0 Introduction

This work focuses on the weight one Eisenstein series derived from the genus theory of imaginary quadratic fields. The relevant background material for this paper is provided in Section 1. Sections 2 and 3 contain a brief review of the genus characters of Gauss, correspondence between the ideals and the quadratic forms of the quadratic fields, and some identities derived from the genus theory. In Sections 4 and 5, we characterize the subspaces of Eisenstein series generated from the genus theory and bring out their connection with the Dedekind zeta function of the genus field of quadratic fields. Finally, in the last section, we derive additional results by twisting the Eisenstein series with primitive Dirichlet characters.

1 Preliminaries

Let $N$ be a positive integer. We say $\chi$ is a Dirichlet character modulo $N$ if, for all integers $m$ and $n$,

(1) $\chi(1)=1$;

(2) $\chi(n+N)=\chi(n)$;

(3) $\chi(mn)=\chi(m)\chi(n)$;

(4) $\chi(n)=0$ if gcd$(n, N)>1$.

Let $N'$ be a positive integer which is divisible by $N$. For any character $\chi$ modulo $N$, we can form a character $\chi'$ modulo $N'$ as follows:

$ \begin{align*} \chi'(k)= \begin{cases} \chi(k), \quad&\text{if gcd} (k, N')=1, \\ 0, \quad&\text{if gcd}(k, N')>1. \end{cases} \end{align*} $

We let $\chi_1$ denote the character such that $\chi_1(n)=1$ for all $n$.

We say that $\chi'$ is induced by the character $\chi$. Let $\chi$ be a character modulo $N$. If there is a proper divisor $d$ of $N$ and a character modulo $d$ which induces $\chi$, then the character $\chi$ is said to be non-primitive; otherwise, it is called primitive.

Let $\chi_{a}(b):=\big(\frac{a}{b}\big)$ denote the Kronecker's extension of the Jacobi symbol [1]³⁵.

Define

$ c_0(\chi) =-\frac{1}{N}\sum\limits_{n=1}^{N-1}n\chi(n). $

Let $D$ be the discriminant of a quadratic field:

$ c_0(\chi_D)= \begin{cases} \dfrac{2h}{w}, \quad&\text{if} \ \ D<0, \\ 0, \quad&\text{if} \ \ D>0, \end{cases} $

where $h$ is the class number of the quadratic field $\mathbb Q(\sqrt{D})$ and $w$ is the number of roots of unity.

It is known that (cf. [2], p. 347 and p. 349): A character $\chi$ is real and primitive if and only if $\chi=\chi_D$ for some $D$, where $D$ is the discriminant of a quadratic field.

Let $\mathbb H=\{\tau: \tau=x+iy, y>0\}$ and let $q={\rm e}^{2i\pi\tau}$, $\tau\in\mathbb H$.

Suppose $\varphi$ and $\psi$ are Dirichlet characters modulo $M$ and $N$, respectively. Define

$ {\frak f}(\tau;\varphi, \psi): =\varphi(0)c_0(\psi)+2\sum\limits_{m, n=1}^{\infty}\varphi(m)\psi(n)q^{mn}. $

We note that if $M=1$, then $\varphi=\chi_1$ and

$ {\frak f}(\tau;\chi_1, \psi) =c_0(\psi)+2\sum\limits_{m, n=1}^{\infty}\psi(n)q^{mn} =c_0(\psi)+2\sum\limits_{n=1}^{\infty}\psi(n)\frac{q^n}{1-q^n}. $

In particular,

$ \begin{equation} {\frak f}(\tau;\chi_1, \chi_D) =c_0(\chi_{D})+2\sum\limits_{m, n=1}^{\infty}\chi_D(n)q^{mn} =c_0(\chi_{D})+2\sum\limits_{n=1}^{\infty}\chi_D(n)\frac{q^n}{1-q^n}. \end{equation} $

(1.1)

Let

$ \Gamma_0(N)= \left\{\!\begin{pmatrix} a&b\\ c&d \end{pmatrix} :a, b, c, d\in\mathbb Z, ad-bc=1, N\mid c\right\}. $

Let $k$ be a non-negative integer, $\psi$ a character modulo $N$, and let ${{\bf{M}}_{{k}}}({\Gamma _0}(N),\psi )$ denote the space of functions $f$ on $\mathbb H$ satisfying the following criteria:

(1) $f$ is analytic on $\mathbb H$;

(2) $f$ is analytic at all cusps of $\Gamma_0(N)$;

(3) $f\big(\frac{a\tau+b}{c\tau+d}\big)=\psi(d)(c\tau+d)^k f(\tau)$, where $\begin{pmatrix} a&b\\ c&d \end{pmatrix}\in\Gamma_0(N)$.

We recall that $\Gamma_1(N)$ is the subgroup of $\Gamma_0(N)$ such that

$ a\equiv d\equiv 1 (\text{mod N}) \quad \text{and} \quad N\mid c. $

Thus, if ${{ f}} \in {{\bf{M}}_k}({\Gamma _{\bf{0}}}(N),\psi )$, then $f \in {{\bf{M}}_{{k}}}({\Gamma _1}(N))$, that is,

$ f\Big(\frac{a\tau+b}{c\tau+d}\Big)=(c\tau+d)^k f(\tau), $

where $\begin{pmatrix} a&b\\ c&d \end{pmatrix}\in\Gamma_1(N)$.

Theorem 1.1 Suppose $\varphi$ and $\psi$ are primitive characters modulo $M$ and $N$, respectively, and $\varphi\psi(-1)=-1$. Then

$ {\frak f}(\tau;\varphi, \psi)\in {\bf{M}}_1(\Gamma_0(MN), \varphi\psi). $

A proof based on Weil's converse theorem for modular forms can be found in [3, Theorem 9.1].

