You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We develop a criterion for a normal basis (Theorem \[criterion\]), and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$ (Theorem \[main\]). This result would be an answer for the Lang-Schertz conjecture on a ray class field with modulus generated by an integer ($\geq2$) (Remark \[S-Rinvariant\]).'
address:
- 'Department of Mathematical Sciences, KAIST'
- 'Department of Mathematical Sciences, KAIST'
- 'Department of Mathematical Sciences, KAIST'
author:
- Ho Yun Jung
- Ja Kyung Koo
- Dong Hwa Shin
title: Normal bases of ray class fields over imaginary quadratic fields
---

[^1]

[^2]

[^3]

Introduction
============

Let $L$ be a finite Galois extension of a field $K$. From the normal basis theorem ([@Waerden]) we know that there exists a normal basis of $L$ over $K$, namely, a basis of the form $\{x^\gamma:\gamma\in\mathrm{Gal}(L/K)\}$ for a single element $x\in
L$.

Okada ([@Okada1]) showed that if $k$ and $q$ ($>2$) are positive integers with $k$ odd and $T$ is a set of representatives for which $(\mathbb{Z}/q\mathbb{Z})^\times=T\cup(-T)$, then the real numbers $(\frac{1}{\pi}\frac{d}{dz})^k(\cot \pi z)|_{z=a/q}$ for $a\in T$ form a normal basis of the maximal real subfield of $\mathbb{Q}(e^{2\pi i/q})$ over $\mathbb{Q}$. Replacing the cotangent function by the Weierstrass $\wp$-function with fundamental period $i$ and $1$, he further obtained in [@Okada2] normal bases of class fields over Gauss’ number field $\mathbb{Q}(\sqrt{-1})$. This result was due to the fact that Gauss’ number field has class number one, which can be naturally extended to any imaginary quadratic field of class number one.

After Okada, Taylor ([@Taylor]) and Schertz ([@Schertz2]) established Galois module structures of rings of integers of certain abelian extensions over an imaginary quadratic field, which are analogues to the cyclotomic case ([@Leopoldt]). They also found normal bases by making use of special values of modular functions. And, Komatsu ([@Komatsu]) considered certain abelian extensions $L$ and $K$ of $\mathbb{Q}(e^{2\pi i/5})$ and constructed a normal basis of $L$ over $K$ in terms of special values of Siegel modular functions.

For any pair $(r_1,r_2)\in\mathbb{Q}^2-\mathbb{Z}^2$ we define the *Siegel function* $g_{(r_1,r_2)}(\tau)$ on $\mathfrak{H}$ (the complex upper half-plane) by the following infinite product $$\begin{aligned}
g_{(r_1,r_2)}(\tau)=-q_\tau^{(1/2)\textbf{B}_2(r_1)}e^{\pi
ir_2(r_1-1)}(1-q_z)\prod_{n=1}^{\infty}(1-q_\tau^nq_z)(1-q_\tau^nq_z^{-1}),\end{aligned}$$ where $\textbf{B}_2(X)=X^2-X+1/6$ is the second Bernoulli polynomial, $q_\tau=e^{2\pi i\tau}$ and $q_z=e^{2\pi iz}$ with $z=r_1\tau+r_2$. It is a modular unit in the sense of [@K-L].

Let $K$ ($\neq\mathbb{Q}(\sqrt{-1}),~\mathbb{Q}(\sqrt{-3})$) be an imaginary quadratic field of discriminant $d_K$ and $\mathcal{O}_K=\mathbb{Z}[\theta]$ be its ring of integers with $$\begin{aligned}
\label{theta}
\theta=\left\{\begin{array}{ll}\sqrt{d_K}/2&\textrm{for}~d_K\equiv0\pmod{4}\vspace{0.1cm}\\
(-1+\sqrt{d_K})/2&\textrm{for}~
d_K\equiv1\pmod{4}.\end{array}\right.\end{aligned}$$ In what follows we denote the Hilbert class field and the ray class field modulo $N\mathcal{O}_K$ for an integer $N$ ($\geq2$) by $H$ and $K_{(N)}$, respectively. We showed in [@J-K-S] that the singular value $x=g_{(0,1/N)}(\theta)^{-12N/\gcd(6,N)}$ is a primitive generator of $K_{(N)}$ over $K$. We achieved this result by showing that the absolute value of $x$ is the smallest one among those of all the conjugates.

