The Space of Binary Theta Series

Ernst Kani

1 Introduction

The purpose of this paper is to study the space Θ_{D} generated by the binary theta series ϑ_{f} attached to the primitive, positive-definite binary quadratic forms f(x, y) = ax^{2} + bxy + cy^{2} of discriminant D = b^{2} − 4ac < 0. It is curious that while much has been written about the space generated by theta series attached to quadratic forms in 2k ≥ 4 variables (cf. [10] and the references therein), the binary case does not seem to have been treated in detail in the literature.

By the work of Weber[25], Hecke[8] and Schoeneberg[18], it is known that Θ_{D} is a subspace of the space M_{1}(|D|, ψ_{D}) of modular forms of weight 1, level |D| and Nebentypus ψ_{D}, where ψ_{D} = (\frac{D}{·}) is the Kronecker-Legendre character. Contrary to the case of higher weight, the space of binary theta series is often a proper subspace of M_{1}(|D|, ψ_{D}) (cf. Remark 16) and so it is of interest to be able to identify it inside the space of modular forms.

As a first step towards this, we explain in this paper how each theta series ϑ_{f} can be expressed as a linear combination of the canonical (extended) Atkin-Lehner basis of M_{1}(|D|, ψ_{D}); cf. Remark 30(b). This will be used in the next paper[12] to give an intrinsic description of the space of theta series.

To this end we first observe that Θ_{D} has a natural basis {ϑ_{χ}} indexed by the characters χ ∈ Cl(D)^{∗} of the class group Cl(D) of forms of discriminant D. It turns out that each ϑ_{χ} is a normalized eigenform with respect to Hecke algebra 𝕋(D); the latter is the algebra generated by all Hecke operators T_{n} with (n, D) = 1.

Theorem 1 The space Θ_{D} is a 𝕋(D)-submodule of M_{1}(|D|, ψ_{D}) of multiplicity one, and has a canonical basis {ϑ_{χ}} consisting of normalized 𝕋(D)-eigenforms. Furthermore, ϑ_{χ} is a cusp form if and only if χ is not a quadratic character.

This theorem implies the following interesting result.

Theorem 2 We have Θ_{D} = Θ_{D}^{E} ⊕ Θ_{D}^{S}, where Θ_{D}^{E} = Θ_{D} ∩ E_{1}(|D|, ψ_{D}) denotes the Eisenstein space part and Θ_{D}^{S} = Θ_{D} ∩ S_{1}(|D|, ψ) denotes the cusp space part of Θ_{D}. Moreover,

(1) dim Θ_{D}^{E} = g_{D} and dim Θ_{D}^{S} = \frac{1}{2}(h_{D} − g_{D}), where h_{D} = |Cl(D)| denotes the number of classes of forms of discriminant D, and g_{D} = [Cl(D) : Cl(D)^{2}] denotes the number of genera.

Note that it follows from (1) that Θ_{D} has no non-zero cusp forms if and only if h_{D} = g_{D}, i.e. if and only if D is an idoneal discriminant. As is explained in Remark 17(b) below, this can be viewed as an alternate version of Theorem 3 of Kitaoka[13].

If D is a fundamental discriminant, i.e. if D = d_{K}, where d_{K} is the discriminant of K := ℚ(\sqrt{D}), then each ϑ_{χ} is a primitive form (newform) (cf. Remark 16), and hence in this case the ϑ_{χ}’s are part of the canonical Atkin-Lehner basis. However, in the general case this is no longer true for every χ ∈ Cl(D)^{∗} because some of the characters χ ∈ Cl(D)^{∗} are not primitive, i.e. they are lifts χ = χ′ ∘ π of characters χ′ ∈ Cl(D′)^{∗} of some “lower level” D′|D via the canonical map π = π_{D,D′} : Cl(D) → Cl(D′). In the general case the situation is as follows.

Theorem 3 Let χ be a character on the class group Cl(D), where D = f_{D}^{2}d_{K}.

(a) There is a unique divisor f_{χ}|f_{D} and a unique primitive character χ_{pr} on the class group Cl(D_{χ}), where D_{χ} = f_{χ}^{2}d_{K}, such that χ = χ_{pr} ∘ π̄_{D,D_{χ}}.

(b) The form ϑ_{χ_{pr}} ∈ Θ_{D_{χ}} is a primitive form of level |D_{χ}| whose L-function is the Hecke L-function associated to a suitable Hecke character χ̃_{pr}, i.e. L(s, ϑ_{χ_{pr}}) = L(s, χ̃_{pr}). Moreover, there exist constants c_{n}(χ) ∈ ℝ such that

ϑ_{χ}(z) = ∑_{n|f̄_{χ}^{2}} c_{n}(χ)ϑ_{χ_{pr}}(nz), (2)

where f̄_{χ} = f_{D}/f_{χ}. Furthermore, the function n ↦ c_{n}(χ) is multiplicative and has the generating function

C(s, χ) := ∑_{n|f̄_{χ}^{2}} c_{n}(χ)n^{−s} = L(s, ϑ_{χ})/L(s, ϑ_{χ_{pr}}) = L(s, ϑ_{χ})/L(s, χ̃_{pr}). (3)

Note that while L(s, ϑ_{χ_{pr}}) = L(s, χ̃_{pr}) is a classical Hecke L-function and hence is well-understood, the L-function L(s, ϑ_{χ}) is more complicated and is, in fact, unknown. Thus, (3) does not help in determining the constants c_{n}(χ). Instead, we calculate C(s, χ) directly in Theorem 28 by using facts about ideals in quadratic orders which are presented in the Appendix (§6). As a consequence, we thus obtain an explicit expression for L-function L(s, χ); cf. Corollary 31. In particular, we obtain

Corollary 4 If pē_{p}||f̄_{χ} denotes the largest power of p dividing f̄_{χ}, then the p-Euler factor L_{p}(s, χ) of L(s, χ) at p|f̄_{χ} is

L_{p}(s, χ) = \frac{1 − p^{(1−2s)ē_{p}}}{1 − p^{1−2s}} + \frac{(1 − \frac{1}{p}ψ_{D_{χ}}(p)) p^{(1−2s)ē_{p}}}{1 − a_{p}(χ_{pr})p^{−s} + ψ_{D_{χ}}(p)p^{−2s}} .

At the end of §5, we present some special cases and numerical examples of these results; cf. Examples 35 and 37.

After the first version of this paper was completed, Norm Hurt drew my attention to the papers of Sun and Williams [22], [23] which are partially related to some of the topics mentioned here. Indeed, Corollaries 31 and 4 can be viewed (for D < 0) as a generalization of many of their results; cf. Remark 32(c) for more details.

2 The basis {ϑ_{χ}} of Θ_{D}

As in the introduction, let f(x, y) = ax^{2} + bxy + cy^{2} be a primitive, positive definite binary quadratic form of discriminant D = b^{2} − 4ac < 0. Thus a, b, c ∈ ℤ and (a, b, c) = 1, and D = f_{D}^{2}d_{K}, where d_{K} is the (fundamental) discriminant of the imaginary quadratic field K = ℚ(\sqrt{D}). For a fixed D, let Q_{D} denote the set of all such forms. The binary theta series attached to f ∈ Q_{D} is the function on the upper half-plane ℌ given by

ϑ_{f}(z) = ∑_{x,y∈ℤ}e^{2πif(x,y)z} = ∑_{n=0}^{∞} r_{n}(f)e^{2πinz},

where r_{n}(f) = #{(x, y) ∈ ℤ^{2} : f(x, y) = n} denotes the number of representations of n by f. The following fact is fundamental for most of this paper.

Proposition 5 If f ∈ Q_{D}, then ϑ_{f} is a modular form of level |D| and Nebentypus ψ_{D} = (\frac{D}{·}). Thus Θ_{D} := ∑_{f∈Q_{D}} ℂϑ_{f} is a subspace of M_{1}(|D|, ψ_{D}).

Proof. See Schoeneberg[19], ch. IX, Theorems 4 and 5, or Miyake[16], Corollary 4.9.5(3).

Remark 6 (a) This result was proved by Weber[25] in 1893 (see his formula §15 (21)), except that he did not verify that ϑ_{f} is holomorphic at all the cusps. In 1926 Hecke[8] proved a more general result from which above result can be deduced (cf. Remark 11(c)). Siegel[21] proved something weaker but for quadratic forms in an arbitrary number m ≥ 2 of variables; cf. [21], Hilfssatz 30 and 31 and also Satz 4, where the case m = 2 is excluded. In 1939 Schoeneberg[18] proved a very general result about theta series which includes the above result as a special case.

(b) The L-function associated to the modular form ϑ_{f} is

Z_{f}(s) = L(s, ϑ_{f}) = ∑_{n≥1} \frac{r_{n}(f)}{n^{s}} = ∑_{(x,y)≠(0,0)} f(x, y)^{−s}, where Re(s) > 1.

This function is often called the Epstein zeta-function of f (cf. e.g. [4]), even though it was introduced by Dirichlet in 1839 (cf. [5], §6.18 (p. 358)) and was studied intensively by him and by Kronecker[14] (and by others) many years before Epstein.

Recall that the group GL_{2}(ℤ) acts on binary quadratic forms by change of coordinates, and that this action preserves the set Q_{D}. It is immediate that r_{n}(fT) = r_{n}(f), for all n ≥ 0 and T ∈ GL_{2}(ℤ), so ϑ_{fT} = ϑ_{f}. We can thus index the theta series by the quotient set

\overline{Q}_{D} = Q_{D}/GL_{2}(ℤ) = Q_{D}/≈,

where ≈ denotes the equivalence relation induced by GL_{2}(ℤ)-equivalence.

