Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Let $\chi$ be an algebraic Hecke character on $K$ with conductor $\mathfrak{f}$ and infinity type $(a,b)$, i.e.
$$ \chi (\mathfrak{a}) = \epsilon(\alpha)\chi_\infty^{-1}(\alpha) = \epsilon(\alpha) \cdot \alpha^a \overline{\alpha}^b $$ where $\mathfrak{a}=(\alpha)$ for all $\alpha \in K^\times$ and $(\mathfrak{a},\mathfrak{f})=1$ and a finite order character $$ \epsilon : (\mathcal{O}_K/\mathfrak{f})^\times \longrightarrow \mathbb{S}^1 $$ One has an associated Hecke $L$-function \begin{equation} L(s,\chi) = \sum\limits_{\substack{0 \neq \mathfrak{a} \lhd \mathcal{O}_K \\ (\mathfrak{a},\mathfrak{f})=1 }} \frac{\chi(\mathfrak{a})}{N(\mathfrak{a})^s} \end{equation} Which is absolutely convergent on $\lbrace z \in \mathbb{C} \, | \, \operatorname{Re}(s) > \frac{a+b}{2}+1 \rbrace$. Let $P_\mathfrak{f}:= \lbrace \mathfrak{a}=(\alpha) \text{ principal fractional ideals } \: | \: \alpha \equiv 1 \: \operatorname{mod}^* \: \mathfrak{f} \rbrace$ the subgroup of $I(\mathfrak{f}):= \lbrace \mathfrak{a} \text{ fractional ideals of } K \: | \: (\mathfrak{a},\mathfrak{f})=1 \rbrace$.

I am reading a paper and the author writes for $a \in \mathbb{N}$, $s > \frac{a}{2}+1$

\begin{align} L(s,\overline{\chi}^a) = \sum\limits_{\substack{0 \neq \mathfrak{a} \lhd \mathcal{O}_K \\ (\mathfrak{a},\mathfrak{f})=1 }} \frac{\overline{\chi}^a(\mathfrak{a})}{N(\mathfrak{a})^s} \underset{(1)}{=}& \frac{1}{\omega_\mathfrak{f}} \sum\limits_{\mathfrak{a} \in I(\mathfrak{f})/P_\mathfrak{f}} \frac{\overline{\chi}^a(\mathfrak{a})}{N(\mathfrak{a})^s} \sum\limits_{\substack{\alpha \in \mathfrak{a}^{-1} \\ \alpha \equiv 1 \: \operatorname{mod}^* \: \mathfrak{f} }}\frac{\overline{\chi}^a(\mathfrak{a})}{|\alpha|^{2s}} \\ \underset{(2)}{=}& \frac{1}{\omega_\mathfrak{f}} \sum\limits_{\mathfrak{a} \in I(\mathfrak{f})/P_\mathfrak{f}} \; \sum\limits_{\gamma \in \mathfrak{a}^{-1}\mathfrak{f}}\frac{(\overline{\chi(\alpha_\mathfrak{a} \mathfrak{a}) + \chi(\mathfrak{a})\gamma})^a}{|\chi(\alpha_\mathfrak{a} \mathfrak{a}) + \chi(\mathfrak{a})\gamma|^{2s}} \end{align}

Here he says that $\omega_\mathfrak{f}$ is the number of roots of unity in $K$ that are congruent to $1$ modulo $\mathfrak{f}$.

I don't understand from where it comes from, nor how he gets the equalities (1) and (2)... If anyone can help explaining, it would be very much appreciated.

P.S: Why does one have $|\chi(\mathfrak{a})|^2=N(\mathfrak{a})$ in (1) ?


1 Answer 1


When $\psi$ is a character of $C_K$ then $$L(s,\psi ) =\sum_{0\ne I\subset O_K} \psi(I)N(I)^{-s}= \sum_{c\in C_K} \psi(c) \sum_{I\subset O_K,I\sim c} N(I)^{-s} $$ $$= \sum_{c\in C_K} \psi(c) \sum_{b\in (J_c-0)^{-1}/O_K^\times} N(J_c b)^{-s} = \sum_{c\in C_K} \psi(c)N(J_c)^{-s} \sum_{b\in (J_c-0)^{-1}/O_K^\times} |N_{K/Q}( b)|^{-s}$$

where $J_c$ is any fixed ideal in the class $c$.

(this is because $I$ is in the same class as $J_c$ iff $I = b J_c$ with $0\ne b\in J_c^{-1} = \{ d\in K, dJ_c\subset O_K\}$)

In your question it works the same way except that $\psi$ is a character of the class group of an order multiplied by a character of infinite place, and $O_K^\times$ is finite because $K$ is an imaginary quadratic field.

The point of doing so is that $\sum_{b\in (J_c-0)^{-1}/O_K^\times} |N_{K/Q}( b)|^{-s}$ is the Mellin transform of some kind of $\theta$ function.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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