In this paper we will show that the conjugates of a high power of $x$ form a normal basis of $K_{(N)}$ over $K$ by applying a criterion for a normal basis (Theorems \[criterion\] and \[main\]). As for the action of $\mathrm{Gal}(K_{(N)}/K)$ on $x$ in the process we adopt the idea of Gee-Stevenhagen ([@Gee], [@Stevenhagen]). Our result is also related to the Lang-Schertz conjecture on the Siegel-Ramachandra invariant to construct ray class fields $K_{(N)}$ (Remark \[S-Rinvariant\]).

A criterion for a normal basis
==============================

In this section we let $L$ be a finite abelian extension of a number field $K$ with $G=\mathrm{Gal}(L/K)=\{\gamma_1=\mathrm{id},\cdots,\gamma_n\}$. Furthermore, we denote by $|\cdot|$ the usual absolute value defined on $\mathbb{C}$.

\[det\] A set of elements $\{x_1,\cdots,x_n\}$ in $L$ is a $K$-basis of $L$ if and only if $$\det(x_i^{\gamma_j^{-1}})_{1\leq i,j\leq n}\neq0.$$

Straightforward.

By $\widehat{G}$ we mean the character group of $G$.

\[Frobenius\] If $f$ is any $\mathbb{C}$-valued function on $G$, then $$\prod_{\chi\in\widehat{G}} \sum_{1\leq i\leq
n}{\chi}(\gamma_i^{-1})f(\gamma_i)=\det(f(\gamma_i
\gamma_j^{-1}))_{1\leq i,j\leq n}.$$

See [@Lang] Chapter 21 Theorem 5.

Combining Lemmas \[det\] and \[Frobenius\] we derive the following proposition.

\[character\] The conjugates of an element $x\in L$ form a normal basis of $L$ over $K$ if and only if $$\sum_{1\leq i\leq n }{\chi}(\gamma_i^{-1})x^{\gamma_i}\neq0
\quad\textrm{for all}~\chi\in\widehat{G}.$$

For an element $x\in L$, set $x_i=x^{\gamma_i}$ ($i=1,\cdots,n$). We establish that $$\begin{aligned}
&&\textrm{the conjugates of $x$ form a normal basis of $L$ over
$K$}\\
&\Longleftrightarrow&\textrm{\{$x_1,\cdots,x_n\}$ is a $K$-basis of
$L$}~\textrm{by the definition of a normal basis}\\
&\Longleftrightarrow& \det(x_i^{\gamma_j^{-1}})_{1\leq i,j\leq
n}\neq0\quad\textrm{by Lemma \ref{det}}\\
&\Longleftrightarrow& \sum_{1\leq i\leq
n}{\chi}(\gamma_i^{-1})x_i\neq0\quad\textrm{for all
$\chi\in\widehat{G}$ by Lemma \ref{Frobenius} with}~f(\gamma_i)=x_i.\end{aligned}$$

Now we present a useful criterion which enables us to determine whether the conjugates of an element $x\in L$ form a normal basis of $L$ over $K$.

\[criterion\] Assume that there exists an element $x\in L$ such that $$\label{smaller}
|x^{\gamma_i}/x|<1\quad\textrm{for $1<i\leq n$}.$$ Then the conjugates of a high power of $x$ form a normal basis of $L$ over $K$.

By the hypothesis (\[smaller\]) we can take a suitably large integer $m$ such that $$\label{smaller2}
|x^{\gamma_i}/x|^m\leq 1/\#G\quad\textrm{for $1<i\leq n$},$$ where $\#G$ is the cardinality of $G$. Then, for any $\chi\in\widehat{G}$ we derive that $$\begin{aligned}
|\sum_{1\leq i\leq n}{\chi}(\gamma_i^{-1})(x^m)^{\gamma_i}| &\geq&
|x^m|(1-\sum_{1<i\leq n}|(x^m)^{\gamma_i}/x^m|)\quad\textrm{by the
triangle inequality}\\
&\geq&|x^m|(1-(1/\#G)(\#G-1))=|x^m|/\#G>0\quad\textrm{by
(\ref{smaller2}).}\end{aligned}$$ Therefore the conjugates of $x^m$ form a normal basis of $L$ over $K$ by Proposition \[character\].

Actions of Galois groups {#section3}
========================

We shall investigate an algorithm to get all conjugates of the singular value of a modular function.