Suppose $-D < 0$ is the discriminant of an imaginary quadratic field. For $M=1$, $N=D$ and $\chi(n)=\chi_{-D}(n)=\big(\frac{-D}{n}\big)$. Since $\chi_{-D}$ is primitive modulo $D$ with $\chi_{-D}(-1)=-1$, we derive, from above theorem, the following result due to Hecke.

Corollary 1.2 If $-D < 0$ is the discriminant of an imaginary quadratic field, then ${\frak f}(\tau; \chi_1, \chi_{-D}) \in {\bf{M}}_1(\Gamma_0(D), \chi_{-D})$.

From (1.1),

$ \begin{equation} {\frak f}(\tau;\chi_1, \chi_{-D})=\frac{2h}{w}+ 2\sum\limits_{n=1}^{\infty}\Big(\frac{-D}{n}\Big)\frac{q^n}{1-q^n}, \end{equation} $

(1.2)

where $h$ is the class number of the quadratic field $\mathbb Q(\sqrt{-D})$ and $w$ is the number of roots of unity. Moreover,

$ \begin{equation} h+w\sum\limits_{n=1}^{\infty}\Big(\frac{-D}{n}\Big)\frac{q^n}{1-q^n} =\sum\limits_{i=1}^h \sum\limits_{m, n=-\infty}^{\infty}q^{Q_i(m, n)}, \end{equation} $

(1.3)

where $Q_i(x, y), i=1, 2, \cdots, h$, are the inequivalent quadratic forms of discriminant $-D$. The proof is given in [4, Theorem 4].

The definition of the class number $h$ and the construction of the inequivalent quadratic forms associated with a given quadratic field will be briefly reviewed in the following sections.

For later use, we remind the reader of the definition of a Mellin transform. For

$ f(q)=\sum\limits_{n=0}^{\infty} a_n q^n, $

the Mellin transform of $f$ is defined as

$ M(s, f): = \frac{{{{(2\pi )}^s}}}{{\Gamma (s)}}\int_0^\infty {(f({\rm{i}}y) - {a_0}){y^s}\frac{{{\rm{d}}y}}{y}.} $

Then

$ M(s, f)=\sum\limits_{n=1}^{\infty}\frac{a_n}{n^s}. $

2 Identities involving Genus characters

Let $\mathbb F$ be a quadratic field and let $\mathbb O$ denote its ring of integers. Let ${\frak a}$ denote an ideal in $\mathbb O$ and $\frak p$ denote a prime ideal. The norm of an ideal ${\frak a}$ is denoted as $N{\frak a}$. We will be dealing with non-zero ideals only.

We say the non-zero ideals ${\frak a}$ and ${\frak b}$ in $\mathbb O$ are equivalent, denoted by ${\frak a}\sim {\frak b}$, if there exist non-zero $\alpha, \beta\in \mathbb O$ such that

$ (\alpha){\frak a}=(\beta){\frak b}, $

where $(\alpha), (\beta)$ denote the principal ideals generated by $\alpha$ and $\beta$.

The equivalence relation $\sim$ partitions the set of ideals in $\mathbb O$ into the ideal classes $\{\bf A_1, \bf A_2, \cdots, \bf A_h\}$ which form a finite commutative group denoted as $\bf A$. The number $h$ is called the class number of the quadratic field $\mathbb F$.

We begin with a lemma.

Lemma 2.1^[5]54 Every discriminant can be written uniquely, modulo permutations, as the product of prime discriminants.

Let $D$ be the discriminant of the quadratic field $\mathbb F$. For a given decomposition of $D$ into a product of two discriminants: $D=d_1d_2$, the number of decompositions is $2^{k-1}$. We define a character on the prime ideals of $\mathbb O$ associated with the given decomposition as:

$ \chi({\frak p})=\chi_{d_1}({\frak p})=\left(\frac{d_1}{N{\frak p}}\right) $

for any prime ideal ${\frak p}\nmid D$. When ${\frak p}|D$, only one of the characters $\left(\frac{d_1}{N{\frak p}}\right), \left(\frac{d_2}{N{\frak p}}\right)$ is zero, and we define $\chi({\frak p})$ to be the one with non-zero value. Since the number of decompositions is $2^{k-1}$, there are exactly $2^{k-1}$ such characters. These are the famous genus characters due to Gauss.

We also introduce $\chi_{d_2}({\frak p})=\left(\frac{d_2}{N{\frak p}}\right)$. For ${\frak p}\nmid D$, the following identity holds ^[5]55 :

$ \chi_{d_1}({\frak p})=\chi_{d_2}({\frak p}). $

For a given decomposition of $D=d_1d_2$, define

$ L_{\mathbb F}(s, \chi) =\sum\limits_{{\frak a}}\frac{\chi({\frak a})}{(N{\frak a}) ^s}. $

Recall that the $L$-series associated with the character $\chi$ is defined as

$ L(s, \chi)=\sum\limits_{n=1}^{\infty}\frac{\chi(n)}{n^s}. $

We have the following result due to Kronecker.

Theorem 2.2

$ L_{\mathbb F}(s, \chi) =L(s, \chi_{d_1})L(s, \chi_{d_2}). $

The proof is given in [5, p. 57].