Proposition 7 The set \{ϑ_{f} : f ∈ \overline{Q}_{D}\} is a basis of the space Θ_{D}. Thus

dim Θ_{D} = \overline{h}_{D} := #\overline{Q}_{D}.

As we shall see, this follows easily from the following basic facts about binary quadratic forms which are due to Dirichlet[5] and Weber[24].

Lemma 8 Let f, f_{1}, f_{2} ∈ Q_{D} be primitive quadratic forms. Then:

(a) There exist infinitely many prime numbers p with r_{p}(f) > 0.

(b) If there is a prime number p ∤ D with r_{p}(f_{i}) > 0, for i = 1, 2, then f_{1} ≈ f_{2}.

Proof. (a) This was first proved by Dirichlet[5] for prime discriminants and by Weber[24] in the general case. A proof using class field theory is given in Cox[3], Theorem 9.12.

(b) Weber[25] states this (elementary) result on p. 259 and points out that it follows easily from his paper [24] (or from a paper of Schering). It is restated and proved (without references) as Satz 1 in Piehler[17].

Proof of Proposition 7. Let f_{1}, …, f_{N} ∈ Q_{D} be a system of representatives of \overline{Q}_{D}; thus, N = \overline{h}_{D}. It is clear from what was said above that ϑ_{f_{1}}, …, ϑ_{f_{N}} generate Θ_{D}. To prove that they are linearly independent, suppose that c_{1}ϑ_{f_{1}} + … + c_{N}ϑ_{f_{N}} = 0, for some c_{1}, …, c_{N} ∈ ℂ. Thus c_{1}r_{n}(f_{1}) + … + c_{N}r_{n}(f_{N}) = 0, for all n ≥ 0.

Fix i. By Lemma 8(a) we know that there is a prime p = p_{i} ∤ D with r_{p}(f_{i}) > 0, and from Lemma 8(b) it follows that r_{p}(f_{j}) = 0 for all j ≠ i. We thus have that c_{i}r_{p}(f_{i}) = 0, so c_{i} = 0, and hence {ϑ_{f_{i}}} is a basis of Θ_{D}, as claimed.

We now introduce another basis of the space Θ_{D}. For this, recall that by Gauss’s theory of composition of forms (cf. Gauss[6]) the set

Cl(D) := Q_{D}/SL_{2}(ℤ) = Q_{D}/∼

has the structure of an abelian group. The identity of Cl(D) is the class of the principal form 1_{D} which is defined by 1_{D}(x, y) = x^{2} + εxy + \frac{ε−D}{4}y^{2}, for D ≡ ε (mod 4) and ε ∈ {0, 1}. Note also that the SL_{2}(ℤ)-equivalence relation ∼ is related to the previous GL_{2}(ℤ)-equivalence relation ≈ by

f_{1} ≈ f_{2} ⇔ f_{1} ∼ f_{2} or f_{1} ∼ f_{2}^{−1}, (4)

where f_{2}^{−1} denotes a representative of the inverse class of f_{2} (in the group Cl(D)). From this it follows easily that

(5) \overline{h}_{D} = \frac{1}{2}(g_{D} + h_{D}), where h_{D} = |Cl(D)| and g_{D} = [Cl(D) : Cl(D)^{2}].

Let χ ∈ Cl(D)^{∗} := Hom(Cl(D), ℂ^{×}) be a character on Cl(D), and put

ϑ_{χ} := \frac{1}{w_{D}} ∑_{f∈Cl(D)} χ(f)ϑ_{f} ∈ Θ_{D}, (6)

where w_{D} = r_{1}(1_{D}). (Thus, as is well known, w_{D} = 2 when D < −4 and w_{−4} = 4 and w_{−3} = 6.) Note that the terms on the right hand side are not linearly independent since we sum over Cl(D) in place of \overline{Q}_{D}. However, by using (4) and noting that χ(f^{−1}) = \overline{χ(f)}, we can re-write (6) in the form

ϑ_{χ} = \frac{1}{w_{D}} ∑_{f̄∈\overline{Q}_{D}} w(f̄)Re(χ(f̄))ϑf̄, (7)

where w(f̄) = #{f ∈ Cl(D) : f ≈ f̄}. (Thus, w(f) = 1, if f^{2} ∼ 1_{D} and w(f) = 2 otherwise.) From this expression we see immediately that ϑ_{χ^{−1}} = ϑ_{χ}, and so it is useful to index the ϑ_{χ}’s by the set \overline{Cl(D)}^{∗} = Cl(D)^{∗}/(χ ↦ χ^{−1}). We then have:

Proposition 9 If f ∈ Q_{D} is a primitive quadratic form, then

ϑ_{f} = \frac{w_{D}}{h_{D}} ∑_{χ∈Cl(D)^{∗}} \overline{χ}(f)ϑ_{χ} = \frac{w_{D}}{h_{D}} ∑_{χ∈\overline{Cl(D)}^{∗}} w(χ)Re(χ(f))ϑ_{χ}, (8)

where w(χ) = #\{χ, χ^{−1}\}. Thus \{ϑ_{χ} : χ ∈ \overline{Cl(D)}^{∗}\} is also a basis of Θ_{D}.

Proof. The second identity of (8) follows immediately from the fact that ϑ_{χ^{−1}} = ϑ_{χ}. To prove the first identity, recall that the orthogonality relations for group characters imply that

∑_{χ∈Cl(D)^{∗}} χ(f) = \{ \begin{matrix}h_{D} & if f ∼ 1_{D}, \\ 0 & otherwise. \end{matrix}

From this, together with the definition (6) of ϑ_{χ}, it follows that

w_{D} ∑_{χ∈Cl(D)^{∗}} \overline{χ}(f)ϑ_{χ} = ∑_{χ∈Cl(D)^{∗}} \overline{χ}(f) ∑_{f_{1}∈Cl(D)} χ(f_{1})ϑ_{f}
= ∑_{χ∈Cl(D)^{∗}} χ(f^{−1}) ∑_{f_{1}∈Cl(D)} χ(f_{1})ϑ_{f_{1}}
= ∑_{f_{1}∈Cl(D)} ϑ_{f_{1}} ∑_{χ∈Cl(D)^{∗}} χ(f^{−1}f_{1}) = h_{D}ϑ_{f},

which proves the first identity of (8) and hence (8) itself.

From (8) we see that the set \{ϑ_{χ} : χ ∈ \overline{Cl(D)}^{∗}\} generates Θ_{D}. Since Cl(D) ≃ Cl(D)^{∗}, it follows that #(\overline{Cl(D)}^{∗}) = #(Cl(D)/(f ↦ f^{−1})) = #\overline{Q}_{D} = dim Θ_{D}, where the last two identities follow from (4) and Proposition 7, respectively. Thus, the set \{ϑ_{χ} : χ ∈ \overline{Cl(D)}^{∗}\} is a basis of Θ_{D}.

We now examine the Fourier coefficients a_{n}(χ) := a_{n}(ϑ_{χ}) of the modular form ϑ_{χ} more closely. For this we shall use a basic result due to Dedekind that the class group Cl(D) can be identified with the group Cl(𝔒_{D}) = Pic(𝔒_{D}) = I(𝔒_{D})/P (𝔒_{D}) of classes of invertible (fractional) ideals of the order 𝔒_{D} of discriminant D in K = ℚ(\sqrt{D}). (Here we use the terminology and notation of Appendix §6.1.) More precisely, if L(f) = aℤ + \frac{−b+\sqrt{D}}{2}ℤ denotes the quadratic lattice in K associated to the form f(x, y) = ax^{2} + bxy + cy^{2} ∈ Q_{D}, then the rule f ↦ L(f) defines an isomorphism

λ_{D} : Cl(D) →^{∼} Cl(𝔒_{D}) := I(𝔒_{D})/P (𝔒_{D});

cf. Cox[3], Theorem 7.7 (p. 137). We observe:

Proposition 10 Let χ ∈ Cl(D)^{∗}. If n ≥ 1, then the n-th Fourier coefficient of ϑ_{χ} is given by the formula

a_{n}(χ) = ∑_{𝔞∈Id_{n}(𝔒_{D})} χ^{∗}(𝔞), (9)

where Id_{n}(𝔒_{D}) denotes the set of invertible ideals of 𝔒_{D} of norm n and χ^{∗} = χ∘λ_{D}^{−1} ∈ Cl(𝔒_{D})^{∗}. Moreover, a_{0}(χ) = 0 except when χ = 1 is trivial, and then a_{0}(1) = \frac{h_{D}}{w_{D}}.

Proof. Recall that if f ∈ Q_{D}, then f(x, y) = ax^{2} + bxy + cy^{2} = N_{K}(ax − β_{f}y)/N(L), where β_{f} = \frac{−b+\sqrt{D}}{2}, N_{K} denotes the field norm and N(L) = a is the norm of the lattice L := L(f); cf. [3], p. 137. We thus obtain (in the notation of the appendix) that

ϑ_{f}(z) = ∑_{x,y∈ℤ}e^{2πif(x,y)z} = ∑_{α∈L(f)} e^{2πiN_{K}(α)z/N(L(f))} = 1 + w_{D} ∑_{{𝔞 ∈ Id(𝔒_{D}) \atop 𝔞L(f) ∈ P (𝔒_{D})}}e^{2πiN(𝔞)z}, (10)

where the last equality follows from the fact that 𝔞 := αL(f)^{−1} is an integral 𝔒_{D}-ideal if and only if α ∈ L(f) (where α ∈ K^{×}), together with the fact that |O_{D}^{×}| = w_{D}. From this it thus follows from the definition that

ϑ_{χ}(z) = \frac{1}{w_{D}} ∑_{f∈Cl(D)} χ(f) (1 + w_{D} ∑_{{𝔞 ∈ Id(𝔒_{D}) \atop 𝔞L(f) ∈ P (𝔒_{D})}}e^{2πiN(𝔞)z}) = c_{χ} + ∑_{𝔞∈Id(𝔒_{D})} χ^{∗}(𝔞)e^{2πiN(𝔞)z},

where c_{χ} = \frac{1}{w_{D}} ∑_{f∈Cl(D)} χ(f). This proves (9). Note that c_{χ} = 0 except when χ = 1; in the latter case clearly c_{χ} = \frac{h_{D}}{w_{D}}.