For each positive integer $N$, let $\mathcal{F}_N$ be the field of modular functions of level $N$ defined over $\mathbb{Q}(\zeta_N)$ with $\zeta_N=e^{2\pi i/N}$. Then, $\mathcal{F}_N$ is a Galois extension of $\mathcal{F}_1=\mathbb{Q}(j)$ ($j=$the elliptic modular function) whose Galois group is isomorphic to $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm1_2\}$ ([@Lang] or [@Shimura]).

Throughout this section we let $K$ be an imaginary quadratic field of discriminant $d_K$ and $\theta$ be as in (\[theta\]).

Under the properly equivalence relation, primitive positive definite binary quadratic forms $aX^2+bXY+cY^2$ of discriminant $d_K$ determine a group $\mathrm{C}(d_K)$, called the *form class group of discriminant $d_K$*. We identify $\mathrm{C}(d_K)$ with the set of all *reduced quadratic forms*, which are characterized by the conditions $$\label{reduced}
-a<b\leq a<c\quad\textrm{or}\quad 0\leq b\leq a=c$$ together with the discriminant relation $$\label{disc}
b^2-4ac=d_K.$$ It is well-known that $\mathrm{C}(d_K)$ is isomorphic to $\mathrm{Gal}(H/K)$ ([@Cox]). For a reduced quadratic form $Q=aX^2+bXY+cY^2$ in $\mathrm{C}(d_K)$ we define a CM-point $$\label{theta_Q}
\theta_Q=(-b+\sqrt{d_K})/2a.$$ Furthermore, we define $\beta_Q=(\beta_p)_p\in\prod_{p~:~\textrm{primes}}\mathrm{GL}_2(\mathbb{Z}_p)$ as

-   Case 1 : $d_K\equiv0\pmod{4}$ $$\begin{aligned}
    \label{u1}
    \beta_p=\left\{\begin{array}{ll}
    \begin{pmatrix}a&b/2\\0&1\end{pmatrix}&\textrm{if}~p\nmid a\vspace{0.1cm}\\
    \begin{pmatrix}-b/2&-c\\1&0\end{pmatrix}&\textrm{if}~p\mid a~\textrm{and}~p\nmid c\vspace{0.1cm}\\
    \begin{pmatrix}-a-b/2&-c-b/2\\1&-1\end{pmatrix}&\textrm{if}~p\mid a~\textrm{and}~p\mid c
    \end{array}\right.\end{aligned}$$

-   Case 2 : $d_K\equiv1\pmod{4}$ $$\begin{aligned}
    \label{u2}
    \beta_p=\left\{\begin{array}{ll}
    \begin{pmatrix}a&(b-1)/2\\0&1\end{pmatrix}&\textrm{if}~p\nmid a\vspace{0.1cm}\\
    \begin{pmatrix}-(b+1)/2&-c\\1&0\end{pmatrix}&\textrm{if}~p\mid a~\textrm{and}~p\nmid c\vspace{0.1cm}\\
    \begin{pmatrix}-a-(b+1)/2&-c-(b-1)/2\\1&-1\end{pmatrix}&\textrm{if}~p\mid a~\textrm{and}~p\mid
    c.
    \end{array}\right.\end{aligned}$$

The next two propositions describe Shimura’s reciprocity law explicitly ([@Lang] or [@Shimura]).

\[Gal(H/K)\] Let $N$ be a positive integer. If $h\in\mathcal{F}_N$ is defined and finite at $\theta$ and $Q$ is a reduced quadratic form in $\mathrm{C}(d_K)$, then the value $h^{\beta_Q}(\theta_Q)$ belongs to $K_{(N)}$. Here, we note that there exists $\beta\in\mathrm{GL}_2^+(\mathbb{Q})\cap\mathrm{M}_2(\mathbb{Z})$ such that $\beta\equiv \beta_p\pmod{N\mathbb{Z}_p}$ for all primes $p$ dividing $N$ by the Chinese remainder theorem. Then the action of $\beta_Q$ on $\mathcal{F}_N$ is understood as that of $\beta$ which is an element of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm1_2\} $ ($\simeq\mathrm{Gal}(\mathcal{F}_N/\mathcal{F}_1)$). Furthermore, we have an isomorphism $$\begin{aligned}
\mathrm{C}(d_K)&\longrightarrow&\mathrm{Gal}(H/K)\\
Q&\mapsto&(h(\theta)\mapsto h^{\beta_Q}(\theta_Q))|_H,\end{aligned}$$ where $h\in\mathcal{F}_N$ is defined and finite at $\theta$.