Applying the inverse Mellin transformation to the above identity, we obtain

$ \begin{equation} \sum\limits_{{\frak a}}\chi({\frak a})q^{N{\frak a}} =\frac12\left({\frak f}(\tau;\chi_{d_1}, \chi_{d_2})-\chi_{d_1}(0)c_0(\chi_{d_2})\right). \end{equation} $

(2.1)

We say that two ideals ${\frak a}, {\frak b}$ in $\mathbb O$ belong to the same genus if $\chi({\frak a})=\chi({\frak b})$ for every genus character $\chi$. Using the genus characters, we can partition the set of ideals in $\mathbb O$ into the disjoint union of $2^{k-1}$ genera: {${\frak G}_i: i=1, 2, \cdots, 2^{k-1}\}={\frak G}$; we shall denote the principal genus as ${\frak G}_1$. If we let $h$ be the class number of $\mathbb F$, then each genus contains $\frac{h}{2^{k-1}}$ elements of the class group of the ideals. Using the properties of ideal multiplication, the set of genera can be made into a group, and the genus characters are characters on ${\frak G}$. We will denote the group of the genus characters by ${\frak G}^*$. For details, see [1, Chapter 13].

For $\chi\in{\frak G}^*$, let

$ {\frak f}^{\circ}(\tau;\chi):= \frac12({\frak f}(\tau;\chi_{d_1}, \chi_{d_2})-\chi_{d_1}(0)c_0(\chi_{d_2})). $

We now re-write (2.1) as

$ \begin{equation} {\frak f}^{\circ}(\tau;\chi) =\sum\limits_{{\frak a}}\chi({\frak a})q^{N{\frak a}} =\sum\limits_{i=1}^{2^{k-1}}\chi({\frak G}_i)\sum\limits_{{\frak a}\in{\frak G}_i}q^{N{\frak a}}. \end{equation} $

(2.2)

3 Modular forms generated by the genus theory of imaginary quadratic fields

Let $\mathbb F$ be an imaginary quadratic field with discriminant $-D < 0$.

Let $-D=d_1d_2$ be a decomposition. Then, $\chi_{d_1}\chi_{d_2}(-1)=\chi_{-D}(-1)=-1$, and $\chi_{d_1}, \chi_{d_2}$ are both primitive modulo $d_1$ and $d_2$, respectively. We derive, from Theorem 1.1, the following theorem.

Theorem 3.1 Let $\chi$ be the character in ${\frak G}^*$ associated with the decomposition $-D=d_1d_2$ and let $ {\frak f}(\tau; \chi)={\frak f}(\tau; \chi_{d_1}, \chi_{d_2})$. Then

$ {\frak f}(\tau;\chi)\in{\bf M}_1(\Gamma_0(D), \chi_{-D}). $

Using the orthogonal relation of the genus characters, we derive from (2.2)

$ \begin{equation} \sum\limits_{{\frak a}\in{\frak G}_k}q^{N{\frak a}} =\frac{1}{2^{k-1}}\sum\limits_{\chi\in{\frak G}^*}\overline{\chi({\frak G}_k)} {\frak f}^{\circ}(\tau;\chi). \end{equation} $

(3.1)

We now give a more explicit expression for the sum on the left hand side of (3.1). First, we need to review some basic facts about associating the ideals with quadratic forms. We will confine the discussion to imaginary quadratic fields.

We now generate a family of $h$ inequivalent quadratic forms

$ Q_1(x, y), Q_2(x, y), \cdots, Q_h(x, y). $

For each $i$, choose an ideal ${\frak a}_i\in \bf A_i$: ${\frak a}_i=[\alpha_i, \beta_i]$, where $\{\alpha_i, \beta\}$ forms a basis of ${\frak a}_i$ considered as a module over $\mathbb Z$. We associate it with a quadratic form $Q_i(x, y)$ as follows:

$ Q_i(x, y)=\frac{N(\alpha_i x+\beta_i y)}{N{{\frak a}_i}}. $

A different choice of ${\frak a}$ will lead to an equivalent quadratic form. The ideal-form correspondence is described in great detail in Chapter 12 of [1].

Let $\bf A=\bf A_i$ for some $1\le i \le h$ and let $\bf A^{-1}$ be its inverse. Fix an ideal ${\frak a}^*\in \bf A^{-1}$. Then for each ${\frak a}\in\bf A$, the ideal ${\frak a}{\frak a}^*=(\xi_{{\frak a}})$ is principal and

$ {\frak a}\shortmid\!\rightarrow (\xi_{{\frak a}}) $

is a bijection between the elements of $\bf A$ and the non-zero principal subideals of ${\frak a}^*$. Let ${\frak a}^*=[\alpha, \beta]$. Then, $\xi_{{\frak a}}=m\alpha+n\beta$ for some integers $m, n$ with $(m, n)\ne(0, 0)$ and

$ N{\frak a}=\frac{N\xi_{{\frak a}}}{N\frak a^*}=\frac{N(m\alpha+n\beta)}{N{\frak a}^*} =Q_{{\frak a}^*}(m, n), $

where $Q_{{\frak a}}$ is the quadratic form generated by the ideal ${\frak a}$.

We note that if $u$ is a unit, $(u\xi)=(\xi)$, then the map from $\mathbb O$ to the set of principal ideals

$ \xi\shortmid\!\rightarrow (\xi) $

is a $w$ to one map. We arrive at a lemma.