Remark 11 (a) The above result shows that the L-function associated to ϑ_{χ} is

L(s, ϑ_{χ}) = ∑_{n≥1} \frac{a_{n}(χ)}{n^{s}} = ∑_{𝔞∈Id(𝔒_{D})} \frac{χ^{∗}(𝔞)}{N(𝔞)^{s}} ,

where χ^{∗} = χ ∘ λ_{D}^{−1}. Thus, L(s, ϑ_{χ}) coincides with the L-function L_{𝔒_{D}}(s, χ^{∗}) defined by the first equation on the bottom p. 280 of the first edition of [15].

(b) For later reference we note that for the trivial character χ = 1 the equations (6) and (9) yield that

a_{n}(1) = \frac{1}{w_{D}} ∑_{f∈Cl(D)} r_{n}(f) = #Id_{n}(𝔒_{D}). (11)

This number will be computed computed below (cf. Remark 32(b)) and is well-known when (n, f_{D}) = 1; cf. (60).

(c) Note that it follows from the first part of (10) that if f ∈ Q_{D}, then

ϑ_{f}(z) = ∑_{i=1}^{f_{D}} ϑ(f_{D}z; α_{i}\sqrt{d_{K}}, 𝔞_{f}, f_{D}\sqrt{d_{K}}),

where ϑ(z; ρ, 𝔞, Q\sqrt{d_{K}}) = ∑_{μ≡ρ(𝔞Q\sqrt{d_{K}})} e^{2πizN_{K}(μ)/(N(𝔞)Q|d_{K}|)} is as in Hecke[8], 𝔞_{f} = L(f)O_{K}, and {α_{i}} is a system of coset representatives of L(f)/f_{D}𝔞_{f}. Thus, Θ_{D} is a subspace of the space generated by dilations of Hecke’s theta-functions.

By using results about ideals in the ring 𝔒_{D} (cf. Appendix, §6.2), we obtain the following important result.

Theorem 12 If χ ∈ Cl(D)^{∗}, then the function n ↦ a_{n}(χ) is multiplicative. Thus ϑ_{χ} is a normalized eigenfunction with eigenvalue a_{n}(χ) with respect to the Hecke operator T_{n} whenever (n, f_{D}) = 1.

Proof. Put χ^{∗} = χ ∘ λ_{D}^{−1} ∈ Cl(𝔒_{D})^{∗}, and write Id_{n} = Id_{n}(𝔒_{D}). If (m, n) = 1, then from (9) together with Proposition 44 of §6.2 we obtain that

a_{mn}(χ) = ∑_{𝔞∈Id_{mn}} χ^{∗}(𝔞) = ∑_{𝔟∈Id_{m}} ∑_{𝔠∈Id_{n}} χ^{∗}(𝔟𝔠) = ∑_{𝔟∈Id_{m}} χ^{∗}(𝔟) ∑_{𝔠∈Id_{n}} χ^{∗}(𝔠) = a_{m}(χ)a_{n}(χ),

which shows that the function n ↦ a_{n}(χ) is multiplicative. Moreover, since a_{1}(χ) = χ^{∗}(𝔒_{D}) = 1, we see that ϑ_{χ} is normalized.

From this it follows from Hecke[9], Satz 42, that ϑ_{χ} is a T_{n}-eigenfunction, at least when (n, |D|) = 1. By using the results of [16], this can be refined to yield the above assertion also for (n, f_{D}) = 1, as we shall now show.

Fix n ≥ 1 with (n, f_{D}) = 1, and consider the function g := (ϑ_{χ})|_{1}T_{n} − a_{n}(χ)ϑ_{χ} ∈ M_{1}(|D|, ψ_{D}). For any m ≥ 1 with (m, n) = 1 we have by [16], Lemma 4.5.14, and the above result that the m-th Fourier coefficient of g is a_{m}(g) = a_{nm}(χ)−a_{n}(χ)a_{m}(χ) = 0. Thus, since ψ_{D} has conductor |d_{K}| and since \frac{|D|}{|d_{K}|} = f_{D}^{2}, we see that g satisfies the hypothesis of Theorem 4.6.8(1) of [16] (with l = n), and so g = 0. This means that ϑ_{χ} is a T_{n}-eigenfunction with eigenvalue a_{n}(χ), as asserted.

3 The Eisenstein and cuspidal parts of Θ_{D}

By the basic theory of modular forms, the space M_{k}(N, ψ) of modular forms of level N, weight k and Nebentypus ψ has a canonical decomposition

M_{k}(N, ψ) = E_{k}(N, ψ) ⊕ S_{k}(N, ψ) (12)

into its Eisenstein part E_{k}(N, ψ) and cuspidal part S_{k}(N, ψ); cf. [16], Theorem 4.7.2 (together with Theorem 2.1.7).

Since Θ_{D} ⊂ M_{1}(|D|, ψ_{D}), we can define its Eisenstein and cuspidal part by

Θ_{D}^{E} := Θ_{D} ∩ E_{1}(|D|, ψ_{D}) and Θ_{D}^{S} := Θ_{D} ∩ S_{1}(|D|, ψ_{D}),

respectively.

We now want to find canonical bases for these spaces. As we shall see, the ϑ_{χ}’s serve this purpose: it turns out that either ϑ_{χ} ∈ Θ_{D}^{E} or ϑ_{χ} ∈ Θ_{D}^{S}; cf. Theorem 14 and Remark 17(a) below.

To verify this, we shall compare the coefficients of ϑ_{χ} to those of a suitable modular form attached to some Hecke character on the ray class group mod f_{D}𝔒_{K} of K = ℚ(\sqrt{D}). To achieve this, recall first that the map 𝔞 ↦ 𝔞∩𝔒_{D} induces an isomorphism

φ_{D} : I_{K}(f_{D})/P_{K,ℤ}(f_{D}) →^{∼} Cl(𝔒_{D}),

where I_{K}(f) is the group of fractional ideals of 𝔒_{K} which are prime to f and P_{K,ℤ}(f) is the subgroup of I_{K}(f) generated by principal ideals of the form α𝔒_{K}, where α ∈ 𝔒_{K} satisfies α ≡ a (mod f𝔒_{K}), for some a ∈ ℤ with (a, f) = 1; cf. Cox[3], Proposition 7.22. We then have:

Proposition 13 Let χ ∈ Cl(D)^{∗}, and let χ̃ := χ ∘ λ_{D}^{−1} ∘ φ_{D} be the associated Hecke character on I_{K}(f_{D}). If n ≥ 1, then

(13) a_{n}(χ) = a_{n}(χ̃) := ∑_{𝔞∈Id_{n}(𝔒_{K})} χ̃(𝔞), provided that (n, f_{D}) = 1.

Proof. By Proposition 45 we know that the map 𝔞 ↦ 𝔞 ∩ 𝔒_{D} induces a bijection

Id_{n}(𝔒_{K}) →^{∼} Id_{n}(𝔒_{D}), whenever (n, f_{D}) = 1,

and so the assertion (13) follows in view of (9).

In the next sections we shall study the precise relation between a_{n}(χ) and a_{n}(χ̃) also in the case that (n, f_{D}) > 1. As we shall see, the formula (13) is in general no longer true in this case, and has to be replaced by the more complicated formula (31) below. Nevertheless, the above relation (13) suffices to derive many useful properties about the ϑ_{χ}’s. To formulate these in a convenient manner, it is useful to introduce the following notation.

Notation. If f_{1} and f_{2} are two modular forms and if N an integer, then we write

f_{1} ∼_{N} f_{2} ⇔^{def} a_{n}(f_{1}) = a_{n}(f_{2}), for all n ≥ 1 with (n, N) = 1.

With this notation, we can then reformulate (13) in terms of the modular form f(z; χ̃) (cf. [16], p. 183) as follows:

ϑ_{χ}(z) ∼_{f_{D}} ∑_{𝔞∈Id(𝔒_{K})∩I_{K}(f_{D})} χ̃(𝔞)e^{2πiN(𝔞)z} ∼_{f_{D}} f(z; χ̃). (14)

Theorem 14 Let χ ∈ Cl(D)^{∗} be a character. If χ is not quadratic, i.e. if χ^{2} ≠ 1, then ϑ_{χ} is a cusp form, and otherwise ϑ_{χ} is in the Eisenstein space.

Proof. Let χ̃ := χ ∘ λ_{D}^{−1} ∘ φ_{D} be the Hecke character on I_{K}(f_{D}) associated to χ.

Assume first that χ^{2} ≠ 1. In this case we have that χ̃ ≠ ψ ∘ N_{K}, for any Dirichlet character ψ mod f_{D}. Indeed, if this were the case, then ψ is necessarily a quadratic character because if (a, f_{D}) = 1, then ψ(a^{2}) = ψ(N_{K}(a𝔒_{K})) = χ̃(a𝔒_{K}) = 1 since a𝔒_{K} ∈ P_{K,ℤ}(f_{D}). Thus χ̃ is also a quadratic character, contrary to the hypothesis.