See [@Gee] or [@Stevenhagen].

\[Gal(K\_[(N)]{}/K)\] Let $\min(\theta,\mathbb{Q})=X^2+BX+C\in\mathbb{Z}[X]$. For each positive integer $N$, the matrix group $$W_{N,\theta}=\bigg\{\begin{pmatrix}t-B s &
-C
s\\s&t\end{pmatrix}\in\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})~:~t,s\in\mathbb{Z}/N\mathbb{Z}\bigg\}$$ gives rise to a surjection $$\begin{aligned}
W_{N,\theta}&\longrightarrow&\mathrm{Gal}(K_{(N)}/H)\\
\alpha&\mapsto&(h(\theta)\mapsto h^\alpha(\theta)),\end{aligned}$$ where $h\in\mathcal{F}_N$ is defined and finite at $\theta$. The action of $\alpha$ on $\mathcal{F}_N$ is the action as an element of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm1_2\}$ ($\simeq\mathrm{Gal}(\mathcal{F}_N/\mathcal{F}_1)$). If $K\neq\mathbb{Q}(\sqrt{-1})$, $\mathbb{Q}(\sqrt{-3})$, then the kernel is $\{\pm1_2\}$.

See [@Gee] or [@Stevenhagen].

Combining the above two propositions we achieve the next result.

\[conjugate\] Let $K$ be an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, and $N$ be a positive integer. Then we have a bijective map $$\begin{aligned}
\begin{array}{cccc}
W_{N,\theta}/\{\pm1_2\}\times\mathrm{C}(d_K)&\longrightarrow&
\mathrm{Gal}(K_{(N)}/K)&\\
\alpha\times Q&\longmapsto&(h(\theta)\mapsto h^{\alpha\cdot
\beta_Q}(\theta_Q)),\end{array}\end{aligned}$$ where $h\in\mathcal{F}_N$ is defined and finite at $\theta$.

See [@J-K-S] Theorem 3.4.

Proposition \[conjugate\] and the following transformation formula of Siegel functions enable us to find all conjugates of the singular value $g_{(0,1/N)}(\theta)^{-12N/\gcd(6,N)}$, which will be used to prove our main theorem.

\[F\_N\] Let $N\geq 2$. For $(v,w)\in\mathbb{Z}^2-N\mathbb{Z}^2$, the function $g_{(v/N,w/N)}(\tau)^{-12N/\gcd(6,N)}$ is determined by $\pm(v/N,w/N)\pmod{\mathbb{Z}^2}$. It belongs to $\mathcal{F}_N$, and $\alpha\in\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm1_2\}$ ($\simeq\mathrm{Gal}(\mathcal{F}_N/ \mathcal{F}_1)$) acts on the function by $$(g_{(v/N,w/N)}(\tau)^{-12N/\gcd(6,N)})^\alpha
=g_{(v/N,w/N)\alpha}(\tau)^{-12N/\gcd(6,N)}.$$

See [@K-S] Proposition 2.4 and Theorem 2.5.

Normal bases of ray class fields
================================

The following lemma is a slight modification of a result in [@J-K-S].

\[newlemma\] Let $d_K$ ($\leq-7$) be the discriminant of an imaginary quadratic field and $\theta$ be as in (\[theta\]). Let $Q=aX^2+bXY+cY^2$ be a reduced quadratic form in $\mathrm{C}(d_K)$ and $\theta_Q$ be as in (\[theta\_Q\]). For each integer $N$ ($\geq2$) we have the inequalities:

-   If $a\geq2$, then $|g_{(s/N,t/N)}(\theta_Q)^{-1}/ g_{(0,1/N)}(\theta)^{-1}|< 1$ for all $(s,t)\in\mathbb{Z}^2-N\mathbb{Z}^2$.

-   If $a=1$, then $|g_{(s/N,t/N)}(\theta_Q)^{-1}/ g_{(0,1/N)}(\theta)^{-1}|< 1$ for all $s,t\in\mathbb{Z}$ with $s\not\equiv0\pmod{N}$.

-   If $a=1$, then $|g_{(0,t/N)}(\theta_Q)^{-1}/g_{(0,1/N)}(\theta)^{-1}|<1$ for all $t\in\mathbb{Z}$ with $t\not\equiv0,\pm1\pmod{N}$.