Lemma 3.2

$ w\sum\limits_{{\frak a}\in\bf A}q^{N{\frak a}} =\sum\limits_{(m, n)\ne(0, 0)}q^{Q_{{\frak a}^*}(m, n)}. $

We say $Q(x, y)\in{\frak G}_k$ if $Q$ is generated by an ideal ${\frak a}$ in an ideal class of ${\frak G}_k$. Since the ideal classes $\bf A$ and $\bf A^{-1}$ always belong to the same genus, we conclude that

$ w\sum\limits_{{\frak a}\in{\frak G}_k}q^{N{\frak a}}=\sum\limits_{Q\in\frak G_k}\sum\limits_{(m, n)\ne(0, 0)}q^{Q(m, n)}. $

From (2.2), we have

Theorem 3.3

$ \sum\limits_{Q\in{\frak G}_k}\sum\limits_{(m, n)\ne(0, 0)}q^{Q(m, n)} =\frac{w}{2^{k-1}}\sum\limits_{\chi\in{\frak G}^*}\overline{\chi({\frak G}_k)} {\frak f}^{\circ}(\tau;\chi). $

We can re-write the left hand side of the above identity as

$ \sum\limits_{n=1}^{\infty}c_k(n)q^n=\sum\limits_{Q\in\frak G_k}\sum\limits_{(m, n)\ne(0, 0)}q^{Q(m, n)}, $

where $c_k(n)$ counts the number of elements in the set

$ \{(x, y)\in\mathbb Z^2\backslash\{(0, 0)\}:n=Q(x, y), Q\in {\frak G}_k\}. $

Each ${\frak f}^{\circ}(\tau; \chi)$ can be expressed as a power series:

$ {\frak f}^{\circ}(\tau;\chi)=\sum\limits_{n=1}^{\infty}q^n \sum\limits_{m|n}\chi_{d_1}(n/m)\chi_{d_2}(m). $

Comparing the coefficients on both sides, we obtain an explicit formula expressing $c_k(n)$ as a sum involving the genus characters. In particular, for the principal genus ${\frak G}_1$, we have

Corollary 3.4

$ \sum\limits_{Q\in{\frak G}_1}\sum\limits_{(m, n)\ne(0, 0)}q^{Q(m, n)}=\frac{w}{2^{k-1}}\sum\limits_{\chi\in{\frak G}^*}{\frak f}^{\circ}(\tau, \chi) $

and

$ c_1(n)=\sum\limits_{\chi\in{\frak G}^*}\sum\limits_{m|n}\chi_{d_1}(n/m)\chi_{d_2}(m). $

As a corollary to the above theorem, we observe that for the important cases in which every genus consists of a single quadratic form, the sum $\sum_{m, n=-\infty}^{\infty}q^{Q(m, n)}$ for each $Q$ can be given explicitly. For the general case that contains more than one form in each genus, readers can consult [6].

We note that, with the exception of $\chi$ associated with the decomposition $d_1=1, d_2=-D$, ${\frak f}^{\circ}(\tau, \chi)=\frac12{\frak f}(\tau; \chi_{d_1}, \chi_{d_2})$. Re-writing the right hand side of (3.1) in terms of ${\frak f}(\tau; \chi)$, we have

$ \begin{equation} \sum\limits_{Q\in{\frak G}_k}\sum\limits_{m, n=-\infty}^{\infty}q^{Q(m, n)} =h^*+w\sum\limits_{{\frak a}\in{\frak G}_k}q^{N{\frak a}} =\frac{w}{2^{k}}\sum\limits_{\chi\in{\frak G}^*}\overline{\chi({\frak G}_k)} {\frak f}(\tau;\chi), \end{equation} $

(3.2)

where $h^*=\frac{h}{2^{k-1}}$ is the number of inequivalent quadratic forms in the principal genus. Since each ${\frak f}(\tau, \chi)\in{\bf M}_1(\Gamma_0(D), \chi_{-D})$, this implies

Corollary 3.5

$ \sum\limits_{Q\in{\frak G}_k}\sum\limits_{m, n=-\infty}^{\infty}q^{Q(m, n)} \in{\bf M}_1(\Gamma_0(D), \chi_{-D}). $

The results in the next section will explain further the role played by the quadratic forms on the structure of the subspace of the Eisenstein series in $ {\bf M}_1(\Gamma_1(D))$.

4 A subspace of the Eisenstein series generated from the genus theory

Let $N$ be a positive integer. We will later choose $N=D$, where $-D < 0$ is the discriminant of an imaginary quadratic field.

It is established in [7 p. 137, Proposition 28] that

$ {\bf M}_k(\Gamma_1(N))=\bigoplus\limits_{\chi} \bf M_k(\Gamma_0(N), \chi), $

where the sum is over all Dirichlet characters $\chi$ modulo $N$ with $\chi(-1)=(-1)^k$.

Each $M_k(\Gamma_0(N), \chi)$ can be decomposed further as:

$ M_k(\Gamma_0(N), \chi)=\bf E_k(\Gamma_0(N), \chi)\oplus \bf S_k(\Gamma_0(N), \chi), $

where $\bf S_k(\Gamma_0(N), \chi)$ is the space of cusp forms and $\bf E_k(\Gamma_0(N), \chi)$ the space of the Eisenstein series.

Choose $k=1$. To describe the structure of $\bf E_1(\Gamma_0(N), \chi)$, we introduce $A_N$ to be the set of triplets $(\varphi, \psi, t)$ such that $\varphi$ and $\psi$ are primitive characters modulo $u$ and $v$ satisfying the condition $\varphi\psi(-1)=-1$ and $tuv|N$, and we define

$ {\frak f}(\tau;\varphi, \psi, t)={\frak f}(t\tau;\varphi, \psi). $

The structure of $\bf E_1(\Gamma_0(N), \chi)$ is given as below (cf. [7, p. 141, Theorem 4.8.1]):

Let $N$ be a positive integer. For any character $\chi$ modulo $N$, the set

$ \{{\frak f}(\tau;\varphi, \psi, t): (\varphi, \psi, t)\in A_N, \chi=\varphi\psi\} $

forms a basis of $\bf E_1(\Gamma_0(N), \chi)$.