Since χ̃ ≠ ψ ∘ N_{K}, we know by [16], Theorem 4.8.2, that f(z; χ̃) ∈ S_{1}(N, ψ_{D}) where N = |d_{K}|N_{K}(f_{D}𝔒_{K}) = |D|; clearly f(z; χ̃) is a 𝕋(N)-eigenfunction. Since ϑ_{χ} ∼_{f_{D}} f(z; χ̃) by (14), it follows from Lemma 15(c) below (together with Theorem 12) that ϑ_{χ} is also a cusp form.

Now suppose that χ ∈ Cl(D)^{∗} is a quadratic character. Then by Gauss’s genus theory there exist fundamental discriminants D_{1}, D_{2} such that D = c^{2}D_{1}D_{2} for some integer c and such that χ̃ = ψ_{D_{1}} ∘ N_{K} = ψ_{D_{2}} ∘ N_{K}; cf. Weber[26], §104 and §109.

Since the pair (ψ_{D_{1}}, ψ_{D_{2}}) satisfies condition (4.7.2)(ii) on p. 176 of [16], there is a form f_{1} := f_{1}(z; ψ_{D_{1}}, ψ_{D_{2}}) ∈ E_{1}(|D_{1}D_{2}|, ψ_{D_{1}D_{2}}) ⊂ E_{1}(|D|, ψ_{D}) such that its L-function is L(s, f_{1}) = L(s, ψ_{D_{1}})L(s, ψ_{D_{2}}); cf. [16], Theorem 4.7.1. We now claim:

ϑ_{χ}(z) ∼_{D} f_{1}(z; ψ_{D_{1}}, ψ_{D_{2}}). (15)

Indeed, since χ̃ = ψ_{D_{2}} ∘ N_{K}, it follows from (13) that for (n, D) = 1 we have that

a_{n}(χ) = a_{n}(χ̃) = ∑_{𝔞∈Id_{n}(𝔒_{K})} ψ_{D_{2}}(N(𝔞)) = ψ_{D_{2}}(n)#Id_{n}(𝔒_{K}) = ψ_{D_{2}}(n) ∑_{t|n} ψ_{d_{K}}(t), (16)

the latter by (60). Since ψ_{d_{K}}(t) = ψ_{D_{1}}(t)ψ_{D_{2}}(t) and ψ_{D_{2}}(nt) = ψ_{D_{2}}(t^{2} \frac{n}{t}) = ψ_{D_{2}}(\frac{n}{t}), we thus see that a_{n}(χ) = ∑_{t|n} ψ_{D_{1}}(t)ψ_{D_{2}}(\frac{n}{t}) = a_{n}(f_{1}; ψ_{D_{1}}, ψ_{D_{2}}), which proves (15). Using this, it follows from Lemma 15(c) below that ϑ_{χ} ∈ E_{1}(|D|, ψ_{D}).

Above we had used the following general fact about 𝕋(N)-eigenfunctions which is proved implicitly on p. 179 of [16]. Here, as usual, 𝕋(N) ⊂ End_{ℂ}(M_{k}(N, ψ)) is the Hecke algebra generated by the Hecke operators T_{n} with (n, N) = 1.

Lemma 15 (a) If f ∈ M_{k}(N, ψ) is a 𝕋(N)-eigenfunction, then either f ∈ S_{k}(N, ψ) or f ∈ E_{k}(N, ψ).

(b) If f_{1}, f_{2} ∈ M_{k}(N, ψ) are two non-zero 𝕋(N)-eigenfunctions which have the same 𝕋(N)-eigenvalues, then either both are cusp forms or both are in E_{k}(N, ψ).

(c) Let f_{1}, f_{2} ∈ M_{k}(N, ψ) be two normalized 𝕋(N)-eigenfunction with f_{1} ∼_{N} f_{2}. Then f_{1} is a cusp form if and only if f_{2} is a cusp form, and similarly f_{1} ∈ E_{k}(N, ψ) if and only if f_{2} ∈ E_{k}(N, ψ).

Proof. (a) If not, then f = f_{1} + f_{2} with f_{1} ∈ S_{k}(N, ψ), f_{2} ∈ E_{k}(N, ψ) and f_{i} ≠ 0 for i = 1, 2. Then as in the proof of Theorem 4.7.2 of [16] on p. 179, f_{i} is a 𝕋(N)- eigenfunction with the same eigenvalues as f and there exist normalized eigenfunctions f̃_{1} ∈ S_{k}(N, ψ) and f̃_{2} = f_{k}(z; χ_{1}, χ_{2}) ∈ E_{k}(N, ψ) with the same 𝕋(N)-eigenvalues as f. Thus L(s, f̃_{1}) ≈_{N} L(s, f̃_{2}), where ≈_{N} means equality except for the Euler factors at the primes p|N. But this is impossible because Γ(s)L(s, f̃_{1}) is entire while Γ(s)L(s, f̃_{2}) has a pole at s = k, as is explained in [16], p. 179.

(b) If false, then f = f_{1} + f_{2} is a 𝕋(N)-eigenfunction which contradicts (a).

(c) Since f_{i} is a normalized eigenfunction, the T_{n}-eigenvalue of f_{i} is a_{n}(f_{1}) = a_{n}(f_{2}). Thus f_{1} and f_{2} satisfy the hypotheses of (b), and so the assertion follows.

Remark 16 In the case that D = d_{K} is a fundamental discriminant, it follows from the above theorem that all the ϑ_{χ}’s are primitive forms in the sense of section 4 below. Indeed, since in this case ψ_{D} = ψ_{d_{K}} is a primitive character, it follows from the general theory of newforms (primitive forms in the terminology of [16], p. 164) that all 𝕋(D)- eigenfunctions of S_{1}(|D|, ψ_{D}) are newforms; cf. [16], Lemma 4.6.9(1) and Theorem 4.6.12. In particular, it follows immediately that the ϑ_{χ} = f(z, χ̃) are newforms when χ^{2} ≠ 1.

A similar result also holds for the Eisenstein series ϑ_{χ} when χ^{2} = 1, but this requires a different argument. Here we use instead the structure theorem of the Eisenstein space E_{1}(|D|, ψ_{D}); cf. [16], p. 179. Specialized to the present situation, it yields that

E_{1}(|d_{K}|, ψ_{d_{K}}) = ⨁_{d_{K}=D_{1}D_{2}} ℂf_{1}(z; ψ_{D_{1}}, ψ_{D_{2}}), (17)

where the sum is over all factorizations d_{K} = D_{1}D_{2} of d_{K} into fundamental discriminants D_{1}, D_{2} (and the factorization d_{K} = D_{2}D_{1} is considered to be the same as the factorization d_{K} = D_{1}D_{2}), and f_{1}(z; ψ_{D_{1}}, ψ_{D_{2}}) is as in the above proof. To verify this from formula (4.7.17) of [16], note first that if ψ_{d_{K}} = ψ_{1}ψ_{2} is any factorization into characters ψ_{i} with conductors M_{i} such that M_{1}M_{2}|d_{K}, then necessarily |d_{K}| = M_{1}M_{2} and (M_{1}, M_{2}) = 1. By adjusting signs, we thus have the factorization d_{K} = D_{1}D_{2} into fundamental discriminants (with |D_{i}| = M_{i}). Moreover, since ψ_{i} is necessarily quadratic (because (M_{1}, M_{2}) = 1), it follows that ψ_{i} = ψ_{D_{i}} = (\frac{D_{i}}{·} ) is the Kronecker-Legendre character. This verifies (17).

From (17) it follows that we can improve the relation (15) to an equality because all 𝕋(D)-eigenspaces of E_{1}(D, ψ_{D}) are 1-dimensional by (17), and so all the ϑ_{χ} with χ^{2} = 1 are “primitive” as well.

We observe that it follows from (17) that

Θ_{d_{K}}^{E} := Θ_{D} ∩ E_{1}(|d_{K}|, ψ_{d_{K}}) = E_{1}(|d_{K}|, ψ_{d_{K}}).

However, the analogous statement for Θ_{d_{K}}^{S} = Θ_{D} ∩ S_{1}(|d_{K}|, ψ_{d_{K}}) is in general not true, as the discussion of Serre[20], §9, and/or [12] shows. In particular, the table on p. 258 of [20] shows that we have Θ_{−p}^{S} ≠ S_{1}(p, ψ_{−p}) for p = 139, 163, 211, 227, 283.

We are now ready to prove Theorems 1 and 2 of the introduction.

Proof of Theorem 1. By Proposition 9 and Theorem 12, the subspace Θ_{D} of M_{1}(|D|, ψ_{D}) has the basis {ϑ_{χ}} consisting of (normalized) 𝕋(D)-eigenfunctions, and hence is a 𝕋(D)-submodule.

To verify that Θ_{D} has multiplicity one, we have to show that ϑ_{χ_{1}} and ϑ_{χ_{2}} belong to different 𝕋(D)-eigenspaces whenever ϑ_{χ_{1}} ≠ ϑ_{χ_{2}}, i.e., whenever χ_{1} ∉ {χ_{2}, χ_{2}^{−1}}.

