4
$\begingroup$

Let $L$ be a number field, $\mathcal{O}_L$ its ring of integers, and $\mathcal{Cl(O}_L)$ its ideal class group. Let's fix an arbitrary class $[c] \in \mathcal{Cl(O}_L)$. By $r(n)=r([c], n)$, I mean the number of ideals of norm $n$, that belong to the class $[c]$,

$$r(n)=r([c], n)= \sharp\bigg\{ \mathfrak{I} \subseteq \mathcal{O}_L: \mathfrak{I} \in [c], N(\mathfrak{I})=n \bigg\}.$$

Dedekind zeta function, is something which is related to all of these $r([c], n)$'s, where $[c]$ varies arbitrarily in $\mathcal{Cl(O}_L)$. I am curious about the situation when we restrict ourselves to only one arbitrary but fixed class $[c] \in \mathcal{Cl(O}_L)$: What are the "well-known" functions in number theory, that are related to $r(n)=r([c], n)$'s?




What I know: Suppose that $L$ is an imaginary quadratic field, $D$ its discriminant, and let $[c]$ be an arbitrary but fixed class in $\mathcal{Cl(O}_L)$, and let $\Theta(z)=\Theta_{[c]}(z)= 1+\sum_{n=1}^{\infty} r(n)q^n$, where $q=e^{2\pi i z}$, and $r(n)=r([c], n)$. Then $\Theta(z)$ is a modular form of weight $1$, with level $N=\vert D \vert$ and charachter $\chi(.)=\left(\dfrac{D}{.}\right)$; i.e. for $z \in \mathcal{H}$ and $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$ we have $\Theta\left(\dfrac{az+b}{cz+d}\right) = \chi(a)\left(cz+d\right) \Theta(z)$. [We can associate a quadratic form the class $[c]$, and here $\Theta(z)$ is the associated theta series.]

$\endgroup$

1 Answer 1

6
$\begingroup$

It sounds like you're looking for something like the function $$\zeta_C(s) = \sum_{\mathfrak{a} \in C} N(\mathfrak{a})^{-s} = \sum_{n \ge 1} r([c], n) n^{-s}.$$ These functions are sometimes called "ideal class zeta functions" and they come up from time to time in the literature. See e.g. this paper in J London Math Soc:

Friedman, Eduardo, The zero near 1 of an ideal class zeta function, J. Lond. Math. Soc., II. Ser. 35, 1-17 (1987). ZBL0583.12010.

The related functions where you sum over a character $\psi$ of the ideal class group, $$L(\psi, s) = \sum_{C \in Cl(L)} \psi(C) \zeta_C(s),$$ are much more commonly studied (since they have an Euler product expansion, which isn't true of the $\zeta_C(s)$ individually). These are examples of Hecke $L$-functions.

$\endgroup$
1
  • $\begingroup$ $L(\psi, s) = \prod_{\mathfrak{P}}(1 - \psi([\mathfrak{P}]) N(\mathfrak{P})^{-s})^{-1}$ (product over prime ideals of $O_L$). $\endgroup$ Jan 9, 2021 at 17:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.