definition to a größencharakter in either sense.

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    $\begingroup$ I just want to bring up a minor detail of terminology. Shouldn't "grössencharacter" be "Größencharakter" with plural "Größencharaktern"? $\endgroup$
    – David R.
    Mar 23, 2018 at 18:02
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    $\begingroup$ @DavidR. You’re quite right, I have probably anglicised the word incorrectly! Indeed the original German should be “größencharakter”, and I think when anglicised people sometimes write “grossencharacter”. But perhaps I shouldn’t have included the umlaut as well in this case? $\endgroup$
    – Alex Saad
    Mar 26, 2018 at 23:07
  • $\begingroup$ Does the following link answer the question completely? en.wikipedia.org/wiki/… $\endgroup$
    – dan_fulea
    Apr 10, 2018 at 19:34
  • $\begingroup$ @dan_fulea unfortunately not. I am not asking for an explanation of the equivalence of both definitions (although the article does not completely explain this), but rather how Zagier's "definition" fits into either of the established ones. For example, what are the modulus $\mathfrak{m}$ and the characters $\chi_f$, $\chi_\infty$ in this situation? $\endgroup$
    – Alex Saad
    Apr 11, 2018 at 10:09

1 Answer 1

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$\psi_N$ is a Größencharakter modulo $\mathfrak{m}:=\mathcal{O}_K$ in the following way. Let $\chi_f$ be the trivial character on the $1$-element group $(\mathcal{O}_K/\mathfrak{m})^\times$, and let $\chi_\infty(z,\overline{z}):=\overline{z}^N$. Then, for any nonzero $a\in\mathcal{O}_K$, we have $\psi_N((a))=\overline{a}^N=\chi_f(a)\chi_\infty(a)$, and we are done.

Note that the condition $w_K\mid N$ ensures that $\psi_N$ is well-defined: if $a,b\in\mathcal{O}_K$ are nonzero, and $(a)=(b)$, then $a=bu$ for some unit $u\in\mathcal{O}_K^\times$, and hence $a^N=(bu)^N=b^N$.

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