For this, we first note that there is an f ∈ Q_{D} such that χ_{1}(f) ≠ χ_{2}(f), \overline{χ}_{2}(f). Indeed, the hypothesis on χ_{1}, χ_{2} implies that ∃f_{i} ∈ Q_{D} such that χ_{1}(f_{1}) ≠ χ_{2}(f_{1}) and χ_{1}(f_{2}) ≠ χ_{2}^{−1}(f_{2}) = \overline{χ}(f_{2}). Now if χ_{1}(f_{1}) ≠ \overline{χ}_{2}(f_{1}), then we can take f = f_{1} and if χ_{1}(f_{2}) ≠ χ_{2}(f_{2}), then we can take f = f_{2}. If neither of these cases holds, then we can take f ∼ f_{1}f_{2} (product in Cl(D)) because here we have that χ_{1}(f_{1}) = \overline{χ}_{2}(f_{1}) and χ_{1}(f_{2}) = χ_{2}(f_{2}), and so χ_{1}(f) = χ_{1}(f_{1})χ_{1}(f_{2}) ≠ χ_{2}(f_{1})χ_{1}(f_{2}) = χ_{2}(f_{1})χ_{2}(f_{2}) = χ_{2}(f), and χ_{1}(f) = χ_{1}(f_{1})χ_{1}(f_{2}) ≠ \overline{χ}_{2}(f_{1})χ_{1}(f_{2}) = \overline{χ}_{2}(f_{1})\overline{χ}_{2}(f_{2}) = \overline{χ}_{2}(f).

With f as above, choose a prime p ∤ D such that r_{p}(f) > 0; cf. Lemma 8(a). Then by (7) and Lemma 8(b) we see that

a_{p}(χ_{i}) = \frac{w(f)}{w_{D}}Re(χ_{i}(f))r_{p}(f), for i = 1, 2.

Since χ_{1}(f) ≠ χ_{2}(f), \overline{χ}_{2}(f) and |χ_{i}(f)| = 1, it follows that Re(χ_{1}(f)) ≠ Re(χ_{2}(f)), and so a_{p}(χ_{1}) ≠ a_{p}(χ_{2}). Since a_{p}(χ_{i}) is the T_{p}-eigenvalues of ϑ_{χ_{i}} by Theorem 12, we see that ϑ_{χ_{1}} and ϑ_{χ_{2}} lie in distinct 𝕋(D)-eigenspaces, so Θ_{D} has multiplicity one.

The last assertion of Theorem 1 follows immediately from Theorem 14.

Proof of Theorem 2. Let V_{E} and V_{S} denote the ℂ-subspace of Θ_{D} generated by {ϑ_{χ} : χ^{2} = 1} and by {ϑ_{χ} : χ^{2} ≠ 1}, respectively. By Theorem 14 we know that V_{E} ⊂ Θ_{D}^{E} and V_{S} ⊂ Θ_{D}^{S}, and by Proposition 9 we know that V_{E} + V_{S} = Θ_{D}. Since Θ_{D}^{E} ∩ Θ_{D}^{S} = {0} by (12), it follows that V_{E} = Θ_{D}^{E} and V_{S} = Θ_{D}^{S} and that hence Θ_{D} = Θ_{D}^{E} ⊕ Θ_{D}^{S}.

Let Cl(D)^{∗}[2] = {χ ∈ Cl(D)^{∗} : χ^{2} = 1} denote the group of quadratic characters. By Proposition 9 we know that the set {ϑ_{χ} : χ ∈ Cl(D)^{∗}[2]} is linearly independent and hence is a basis of Θ_{D}^{E} = V_{E}. Thus dim(Θ_{D}^{E}) = |Cl(D)^{∗}[2]| = [Cl(D) : Cl(D)^{2}] = g_{D}, and hence dim(Θ_{D}^{S}) = dim(Θ_{D}) − dim(Θ_{D}^{E}) = h̄_{D} − g_{D} = \frac{1}{2}(h_{D} − g_{D}), the latter by Proposition 7 and equation (5), respectively. This proves (1).

Remark 17 (a) For later reference we note that the above proof of Theorem 2 shows more precisely that {ϑ_{χ} : χ ∈ Cl(D)^{∗}[2]} is a basis of Θ_{D}^{E} and that {ϑ_{χ} : χ ∈ \overline{Cl(D)}^{∗}, χ^{2} ≠ 1\} is a basis of Θ_{D}^{S}.

(b) It follows immediately from (1) that

Θ_{D} ⊂ E_{1}(|D|, ψ_{D}) ⇔ Θ_{D}^{S} = {0} ⇔ h_{D} = g_{D}. (18)

The discriminants D < 0 which satisfy the last condition (or, equivalently, the condition that Cl(D) is an elementary abelian 2-group) are called idoneal discriminants because a number n ≥ 1 is idoneal (in the sense of Euler) if and only if −4n is an idoneal discriminant; cf. [11] for a recent survey about idoneal numbers.

The above assertion (18) can be viewed as an alternate version of Theorem 3 of Kitaoka[13], which states that

ϑ_{1_{D}} ∈ E_{1}(|D|, ψ_{D}) ⇔ h_{D} = g_{D}. (19)

Indeed, if h_{D} = g_{D}, then clearly ϑ_{1_{D}} ∈ θ_{D} ⊂ E_{1}(|D|, ψ_{D}) by (18). To prove the converse, note first that by (8) we have that

ϑ_{1_{D}} = \frac{1}{w_{D}} ∑_{χ∈\overline{Cl(D)}^{∗}} w(χ)ϑ_{χ}

(because χ(1_{D}) = 1 for all χ ∈ Cl(D)^{∗}). Thus, by part (a) we see that ϑ_{1_{D}} ∈ E_{1}(|D|, ψ_{D}) ⇔ χ^{2} = 1, ∀χ ∈ Cl(D)^{∗} ⇔ Cl(D)^{2} = {1} ⇔ h_{D} = g_{D}, which proves (19).

The above results immediately imply the following corollary which generalizes the work of Siegel[21] on theta-series attached to forms in m = 2k > 4 variables (cf. [21], particularly pp. 577-581) to the case of binary forms. Note that this gives an alternate proof (for binary forms) of Kitaoka’s extension of Siegel’s work to the case m ≥ 2; cf. [13], Lemma 1.

Corollary 18 (a) Let f_{1}, f_{2} ∈ Q_{D} be two primitive forms. Then ϑ_{f_{1}} − ϑ_{f_{2}} is a cusp form if and only if f_{1} and f_{2} are genus-equivalent, i.e.

ϑ_{f_{1}} − ϑ_{f_{2}} ∈ S_{1}(|D|, ψ_{D}) ⇔ χ(f_{1}) = χ(f_{2}), ∀χ ∈ Cl(D)^{∗}[2] ⇔ f_{1}^{−1}f_{2} ∈ Cl(D)^{2}. (20)

(b) For any f ∈ Q_{D} we have that

F (z, f) := \frac{g_{D}}{h_{D}} ∑_{f_{1}∈Cl(D)^{2}} ϑ_{ff_{1}}(z) = \frac{w_{D}}{h_{D}} ∑_{χ∈Cl(D)^{∗}[2]} χ(f)ϑ_{χ}(z) ∈ Θ_{D}^{E}. (21)

Thus, F (z, f) ∈ Θ_{D}^{E} is the Eisenstein component of ϑ_{f}(z), and ϑ_{f}(z) − F (z, f) ∈ Θ_{D}^{S} is its cuspidal component.

Proof. (a) By (8) we have ϑ_{f_{1}} − ϑ_{f_{2}} = \frac{h_{D}}{w_{D}} ∑_{χ∈\overline{Cl(D)}^{∗}} w(χ)Re(χ(f_{1}) − χ(f_{2}))ϑ_{χ}, and so it follows from Remark 17(a) that ϑ_{f_{1}} − ϑ_{f_{2}} ∈ S_{1}(|D|, ψ_{D}) ⇔ Re(χ(f_{1}) − χ(f_{2})) = 0, ∀χ ∈ Cl(D)^{∗}[2] ⇔ χ(f_{1}) = χ(f_{2}), ∀χ ∈ Cl(D)^{∗}[2] because all χ ∈ Cl(D)^{∗}[2] are real-valued. This proves the proves the first equivalence of (20) and hence also (20) because the second equivalence is obvious.

(b) By (8) we have that F (z, f) = \frac{g_{D}w_{D}}{h_{D}^{2}} ∑_{χ∈Cl(D)^{∗}} c(f, χ)ϑ_{χ}(z) with

c(f, χ) = ∑_{f_{1}∈Cl(D)^{2}} \overline{χ}(ff_{1})) = \overline{χ}(f) ∑_{f_{1}∈Cl(D)^{2}} \overline{χ}(f_{1})) = \{ \begin{matrix}\overline{χ}(f)\frac{h_{D}}{g_{D}} & if χ^{2} = 1, \\ 0 & otherwise \end{matrix}

where the last identity follows from a suitable orthogonality relation and the fact that |Cl(D)^{2}| = \frac{h_{D}}{g_{D}}. From this the asserted identity (21) follows immediately.

By (21) and Theorem 14 it is clear that F (z, f) ∈ Θ_{D}^{E}. Moreover, by (8) and (21) we have that ϑ_{f}(z) − F (z, f) = \frac{w_{D}}{h_{D}} ∑_{χ^{2}≠1} \overline{χ}(f)ϑ_{χ}(z) ∈ Θ_{D}^{S}, the latter by Theorem 14 again. This proves the last assertion.