See [@J-K-S] Lemmas 4.2$\sim$4.3.

Now we are ready to prove our main theorem concerning a normal basis of $K_{(N)}$ over $K$.

\[main\] Let $K$ be an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, and $\theta$ be as in (\[theta\]). Let $N$ ($\geq2$) be an integer. Then, for a suitably large integer $m$, the conjugates of the singular value $$g_{(0,1/N)}(\theta)^{-12Nm/\gcd(6,N)}$$ form a normal basis of $K_{(N)}$ over $K$.

For simplicity we set $x=g_{(0,1/N)}(\theta)^{-12N/\gcd(6,N)}$, which belongs to $K_{(N)}$ by Proposition \[F\_N\] and the main theorem of complex multiplication ([@Lang] or [@Shimura]). It suffices to show by Theorem \[criterion\] that $$\label{tobeshown}
|x^\gamma/x|<1\quad\textrm{for all
$\gamma\in\mathrm{Gal}(K_{(N)}/K),~\gamma\neq\textrm{id}$}.$$ To this end we consider by Propositions \[conjugate\] and \[F\_N\] a conjugate $x^\gamma$ of $x$ which is of the form $$x^\gamma=(g_{(0,1/N)}(\tau)^{-12N/\gcd(6,N)})^{\alpha\cdot
\beta_Q}(\theta_Q)=g_{(s/N,t/N)\beta_Q}(\theta_Q)^{-12N/\gcd(6,N)}$$ for some $\alpha=\pm\left(\begin{smallmatrix}t-Bs&-Cs\\s&t\end{smallmatrix}\right)\in
W_{N,\theta}/\{\pm1_2\}$, where $\min(\theta,\mathbb{Q})=X^2+BX+C$, and $Q=aX^2+bXY+cY^2$ is a reduced quadratic form in $\mathrm{C}(d_K)$. If $a\geq2$, then the inequality (\[tobeshown\]) holds by Lemma \[newlemma\](i). If $a=1$, then we derive from the conditions (\[reduced\]) and (\[disc\]) for a reduced quadratic form that $$\begin{aligned}
Q=\left\{\begin{array}{ll} X^2-(d_K/4)Y^2 &
\textrm{for}~d_K\equiv0\pmod{4}\vspace{0.1cm}\\
X^2+XY+((1-d_K)/4)Y^2 & \textrm{for}~
d_K\equiv1\pmod{4},\end{array}\right.\end{aligned}$$ which yields $\beta_Q\equiv1_2\pmod{N}$ by the definitions (\[u1\]) and (\[u2\]). Thus $x^\gamma=g_{(s/N,t/N)}(\theta_Q)^{-12N/\gcd(6,N)}$. Moreover, if $(s,t)\not\equiv(0,\pm1)\pmod{N}$, then the inequality (\[tobeshown\]) is also true by Lemma \[newlemma\](ii) and (iii). It is not necessary to consider the remaining case when $a=1$ and $(s,t)\equiv(0,\pm1)\pmod{N}$, because $\gamma=\mathrm{id}\in\mathrm{Gal}(K_{(N)}/K)$ in this case. Therefore (\[tobeshown\]) holds true for all $\gamma\in\mathrm{Gal}(K_{(N)}/K)$, $\gamma\neq\mathrm{id}$, as desired. This completes the proof.

\[S-Rinvariant\] Let $K$ be an imaginary quadratic field and $\mathfrak{f}$ be a nontrivial integral ideal of $K$. We denote by $\mathrm{Cl}(\mathfrak{f})$ the ray class group modulo $\mathfrak{f}$ and write $C_0$ for its unit class. By definition the ray class field $K_\mathfrak{f}$ modulo $\mathfrak{f}$ is a finite abelian extension of $K$ whose Galois group is isomorphic to $\mathrm{Cl}(\mathfrak{f})$ via the Artin map. For $C\in\mathrm{Cl}(\mathfrak{f})$ we take an integral ideal $\mathfrak{c}$ in $C$ so that $\mathfrak{f}\mathfrak{c}^{-1}=[z_1,z_2]$ with $z=z_1/z_2\in\mathfrak{H}$. We define the *Siegel-Ramachandra invariant* $g_\mathfrak{f}(C)$ by $$g_\mathfrak{f}(C)=g_{(a/N,b/N)}(z)^{12N},$$ where $N$ is the smallest positive integer in $\mathfrak{f}$ and $a$, $b$ are integers such that $1=(a/N)z_1+(b/N)z_2$. This value depends only on the class $C$ and belongs to $K_\mathfrak{f}$ ([@K-L]).