Decompose $\bf E_1(\Gamma_0(N), \chi)$ further into

$ \bf E_1(\Gamma_0(N), \chi)=\bf E_1^{\rm old}(\Gamma_0(N), \chi)\oplus\bf E_1^{\rm new}(\Gamma_0(N), \chi), $

where $\bf E_1^{\rm new}(\Gamma_0(N), \chi)$ is the vector space generated by the span of ${\frak f}(\tau; \varphi, \psi)$ with $uv=N$ and $\varphi, \psi$ primitive modulo $u$ and $v$, respectively.

We now take $N=|D|$ and $\chi=\chi_D$. Note that if $N=D>0$, the discriminant of a real quadratic field, then $\chi_D(-1)=1$, since $\varphi\psi=\chi_D$, the requirement that $\varphi\psi(-1)= -1$ can not be met, thus $\bf E_1(\Gamma_0(D), \chi_D)=\{0\}$. On the other hand, if $-D < 0$ is the discriminant of an imaginary quadratic field, we have:

Theorem 4.1 Let $-D$ be the discriminant of an imaginary quadratic field. Suppose the unique factorization of $-D$ into the product prime discriminants contains $k$ factors. Then the set ${\frak f}(\tau; \chi_{d_1}, \chi_{d_2})$ with $(d_1, d_2)$ ranging over all $2^{k-1}$ pairs of discriminants satisfying $-D=d_1d_2$, described in Section 7, forms a basis for $\bf E_1(\Gamma_0(D), \chi_{-D})$.

Hence, the dimension of $ \bf E_1(\Gamma_0(D), \chi_{-D})$ is $2^{k-1}$.

To prove this result, we recall: A real character $\phi$ is primitive if and only if $\phi=\chi_D$ for some $D$ that is the discriminant of a quadratic field.

We now prove the theorem.

Proof Suppose $\varphi, \psi$ are, respectively, modulo $u$ and $v$ with $D=tuv$. Then, $\varphi$, $\psi$ are both modulo $uv$ and since $\chi_{-D}=\varphi\psi$, $\chi_{-D}$ is also a character modulo $uv$. Since $\chi_{-D}$ is primitive modulo $D$, $t=1$. Thus, $\bf E_1(\Gamma_0(D), \chi_{-D})=\bf E_1^{\rm new}(\Gamma_0(D), \chi_{-D})$.

To establish the conclusion, it suffices to show that if $\varphi$ and $\psi$ are primitive modulo $u$ and $v$ with $uv=-D$, then $u=d_1$, $v=d_2$ for some decomposition $-D=d_1d_2$ and $\varphi=\chi_{d_1}, \psi=\chi_{d_2}$.

We begin with the observation:

$ \varphi^2(n)\psi^2(n)=\chi_{_{-D}}^2(n)=1 $

for all $n$ with gcd$(n, D)=1$.

From this observation, we will deduce that both $\varphi$ and $\psi$ are real. This is the same as showing: $\varphi^2(n)=1$, gcd$ (n, d_1)=1$ and $\psi^2(n)=1$, gcd$ (n, d_2)=1$.

Consider first the case $-D\equiv 1$ mod 4. Since $-D$ is the product of odd prime discriminants, $-D=uv$ implies gcd$(u, v)=1$ and $u=d_1$, $v=d_2$.

Since $\psi$ is of modulo $d_2$, $\psi(n)=1$ if $n\equiv 1$ mod $d_2$. Thus, $ \varphi^2(n)=1 $ for all $n\equiv 1$ mod $d_2$ and gcd$(n, d_1)=1$; since $\varphi (n)=1$ for all $n\equiv 1$ mod $d_1$, together, we have $\varphi^2(n)=1$ if gcd$(n, d_1)=1$, $n\equiv 1$ mod $d_2$ or $n\equiv 1$ mod $d_1$. Since gcd$(d_1, d_2)=1$, every integer $m=m_1d_1+m_2d_2$ for some integers $m_1$ and $m_2$. In particular, for any integer $n$ such that gcd$(n, d_1)=1$, it can be expressed as

$ n=n_1d_1+n_2d_2+1. $

It is easy to check that gcd$(n_2d_2+1, d_1)=1$. Then

$ \varphi^2(n)=\varphi^2(n_1d_1+n_2d_2+1) =\varphi^2(n_2d_2+1)=1. $

This shows that $\varphi$ is a real character.

The identical argument also establishes that $\psi$ is real.

From the assumption that $\varphi$ and $\psi$ are primitive modulo $d_1$ and $d_2$, respectively, we conclude that

$ \varphi=\chi_{d_1}, \quad \psi=\chi_{d_2}. $

We now move on to the remaining case: $-D\equiv 0$ mod 4. Then

$ -D=p_1p_2\cdots p_k, $

where $p_1$ denotes the prime discriminant for the even prime 2: $p_1= -4$ or $\pm 8$. Examples are: $-40=(-8)5, -56=8(-7)$. The characters $\chi_{-4}(n)=\left(\frac{-4}{n}\right)$, $\chi_8(n)=\left(\frac{8}{n}\right)$, $\chi_{-8}(n)=\left(\frac{-8}{n}\right)$ are primitive modulo 4, 8 and 8, respectively.

There are two cases to consider.

Case 1. $-D= MN$ and gcd$(M, N)=1$. Then the same argument as presented above holds.