Remark 19 The Fourier coefficients a_{n}(F (z, f)) of the Eisenstein series F (z, f) are easily determined, at least when (n, D) = 1. Indeed, we have

a_{n}(F (z, f)) = \frac{w_{D}g_{D}}{h_{D}} ε(f, n)#Id_{n}(𝔒_{K}), (22)

where ε(f, n) = 1 if n ≡ f(x, y) (mod |D|), for some x, y ∈ ℤ, and ε(f, n) = 0 otherwise. To see this, note first that a_{n}(F (z, f)) = \frac{w_{D}}{h_{D}} ∑_{χ∈Cl(D)^{∗}[2]} χ(f)a_{n}(χ) by (21). Thus a_{n}(F (z, f)) = 0 = ε(f, n), if Id_{n}(𝔒_{K}) = ∅ (cf. (9)), so assume that ∃𝔞_{n} ∈ Id_{n}(𝔒_{K}). If f_{n} ∈ Q_{D} is such that λ_{D}(f_{n}) = φ_{D}(𝔞_{n}), then there exist x, y ∈ ℤ such that f(x, y) = n. Now for any χ ∈ Cl(D)^{∗}[2] we have by (16) that a_{n}(χ) = χ̃(𝔞_{n})#Id_{n}(𝔒_{K}) = χ(f_{n})#Id_{n}(𝔒_{K}), so we obtain that

a_{n}(F (z, f)) = \frac{w_{D}}{h_{D}} #Id_{n}(𝔒_{K}) ∑_{χ∈Cl(D)^{∗}[2]} χ(ff_{n}).

By an orthogonality relation we have that ∑_{χ^{2}=1} χ(ff_{n}) = |Cl(D)^{∗}[2]| = g_{D} if ff_{n} ∈ Cl(D)^{2} and equals 0 otherwise. By genus theory (cf. [3], §3.B), this means that this sum equals g_{D}ε(f, n), and so (22) follows.

As another application of the above results we compute the trace tr(T_{n}|V) of the Hecke operator T_{n} on the vector spaces V = Θ_{D}^{E} and V = Θ_{D}^{S}.

Corollary 20 If (n, f_{D}) = 1, then the traces of the Hecke operator T_{n} on the spaces Θ_{D}^{E} and Θ_{E}^{S} are given by

tr(T_{n}|Θ_{D}^{E}) = \frac{h_{D}}{w_{D}} a_{n}(F (z, 1_{D})) and tr(T_{n}|Θ_{D}^{S}) = \frac{h_{D}}{2w_{D}} (r_{n}(1_{D}) − a_{n}(F (z, 1_{D}))). (23)

Proof. By Remark 17(a), Theorem 12, and (21) we have that

tr(T_{n}|Θ_{D}^{E}) = ∑_{χ∈Cl(D)^{∗}[2]} a_{n}(χ) = \frac{h_{D}}{w_{D}} a_{n}(F (z, 1_{D})).

This proves the first equation of (23). To prove the second equation, put X = {χ ∈ \overline{Cl(D)}^{∗} : χ^{2} ≠ 1\}. Then by Remark 17(a) and Theorem 12 we have

tr(T_{n}|Θ_{D}^{S}) = ∑_{χ∈X} a_{n}(χ) = \frac{1}{2} ∑_{χ^{2}≠1} a_{n}(χ) = \frac{1}{2} ∑_{χ} a_{n}(χ) − \frac{1}{2} ∑_{χ^{2}=1} a_{n}(χ).

Since the first sum equals \frac{h_{D}}{w_{D}} r_{n}(1_{D}) by (8) and since (as above) the second sum equals \frac{h_{D}}{w_{D}} a_{n}(F (z, 1_{D})) by (21), the formula (23) follows.

4 Primitive characters and primitive forms

We now want to study the relation between the theta-series ϑ_{χ} and the modular form f(z; χ̃) attached to the associated Hecke character χ̃ = χ ∘ λ_{D}^{−1} ∘ φ_{D} in more detail. For this, it is useful to introduce the following terminology.

Definition. If χ ∈ Cl(D)^{∗} is a character on the class group Cl(D), then we say that χ is primitive if we have that Ker(\overline{π}_{D,D/c^{2}}) ⊄ Ker(χ), for all divisors c|f_{D} with c > 1. Here \overline{π}_{D,D/c^{2}} : Cl(D) → Cl(D/c^{2}) is the homomorphism induced by the map \overline{π}_{𝔒_{D},𝔒_{D/c^{2}}} : Cl(𝔒_{D}) → Cl(𝔒_{D/c^{2}}) which is defined in Remark 39 of the Appendix; in other words, \overline{π}_{D,D′} = λ_{D′}^{−1} ∘ \overline{π}_{𝔒_{D},𝔒_{D′}} ∘ λ_{D}, for D′ = D/c^{2}.

Moreover, the conductor f_{χ} of χ ∈ Cl(D)^{∗} is defined by

f_{χ} = gcd(f : f|f_{D} and Ker(\overline{π}_{D,f^{2}d_{K}}) ⊂ Ker(χ)).

Thus, if χ ∈ Cl(D)^{∗} is primitive, then clearly f_{χ} = f_{D}. Moreover, the converse is also true, as the following result shows.

Proposition 21 Let χ ∈ Cl(D)^{∗} be a character with conductor f_{χ}, and let f|f_{D}. Then Ker(\overline{π}_{D,f^{2}d_{K}}) ⊂ Ker(χ) ⇔ f_{χ}|f. (24)

Thus, if D_{χ} := f_{χ}^{2}d_{K}, then χ = χ_{pr} ∘ \overline{π}_{D,D_{χ}}, for a unique character χ_{pr} ∈ Cl(D_{χ})^{∗}, and χ_{pr} is primitive.

Proof. From the definition of D_{χ} and (52) we have that 𝔒_{D_{χ}} = ∏_{f} 𝔒_{f^{2}d_{K}}, where the product is over all f|f_{D} with Ker(\overline{π}_{D,f^{2}d_{K}}) ⊂ Ker(χ). It thus follows from Corollary 41 that Ker(\overline{π}_{D,D_{χ}}) ⊂ Ker(χ), and so the assertion (24) is obvious. Moreover, since \overline{π}_{D,D_{χ}} : Cl(D) → Cl(D_{χ}) is surjective (cf. Proposition 38), there is a unique character χ_{pr} on Cl(D_{χ}) such that χ = χ_{pr} ∘ \overline{π}_{D,D_{χ}}. Finally, χ_{pr} is primitive because if Ker(\overline{π}_{D_{χ},D_{χ}/c^{2}}) ⊂ Ker(χ_{pr}), for some c|f_{Dχ}, then Ker(\overline{π}_{D,(f_{χ}/c)^{2}d_{K}}) = \overline{π}_{D,D_{χ}}^{−1}(Ker(\overline{π}_{D_{χ},D_{χ}/c^{2}})) ⊂ \overline{π}_{D,D_{χ}}^{−1}(Ker(χ_{pr})) = Ker(χ), so f_{χ}|\frac{f_{χ}}{c} by (24) and hence c = 1, i.e. χ_{pr} is primitive.

Using the results of the Appendix, we can now compute the Fourier coefficients a_{n}(χ) of ϑ_{χ} in the case that n|f_{D}^{2}.

Proposition 22 Let χ be a character on Cl(D) with conductor f_{χ}, and put f̄_{χ} = f_{D}/f_{χ}. If n|f_{D}^{2}, then

(25) a_{n}(χ) = \{ \begin{matrix}c ∏_{p|c} (1 − \frac{1}{p}ψ_{D/c^{2}}(p)) & if n = c^{2} and c|f̄_{χ}, \\ 0 & otherwise. \end{matrix}

Proof. If n isn’t a square, then Id_{n}(𝔒_{D}) = ∅ by Proposition 48, so by (9) we have that a_{n}(χ) = 0 in this case. Thus, assume that n = c^{2}, and put χ^{∗} = χ ∘ λ_{D}^{−1}. Then it follows from (9) and Proposition 48 that

a_{n}(χ) = ∑_{𝔞∈cKer(π_{𝔒_{D},𝔒_{D/c^{2}}} )}χ^{∗}(𝔞) = ∑_{𝔞∈Ker(π_{𝔒_{D},𝔒_{D/c^{2}}} )}χ^{∗}(𝔞).

Now from (55) and (24) we see that Ker(π_{𝔒_{D},𝔒_{D/c^{2}}})P (𝔒_{D}) ≤ Ker(χ^{∗}) ⇔ f_{χ}|\frac{f_{D}}{c} ⇔ c|f̄_{χ}. If this is not the case, i.e. if χ^{∗} is non-trivial on Ker(π_{𝔒_{D},𝔒_{D/c^{2}}}), then this sum equals 0 by an orhogonality relation. On the other hand, if c|f̄_{χ}, then we have that a_{n}(χ) = |Ker(π_{𝔒_{D},𝔒_{D/c^{2}}})| and so the assertion follows by using (57).

We now turn our attention to the modular form f(z; χ̃), where, as before, χ̃ = χ ∘ λ_{D}^{−1} ∘ φ_{D}. To determine its properties, we need to know the conductor cond(χ̃) of the Hecke character χ̃; here cond(χ̃) ∈ Id(𝔒_{K}) is as defined on p. 91 of [16].

Theorem 23 The conductor of the Hecke character χ̃ = χ ∘ λ_{D}^{−1} ∘ φ_{D} is cond(χ̃) = f_{χ}𝔒_{K}. In particular, χ is primitive if and only if χ̃ is a primitive Hecke character.

Proof. As usual, we view χ̃ as a homomorphism χ̃ : I_{K}(f) → ℂ^{×} with P_{K,ℤ}(f) ⊂ Ker(χ̃), where f = f_{D}. By definition (cf. [16], p. 91), 𝔪χ̃ := cond(χ̃) is the largest 𝔒_{K}-ideal 𝔪 such that P_{K,1}(𝔪)∩I_{K}(f) ⊂ Ker(χ̃). Here P_{K,1}(𝔪) = {α𝔒_{K} : α ∈ K_{1}(𝔪)}, where K_{1}(𝔪) = 1 + 𝔪𝔒_{𝔪} and 𝔒_{𝔪} = {λ ∈ K : v_{𝔭}(λ) ≥ 0, ∀𝔭|𝔪} denotes the semi-local ring of K defined by the prime divisors of 𝔪. (Note that it follows from the Strong Approximation Theorem that 𝔒_{𝔪} = (𝔒_{K})_{1+𝔪} is the localization of 𝔒_{K} with respect to the multiplicative set 1 + 𝔪, and so we see that K_{1}(𝔪) = ⟨1 + 𝔪⟩ is the group generated by 1+𝔪. Thus, this definition of P_{K,1}(𝔪) coincides with that of [3], p. 160.)