Ramachandra showed in [@Ramachandra] that $K_\mathfrak{f}$ can be generated over $K$ by certain elliptic unit, but his invariant involves overly complicated product of high powers of singular values of the Klein forms and singular values of the $\Delta$-function for practical use. Thus, Lang proposed in his book ([@Lang] p.292) to find a simpler one by utilizing Siegel-Ramachandra invariants, and Schertz ([@Schertz]) showed that $g_\mathfrak{f}(C_0)$ (so any $g_\mathfrak{f}(C)$) generates $K_\mathfrak{f}$ over $K$ under some conditions on $\mathfrak{f}$. He further conjectured that $g_\mathfrak{f}(C_0)$ is always a ray class invariant.

On the other hand, the modulus $\mathfrak{f}$ can be enlarged by class field theory so that we may assume $\mathfrak{f}=N\mathcal{O}_K$ for an integer $N$ ($\geq2$). By definition we see that $g_\mathfrak{f}(C_0)=g_{(0,1/N)}(\theta)^{12N}$. If $K\neq\mathbb{Q}(\sqrt{-1}),~\mathbb{Q}(\sqrt{-3})$, then Theorem \[main\] indicates that $g_\mathfrak{f}(C_0)$ is indeed a primitive generator of $K_\mathfrak{f}$ over $K$, which would be an answer for the Lang-Schertz conjecture.

Let $K=\mathbb{Q}(\sqrt{-5})$ and $N=6$, then $d_K=-20$ and $\theta=\sqrt{-5}$. We get $$\begin{aligned}
&&\mathrm{C}(d_K)=\{Q_1=X^2+5Y^2,Q_2=2X^2+2XY+3Y^2\}\\
&&\theta_{Q_1}=\sqrt{-5},~\theta_{Q_2}=(-1+\sqrt{-5})/2,~
\beta_{Q_1}=\left(\begin{smallmatrix}
1&0\\0&1\end{smallmatrix}\right),~
\beta_{Q_2}=\left(\begin{smallmatrix}1&5\\3&2\end{smallmatrix}\right)\\
&&W_{N,\theta}/\{\pm1_2\}=
\{\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right),
\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right),
\left(\begin{smallmatrix}2&3\\3&2\end{smallmatrix}\right),
\left(\begin{smallmatrix}3&2\\2&3\end{smallmatrix}\right)\}.\end{aligned}$$ By Propositions \[conjugate\] and \[F\_N\], the conjugates of $x=g_{(0,1/6)}(\sqrt{-5})^{-12}$ are $$\begin{aligned}
\begin{array}{ll}
x_1=x, & x_2=g_{(1/6,0)}(\sqrt{-5})^{-12}\\
x_3=g_{(3/6,2/6)}(\sqrt{-5})^{-12}, &
x_4=g_{(2/6,3/6)}(\sqrt{-5})^{-12}\\
x_5=g_{(3/6,2/6)}((-1+\sqrt{-5})/2)^{-12}, &
x_6=g_{(1/6,5/6)}((-1+\sqrt{-5})/2)^{-12}\\
x_7=g_{(3/6,1/6)}((-1+\sqrt{-5})/2)^{-12}, &
x_8=g_{(5/6,4/6)}((-1+\sqrt{-5})/2)^{-12}
\end{array}\end{aligned}$$ possibly with multiplicity. By using a computer program (such as MAPLE) one can find that $$\begin{aligned}
|x_i/x_1|<10^{-4}<1/\#\mathrm{Gal}(K_{(6)}/K)=1/8\quad\textrm{for}~i=2,\cdots,8.\end{aligned}$$ Therefore $\{x_1,\cdots,x_8\}$ becomes a normal basis of $K_{(6)}$ over $K$ by Theorem \[criterion\]. Moreover, one can show that the minimal polynomial of $x$ would be $$\begin{aligned}
(X-x_1)\cdots(X-x_8)&=&X^8-1263840X^7+42016796X^6+72894400X^5\\
&&+15056640X^4-4525280X^3+167196X^2-1280X+1\end{aligned}$$ with integer coefficients ([@K-S] $\S$3).

[10]{}

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