Case 2. gcd$(M, N)=2$.

We look at the sub-case $p_2=-4$ and show that it leads to contradiction.

Then, $\varphi$ and $\psi$ are characters modulo $2d_1$ and $2d_2$, respectively, with gcd$(d_1, d_2)=1$. The same reasoning leads to the conclusion: $\varphi^2(n)=1$ if gcd$(n, 2d_1)=1$, $n\equiv 1$ mod $2d_2$ or $n\equiv 1$ mod $2d_1$. Since gcd$(d_1, d_2)=1$, every even integer $2m=2m_1d_1+2m_2d_2$. In particular, every odd integer $n$ with gcd$(n, d_1)=1$ can be expressed as

$ n=2n_1d_1+2n_2d_2+1. $

Again, it can be easily verified that gcd$(n, 2n_2d_2+1)=1$; hence

$ \varphi^2(n)=\varphi^2(2n_1d_1+2n_2d_2+1) =\varphi^2(2n_2d_2+1)=1. $

This shows that $\varphi$ is a real character modulo $2d_1$.

Similarly, $\psi$ is also real.

Since $\varphi$ is primitive $2d_1$, this implies that $d=2d_1$ is the discriminant of a quadratic field and since it is even, $2d_1\equiv 0$ mod 4. However, it leads to contradiction, because $d_1$ is odd, $2d_1$ cannot be congruent to 0 mod 4.

We now examine the case $p_2=\pm 8$. Assume, with no loss of generality, $\varphi$ and $\psi$ are modulo $2d_1$, $4d_2$, respectively, and gcd$(d_1, d_2)=1$.

Then, as before, we deduce that $\varphi^2(n)=1$ if $(n, d_1)=1$, $n\equiv 1$ mod $4d_2$ or $n\equiv 1$ mod $2d_1$. Using the fact that every odd integer $n$ with gcd$(n, d_1)=1$ can be expressed as

$ n=2n_1d_1+4n_2d_2+1, $

we conclude that $\varphi$ is real. This, again, leads to contradiction using the identical reasoning as in Case 1.

We remark that without modification the above proof shows:

Let $D$ be a discriminant of a quadratic field. Then $\chi_D$ can be expressed as a product of a pair of primitive characters: $\chi=\varphi\psi$ if and only if $\varphi=\chi_{d_1}$ and $\psi=\chi_{d_2}$ for some discriminants $d_1$, $d_2$ with $D=d_1d_2$.

The reason we confine to the case of negative discriminants is because, as mentioned earlier, $\bf E_1(\Gamma_0(D), \chi_D)=\{0\}$ for $D>0$.

We also have, from (3.2) and Corollary 3.4, the following result.

Corollary 4.2 For each ${\frak G}_k$, let

$ {\frak q}_k(\tau)=\sum\limits_{Q\in{\frak G}_k}\sum\limits_{m, n=-\infty}^{\infty}q^{Q(m, n)}. $

Then

$ \{{\frak q}_k(\tau): k= 1, 2, \cdots, 2^{k-1}\} $

forms a basis of $\bf E_1(\Gamma_0(D), \chi_{-D})$.

This clarifies the role of weight one modular forms generated from the genus theory of the imaginary quadratic fields in the scheme of weight one modular forms.

5 Genus fields

To acquire a more complete picture of what has been developed so far, it seems fitting to bring out the relevance of the Eisenstein series ${\frak f}(\tau; \chi_{d_1}, \chi_{d_2})$, encountered earlier in Theorem 3.1, with the genus field associated with an imaginary quadratic field. For this, we need to recall from Lemma 2.1 that a given discriminant $D$ can be factored uniquely, modulo the permutations, into a product of prime discriminants.

(1) $D=P_1P_2\cdots P_k$, if $D$ is odd, where $P_i, i=1, 2, \cdots, k$, are odd prime discriminants;

(2) $D=\pm 8 P_2\cdots P_k$, if $D/4\equiv 2$ mod 4, where $P_i, i=2, \cdots, k$, are odd prime discriminants;

(3) $D=\pm 4 P_2\cdots P_k$, if $D/4\equiv 3$ mod 4, where $P_i, i=2, \cdots, k$, are odd prime discriminants.

We define the genus field $\mathbb K$ of the quadratic field $\mathbb F:=\mathbb Q(\sqrt D)$ as

$ \mathbb K:=\mathbb Q(\sqrt D, \sqrt P_2, \cdots, \sqrt P_k). $

We refer readers to [8, p. 121, Theorem 6.1] for properties of the genus field.

Let $\zeta_{\mathbb K}(s)$ be the Dedekind zeta function for the genus field $\mathbb K$

$ \zeta_{\mathbb K}(s):=\sum\limits_{{\frak a}}\frac{1}{N^s{\frak a}}, $

where ${\frak a}$ are the non-zero integral ideals in $\mathbb K$.