We first observe that f_{χ}𝔒_{K} ⊂ 𝔪χ̃. Indeed, since χ is the lift of χ_{pr} ∈ Cl(D_{χ}), we have that Ker(χ̃) = Ker(χ̃_{pr}) ∩ I_{K}(f) ⊃ P_{K,ℤ}(f_{χ}) ∩ I_{K}(f) ⊃ P_{K,1}(f_{χ}) ∩ I_{K}(f), and so f_{χ}𝔒_{K} ⊂ 𝔪χ̃ by definiton.

We next verify that f_{χ}𝔒_{K} = 𝔪χ̃. For this, we may assume without loss of generality that χ is primitive, i.e. that f_{χ} = f, because if we replace χ by χ_{pr} (using Proposition 21), then f_{χ} = f_{χ_{pr}} and cond(χ̃) = cond(χ̃_{pr}).

Thus, assume that f_{χ} = f. If 𝔪χ̃ ≠ 𝔪 := f𝔒_{K}, then there is a prime ideal 𝔭|𝔪 such that 𝔪 ⊂ 𝔭𝔪χ̃. This means that Ker(χ̃) ⊃ P_{K,1}(𝔪_{χ})∩I_{K}(f) ⊃ P_{K,1}(𝔭^{−1}𝔪)∩I_{K}(f). We observe that if P_{K}^{ℤ}(f) := \{x𝔒_{K} : x ∈ ℚ_{f}^{×}\}, where ℚ_{f}^{×} = \{\frac{a}{b} : a, b ∈ ℤ and (ab, f) = 1\}, then P_{K}^{ℤ}(f) ⊂ P_{K,ℤ}(f), and so P_{K}^{ℤ}(f)(P_{K,1}(𝔭^{−1}𝔪) ∩ I_{K}(f)) ⊂ Ker(χ̃). But since

P_{K,ℤ}(fp^{−1}) ∩ I_{K}(f) ⊂ P_{K}^{ℤ}(f)(P_{K,1}(f𝔭^{−1}) ∩ I_{K}(f)), ∀𝔭 | p | f, (26)

as will be shown below, and since it is easy to see that

P_{K,ℤ}(m) ∩ I_{K}(f) = P_{K}^{ℤ}(f)(P_{K,1}(m𝔒_{K}) ∩ I_{K}(f)), if m|f, (27)

we obtain that P_{K,ℤ}(\frac{f}{p}) ∩ I_{K}(f) = P_{K}^{ℤ}(f)(P_{K,1}(\frac{f}{p}𝔒_{K}) ∩ I_{K}(f)) ⊂ P_{K}^{ℤ}(f)(P_{K,1}(f𝔭^{−1}) ∩ I_{K}(f)) ⊂ Ker(χ̃). But this means that f = f_{χ}|\frac{f}{p}, contradiction.

It remains to verify (26). If p is inert in K, then fp^{−1}𝔒_{K} = f𝔭^{−1}, and so (26) follows from (27). Thus, assume that p splits or is ramified in K. Then we have that

ℚ_{f}^{×}K_{1}(f𝔭^{−1}) ∩ 𝔒_{f𝔒_{K}}^{×} = ℚ_{f}^{×}K_{1}(fp^{−1}𝔭) ∩ 𝔒_{f𝔒_{K}}^{×}. (28)

Indeed, if p𝔒_{K} = 𝔭^{2} is ramified, then this is clear because p^{−1}𝔭 = 𝔭^{−1}, so assume that p𝔒_{K} = 𝔭\overline{𝔭}, i.e. that p^{−1}𝔭 = \overline{𝔭}^{−1}. If λ ∈ ℚ_{f}^{×}K_{1}(fp^{−1}𝔭) ∩ 𝔒_{f𝔒_{K}}^{×}, then N(λ) = λ\overline{λ} ∈ ℚ_{f}^{×} and \overline{λ} ∈ ℚ_{f}^{×}K_{1}(fp^{−1}\overline{𝔭}) ∩ 𝔒_{f𝔒_{K}}^{×}, so λ = N(λ)\overline{λ}^{−1} ∈ ℚ_{f}^{×}K_{1}(fp^{−1}\overline{𝔭}) ∩ 𝔒_{f𝔒_{K}}^{×}. Thus, by symmetry, ℚ_{f}^{×}K_{1}(fp^{−1}𝔭) ∩ 𝔒_{f𝔒_{K}}^{×} = ℚ_{f}^{×}K_{1}(fp^{−1}\overline{𝔭}) ∩ 𝔒_{f𝔒_{K}}^{×}, and so (28) follows.

From this, (26) follows readily. Indeed, let 𝔞 ∈ P_{K,1}(fp^{−1}𝔒_{K}) ∩ I_{K}(f). Then 𝔞 = λ𝔒_{K} with λ ∈ K_{1}(fp^{−1}𝔒_{K})∩𝔒_{f𝔒_{K}}^{×}, and so by (28) we see that 𝔞 ∈ P_{K}^{ℤ}(f)(P_{K,1}(f𝔭^{−1})∩ I_{K}(f)). Thus P_{K,1}(fp^{−1}𝔒_{K})∩I_{K}(f) ⊂ P_{K}^{ℤ}(f)(P_{K,1}(f𝔭^{−1})∩I_{K}(f)), and so (26) follows in view of (27).

From the above theorem it follows easily that if χ ∈ Cl(D)^{∗} is a primitive character, then f(z; χ̃) is a primitive form in the sense of the following definition.

Definition. A modular form f ∈ M_{k}(N, ψ) is called a primitive form if either f ∈ S_{k}(N, ψ) is a normalized newform of some level M|N (so f is a primitive form in the sense of [16], §4.6) or if f = f_{k}(z; ψ_{1}, ψ_{2}) is one of the Eisenstein series defined on p. 178 of [16].

Remark 24 It follows from the theory of newforms (cf. [16], §4.6) and the theory of Eisenstein forms (cf. [16], §4.7) that each 𝕋(N)-eigenspace of M_{k}(N, ψ) contains a unique primitive form. Thus, there is natural bijection between primitive forms and 𝕋(N)-eigenspaces of M_{k}(N, ψ).

Corollary 25 If χ ∈ Cl(D)^{∗} is a primitive character, then f(z; χ̃) is a primitive form of level |D|.

Proof. Suppose first that χ^{2} ≠ 1. Then from the proof of Theorem 14 we know that χ̃ ≠ χ′ ∘ N_{K} for all Dirichlet characters χ′, and by Theorem 23 we know that χ̃ is a primitive Hecke character. It thus follows from Theorem 4.8.2 of [16] that f(z; χ̃) is a primitive cusp form (of level |D|).

Now suppose that χ^{2} = 1. Since χ̃ is a primitive Hecke character by Theorem 23 and since f(z; χ̃) is not a cusp form by (the proof of) Theorem 14, it follows from the last part of the proof of Theorem 4.8.2 of [16] that there exist Dirichlet characters χ_{1}, χ_{2} with χ_{1}χ_{2} = ψ_{D} such that L(s, f(z; χ̃)) = L(s, χ_{1})L(s, χ_{2}). Since the latter equals L(s, f_{1}(z; χ_{1}, χ_{2})) by [16], Theorem 4.7.1, we conclude that f(z; χ̃) = f_{1}(z; χ_{1}, χ_{2}), and hence f(z; χ̃) is primitive of level |D|.

Remark 26 If χ ∈ Cl(D)^{∗}[2] is a primitive quadratic character, then by combining the above proof with that of Theorem 14, we obtain the formula

f(z; χ̃) = f_{1}(z; ψ_{D_{1}}, ψ_{D_{2}}), where χ̃ = ψ_{D_{1}} ∘ N_{K} = ψ_{D_{2}} ∘ N_{K}, (29)

and D = D_{1}D_{2} is a suitable factorization of D into fundamental discriminants. Indeed, by the proof of Corollary 25 we know that (29) holds for some pair of characters χ_{1}, χ_{2}, and the proof of Theorem 14, particularly equations (15) and (16), show that χ_{i} = ψ_{D_{i}} for i = 1, 2, where D = D_{1}D_{2} is the fundamental factorization associated to χ.

We can now prove Theorem 3 of the introduction.

Proof of Theorem 3. (a) This follows immediately from Proposition 21.

(b) Since χ_{pr} ∈ Cl(D_{χ})^{∗} is primitive by part (a), it follows from Corollary 25 that f(z; χ̃_{pr}) ∈ M_{1}(|D_{χ}|, ψ_{D_{χ}}) is a primitive form of level |D_{χ}|. By Theorem 12 and (14) we know that ϑ_{χ_{pr}} is a normalized 𝕋(D_{χ})-eigenfunction which has the same eigenvalues as f(z; χ̃_{pr}). Since f(z; χ̃_{pr}) is primitive of level |D_{χ}|, its associated eigenspace in M_{1}(|D_{χ}|, ψ_{D_{χ}}) is one-dimensional (cf. [16], Theorems 4.6.12 and 4.7.2) and so

ϑ_{χ_{pr}}(z) = f(z; χ̃_{pr}) (30)

because both forms are normalized. Thus ϑ_{χ_{pr}} is primitive and L(s, ϑ_{pr}) = L(s, χ̃_{pr}).