For each $\chi\in{\frak G}^*$ associated with the decomposition $D=d_1d_2$, define

$ L_{\mathbb F}(s, \chi):=L(s, \chi_{d_1})L(s, \chi_{d_2}). $

Then, from the properties of the Hilbert class field theory (cf. [9, p. 70]), we derive the following identity:

$ \zeta_{\mathbb K}(s)=\prod\limits_{\chi\in{\frak G}^*}L_{\mathbb F}(s, \chi). $

All the above facts are valid for any quadratic field. We now focus on imaginary quadratic fields. Let $-D$ be the discriminant of an imaginary quadratic field. The Mellin transform of the Eisenstein series ${\frak f}(\tau; \chi)$ yields

$ M(s, {\frak f}(\tau;\chi) )=L(s, \chi_{d_1})L(s, \chi_{d_2})=L_{\mathbb F}(s, \chi). $

The relevance of the space $\bf E_1(\Gamma_0(D), \chi_{-D})$ and the genus field $\mathbb K$ of the imaginary quadratic field $\mathbb F$ is neatly summarized as the following identity:

Theorem 5.1

$ \zeta_{\mathbb K}(s)=\prod\limits_{\chi\in{\frak G}^*}M(s, {\frak f}(\tau;\chi) ). $

6 Comments

We recall

$ {\frak f}(\tau;\varphi, \psi): =\varphi(0)c_0(\psi)+2\sum\limits_{m, n=1}^{\infty}\varphi(m)\psi(n)q^{mn}. $

Let $\chi$ be a Dirichlet character and let ${\frak f}_{\chi}(\tau; \varphi, \psi)$ denote the twist of ${\frak f}(\tau; \varphi, \psi)$ by $\chi$:

$ {\frak f}_{\chi}(\tau;\varphi, \psi): =\chi(0)\varphi(0)c_0(\psi)+2\sum\limits_{m, n=1}^{\infty}\chi(mn)\varphi(m)\psi(n)q^{mn}. $

We remind our readers that the symbol $\chi$ here denotes a Dirichlet character (not the genus character).

We begin by stating a result which follows readily from Theorem 1.1 and Proposition 17(b) of [9, p. 127].

Theorem 6.1 Suppose $\varphi$ and $\psi$ are primitive characters modulo $M$ and $N$, respectively, $\varphi\psi(-1)=-1$ and $\chi$ is a primitive character modulo $D$. Then ${\frak f}_{\chi}(\tau; \varphi, \psi)\in {\bf{M}}_1(\Gamma_0(MND^2), \chi^2\varphi\psi)$.

Here we do not require gcd$(D, MN)=1$.

In particular, we have:

Corollary 6.2 If $\varphi$ is primitive modulo $M$ and $-D < 0 $ is the discriminant of an imaginary quadratic field, then

$ {\frak f}_{\varphi}(\tau;\chi_1, \chi_{-D})\in {\bf{M}}_1(\Gamma_0(M^2D), \varphi^2\chi_{-D}). $

We now examine the quantity ${\frak f}_{\varphi}(\tau; \chi_1, \chi_{D})$ more closely.

Let $\mathbb F$ be a quadratic field with the discriminant $D$. Let $\varphi$ be a character modulo $M$ and

$ L_{\mathbb F}(s, \varphi\circ N)=\sum\limits_{{\frak a}}\frac{\varphi(N({\frak a}))}{N^s({\frak a})}. $

Theorem 6.3

$ L_{\mathbb F}(s, \varphi\circ N)=L(s, \varphi)L(s, \varphi\chi_{D}), $

where $\chi_D(n)=\big(\frac{D}{n}\big)$.

This identity holds regardless whether $\varphi$ is primitive or not.

Proof As before, ${\frak p}$ denotes a prime ideal of $\mathbb F$ and $p$ is the prime in the rational field $\mathbb Q$ such that $p=N{\frak p}$.

Since both $\varphi$ and $N$ are multiplicative, we have:

$ \begin{align*} \sum\limits_{{\frak a}}\frac{\varphi(N{\frak a})}{N^s({\frak a})} =\, &\prod\limits_{{\frak p}}\Big(1-\frac{\varphi(N{\frak p})}{N^s \frak p}\Big)^{-1} \\ =\, &\prod\limits_{\chi_D(p)=1}\Big(1-\frac{\varphi(N{\frak p})}{N^s {\frak p}}\Big)^{-1}\prod\limits_{\chi_D(p)=-1} \Big(1-\frac{\varphi(N{\frak p})}{N^s {\frak p}}\Big)^{-1}\prod\limits_{\chi_D(p)=0} \Big(1-\frac{\varphi(N{\frak p})}{N^s {\frak p}}\Big)^{-1}\\ \nonumber =\, &\prod\limits_{\chi_D(p)=1}\Big(1-\frac{\varphi(p)}{p^s}\Big)^{-2} \prod\limits_{\chi_D(p)=-1}\Big(1-\frac{\varphi^2(p)}{p^{2s}}\Big)^{-1}\prod\limits_{\chi_D(p)=0}\Big(1-\frac{\varphi(p)}{p^s}\Big)^{-1} \\ =\, &\prod\limits_{\chi_D(p)=1} \Big(1-\frac{\varphi(p)\chi_D(p)}{p^s }\Big)^{-1} \prod\limits_{\chi_D(p)=-1} \Big(1-\frac{\varphi(p)\chi_D(p)}{p^{s}}\Big)^{-1}\\ &\times \prod\limits_{\chi_D(p)=1} \Big(1-\frac{\varphi(p)}{p^{s}}\Big)^{-1} \prod\limits_{\chi_D(p)=-1} \Big(1-\frac{\varphi(p)}{p^{s}}\Big)^{-1}\prod\limits_{\chi_D(p)=0}\Big(1-\frac{\varphi(p)}{p^s}\Big)^{-1}\\ =\, &L(s, \varphi\chi_{D})L(s, \varphi). \end{align*} $

Since we can re-write the above identity as

$ \sum\limits_{{\frak a}}\frac{\varphi(N({\frak a}))}{N^s({\frak a})} =\sum\limits_{m, n=1}^{\infty}\frac{\varphi(mn)\chi_D(n)}{(mn)^s}, $

the inverse Mellin transform yields

$ \begin{align} \sum\limits_{{\frak a}}\varphi(N({\frak a}))q^{N(\frak a)}&=\sum\limits_{m, n=1}^{\infty}\varphi(mn)\chi_D(n)q^{mn} \nonumber\\ &=\sum\limits_{n=1}^{\infty}q^n\sum\limits_{k|n}\varphi(k)\chi_D(n/k). \end{align} $

(6.1)

Thus,

$ \begin{equation} {\frak f}_{\varphi}(\tau;\chi_1, \chi_D)=\varphi(0)c_0(\chi_D)+2\sum\limits_{{\frak a}}\varphi(N({\frak a}))q^{N({\frak a})}. \end{equation} $

(6.2)

In particular, if $-D$ is the discriminant of an imaginary quadratic field, then

$ {\frak f}_{\varphi}(\tau;\chi_1, \chi_{-D})=\frac{2h}{w}\varphi(0)+2\sum\limits_{{\frak a}}\varphi(N({\frak a}))q^{N({\frak a})}. $

Let $\varphi_i, i=1, 2, \cdots, \phi(M)$, be the complete set of characters for the reduced residue system $\mathbb Z_M^*$, where $\phi$ denotes Euler's totient and $I_k(M)=\{{\frak a}:N({\frak a})\equiv k (\text{mod} M)\}$.

We deduce the following theorem.

Theorem 6.4 Suppose gcd$(k, M)=1$, then

$ \sum\limits_{{\frak a}\in I_k(M)} q^{N({\frak a})} =\sum\limits_{j=1}^{\infty}\chi_D(j)\frac{q^{jk}}{1-q^{jM}}. $

Proof We note that the identity holds for $M=1$.

Assume $M>1$, then $\phi(0)=0$ and, from (6.1) and (6.2),

$ \frac12{\frak f}_{\varphi_i}(\tau;\chi_1, \chi_D) =\sum\limits_{k\in\mathbb Z^*_M}\varphi_i(k)\sum\limits_{{\frak a}\in I_k(M)} q^{N({\frak a})}=\sum\limits_{n=1}^{\infty}q^n\sum\limits_{k|n}\varphi_i(k)\chi_D(n/k). $

From (6.1) and the orthogonal relation of $\varphi_i, i=1, 2, \cdots, \phi(M)$,

$ \begin{align} \sum\limits_{{\frak a}\in I_k(M)} q^{N(\frak a)}&=\frac{1}{2\phi(M)}\sum\limits_{i=1}^{\phi(M)}\overline{\varphi(k)} {\frak f}_{\varphi_i}(\tau;\chi_1, \chi_D) \nonumber\\ &=\frac{1}{\phi(M)}\sum\limits_{n=1}^{\infty} \Big(\sum\limits_{m|n}\sum\limits_{i=1}^{M-1}\varphi_i(m)\overline{\varphi_i(k)}\chi_D(n/m)\Big)q^n. \end{align} $

(6.3)

We note

$ \sum\limits_{i=1}^{M-1}\varphi_i(m)\overline{\varphi_i(k)}=\begin{cases} \phi(M), \quad&\text{if} \ \ m\equiv k~(\text{mod} M), \\[0.1in] 0, \quad&\text{otherwise.} \ \ \end{cases} $

Now let $m=lM+k$ and $n=jm$, where $l$ ranges over all non-negative integers and $j$ all positive integers. Then

$ \begin{align} \sum\limits_{{\frak a}\in I_k(M)} q^{N({\frak a})} &=\sum\limits_{l=0}^{\infty}\sum\limits_{j=1}^{\infty}\chi_D(j) q^{j(lM+k)} \nonumber \\ &=\sum\limits_{j=1}^{\infty}\chi_D(j)q^{jk}\sum\limits_{l=1}^{\infty}q^{jlM} \nonumber \\ &=\sum\limits_{j=1}^{\infty}\chi_D(j)\frac{q^{jk}}{1-q^{jM}}. \end{align} $

(6.4)

This is a Lambert series for a generating function counting the number of integral ideals with their norms congruent to $k$ mod $M$. It is valid for any quadratic field. For an imaginary quadratic field and $M=1$, it reduces to the identity (1.3).

Corollary 6.5 Let $-D < 0$ be the discriminant of a quadratic field. Suppose $p\ge 3$ is prime; then $\sum_{{\frak a}\in I_k(p)} q^{N({\frak a})}\in M_1(\Gamma_1(p^2D))$ for each $k = 1,2, \cdots ,p - 1$.

We note that every $\varphi_i, i=1, 2, \cdots, p-1$, is primitive modulo $p$. From Corollary 6.2, ${\frak f}_{\varphi_i}(\tau; \chi_1, \chi_{-D})\in{\bf M}_1(\Gamma_0(p^2D), \varphi_i^2\chi_{-D})$ and, in turn, implies that ${\frak f}_{\varphi_i}(\tau; \chi_1, \chi_{-D})\in{\bf M}_1(\Gamma_1(p^2D))$ for each $i$. From (6.3), we see that $\sum_{{\frak a}\in I_k(p)} q^{N({\frak a})}\in M_1(\Gamma_1(p^2D))$ for each $k=1, 2, \cdots, p-1$.

Acknowledgements The work constitutes part of a series of lectures the author delivered at Luo Yang Normal University and East China Normal University on June 16 and 26, 2017. The author would like to thank Professors Hong-cun Zhai and Zhi-guo Liu for their support and hospitality. A special gratitude goes to Dan Dan Chen for her assistance in reformatting the original manuscript according to the style of the journal.