Now by Theorem 12 we know that ϑ_{χ} is a 𝕋(D)-eigenfunction which lies in the eigenspace defined by the primitive form ϑ_{χ_{pr}}(z) = f(z; χ̃_{pr}) because by (14) we have that ϑ_{χ} ∼_{f_{D}} f(z; χ̃) ∼_{f_{D}} f(z; χ̃_{pr}). Since D/D_{χ} = (f_{D}/f_{χ})^{2} = f̄_{χ}^{2}, it follows from the description of the 𝕋(D)-eigenspaces of M_{1}(|D|, ψ_{D}) given in Corollary 4.6.20 and (implicitly) in Theorem 4.7.2 of [16], that there exist constants c_{n}(χ) ∈ ℂ with n|f̄_{χ}^{2} such that (2) holds. Thus, if we put c_{n}(χ) = 0 when n ∤ f̄_{χ}^{2}, then we have

a_{m}(χ) = ∑_{{n|f̄_{χ}^{2} \atop n|m}} c_{n}(χ)a_{m/n}(χ_{pr}) = ∑_{n|m} c_{n}(χ)a_{m/n}(χ_{pr}), for all m ≥ 1, (31)

and so it follows that L(s, ϑ_{χ}) = (∑_{n≥1} c_{n}(χ)n^{−s})L(s, ϑ_{χ_{pr}}); cf. [7], Theorem 284. Thus (3) holds. Moreover, it follows that n ↦ c_{n}(χ̃) is multiplicative because both L(s, ϑ_{χ}) and L(s, ϑ_{χ_{pr}}) have Euler products by Theorem 12, and hence so does

C(s, χ) := ∑_{n≥1} \frac{c_{n}(χ)}{n^{s}} = \frac{L(s, ϑ_{χ})}{L(s, ϑ_{χ_{pr}})} = \frac{L(s, ϑ_{χ})}{L(s, χ̃_{pr})}. (32)

This means that n ↦ c_{n}(χ) is multiplicative.

Finally, we note that it follows from (32) that all the c_{n}(χ) are real because the Fourier coefficients a_{n}(χ) and a_{n}(χ_{pr}) are real; cf. (7).

5 The Dirichlet series C(s, χ)

In the previous section we had seen that each theta-series ϑ_{χ} can be expressed as a linear combination of “shifted” modular forms f(nz; χ̃_{pr}) = ϑ_{χ_{pr}}(nz) associated to the primitive Hecke character χ̃_{pr}; cf. Theorem 3. We now want to obtain precise formulae for the coefficients c_{n}(χ) of this linear combination. We thus study the (finite) Dirichlet series C(s, χ) of equation (32) in more detail.

As in Hardy/Wright[7], §17.4, we let F_{p}(s) denote the p-Euler factor of a Dirichlet series F (s) at the prime p, i.e.

F_{p}(s) = ∑_{k≥0} a_{p^{k}}p^{−ks}, if F (s) = ∑_{n≥1} a_{n}n^{−s}.

The p-Euler factors C_{p}(s, χ) and L_{p}(s, χ) of the Dirichlet series C(s, χ) and L(s, χ) := L(s, ϑ_{χ}) are related as follows.

Proposition 27 If χ ∈ Cl(D)^{∗}, then for any prime p we have that

C_{p}(s, χ) = L_{p}(s, χ)(1 − a_{p}(χ_{pr})p^{−s} + ψ_{D_{χ}}(p)p^{−2s}). (33)

Proof. By taking m = p^{k} (for k ≥ 0) in (31) we see that L_{p}(s, χ) = C_{p}(s, χ)L_{p}(s, χ_{pr}), and so (33) follows because

L_{p}(s, χ_{pr}) = (1 − a_{p}(χ_{pr})p^{−s} + ψ_{D_{χ}}(p)p^{−2s})^{−1} . (34)

Indeed, since ϑ_{χ_{pr}} = f(z; χ̃_{pr}) is a primitive form of level |D_{χ}| by Theorem 3(b), equation (34) follows from Corollary 4.6.22 and (4.7.16) of [16].

We are now ready to determine C(s, χ) explicitly.

Theorem 28 Let χ ∈ Cl(D)^{∗} be a character of conductor f_{χ}|f_{D}, and let χ_{pr} ∈ Cl(D_{χ}) be the associated primitive character on Cl(D_{χ}), where D_{χ} = f_{χ}^{2}d_{K}. Then

C(s, χ) = ∏_{p|f̄_{χ}} C_{p}(s, χ), (35)

where f̄_{χ} = f_{D}/f_{χ}. Moreover, if p|f̄_{χ} and if pē_{p}||f̄_{χ} is the highest power of p dividing f̄_{χ}, then

C_{p}(s, χ) = \frac{1 − p^{(1−2s)ē_{p}}}{1 − p^{1−2s}} (1 − a_{p}(χ_{pr})p^{−s} + ψ_{D_{χ}}(p)p^{−2s})+(1 − \frac{1}{p}ψ_{D_{χ}}(p)) p^{(1−2s)ē_{p}}; (36)

in other words, we have for k ≥ 1 that

(37) c_{p^{k}}(χ) = \{ \begin{matrix}−a_{p}(χ_{pr})p^{(k−1)/2} & if k ≡ 1 (mod 2) and k < 2ē_{p}, \\ p^{k/2} + p^{k/2−1}ψ_{D_{χ}}(p) & if k ≡ 0 (mod 2) and k < 2ē_{p}, \\ pē_{p} & if k = 2ē_{p} \\ 0 & if k > 2ē_{p} \end{matrix}

Proof. The first assertion (35) follows immediately from the multiplicativity of the c_{n}(χ)’s; cf. Theorem 3(b). To prove (36), we first observe that if 0 ≤ k ≤ 2ē_{p}, then we obtain from (25) that

(38) a_{p^{k}}(χ) = \{ \begin{matrix}0 & if k ≡ 1 (mod 2), \\ p^{k/2} & if k ≡ 0 (mod 2) and k < 2ē_{p}, \\ pē_{p}(1 − \frac{1}{p}ψ_{D_{χ}}(p)) & if k = 2ē_{p}. \end{matrix}

Indeed, if k is odd, then this is clear, so assume that k is even. Then (25) yields that a_{p^{k}}(χ) = p^{k/2}(1 − \frac{1}{p}ψ_{D/p^{k}}(p)). Now if k < 2ē_{p}, then p|\frac{D}{p^{k}} because p^{2ē_{p}}|D. Thus ψ_{D/p^{k}}(p) = 0, and hence (38) holds in this case. On the other hand, if k = 2ē_{p}, then \frac{D}{p^{k}} = D_{χ}f_{1}^{2}, where f_{1} = f̄_{χ}p^{−ē_{p}}. Since (f_{1}, p) = 1, we have that ψ_{D/p^{k}}(p) = ψ_{D_{χ}}, and so (38) holds in all cases.

Note that by using the identity ∑_{k=0}^{ē_{p}−1} p^{k}X^{2} = \frac{1−(pX^{2})ēp}{1−pX^{2}} we can re-write (38) in the form

∑_{k=0}^{2ē_{p}} a_{p^{k}}(χ)X^{k} = \frac{1 − (pX^{2})ē_{p}}{1 − pX^{2}} + (1 − \frac{1}{p}ψ_{D_{χ}}(p))(pX^{2})ē_{p}. (39)

Thus, viewing (33) as an identity of power series in X = p^{−s}, it follows from (33) and (39) that

C_{p}(s, χ) ≡ \frac{1 − (pX^{2})ē_{p}}{1 − pX^{2}} (1−a_{p}(χ_{pr})X+ψ_{D_{χ}}(p)X^{2})+(1−\frac{1}{p}ψ_{D_{χ}}(p))(pX^{2})ē_{p} (mod X^{2ē_{p}+1}).

Now since C_{p}(s, χ) is a polynomial of degree ≤ 2ē_{p} (because c_{p^{k}}(χ) = 0 by definition when k > 2ē_{p}), and since the same is true for the right hand side, it follows that these two polynomials are equal, and so (36) follows (by replacing X by p^{−s}). The last assertion (37) follows immediately from (36) by comparing the coefficients in X = p^{−s}.

Corollary 29 In the situation of Theorem 28 we have that

cf̄_{χ}^{2}(χ) = f̄_{χ}. (40)

Thus, ϑ_{χ} has exact level |D|, and C(s, χ) = 1 if and only if χ is primitive. In particular, ϑ_{χ} is a primitive form if and only if χ is a primitive character.

Proof. Since f̄_{χ} = ∏_{p|f̄_{χ}} pē_{p} and c_{n}(χ) is multiplicative, we have by (37) that

cf̄_{χ}^{2}(χ) = ∏_{p|f̄_{χ}} c_{p^{2ēp}}(χ) = ∏_{p|f̄_{χ}} pē_{p} = f̄_{χ},

which proves (40). From this and (2) it follows that ϑ_{χ} has exact level |D|.

If χ is primitive, i.e. if f̄_{χ} = 1, then clearly C(s, χ) = 1 by definition (cf. (3)). Conversely, if C(s, χ) = 1, i.e., if c_{n}(χ) = 0 for all n > 1, then (40) forces that f̄_{χ} = 1, which means that χ is primitive.

To prove the last assertion, recall first that if χ = χ_{pr} is a primitive character, then ϑ_{χ} is a primitive form by Theorem 3(b). Conversely, suppose that ϑ_{χ} is a primitive form. Then ϑ_{χ} = ϑ_{χ_{pr}} because both are in the same 𝕋(D)-eigenspace of M_{1}(|D|, ψ_{D}), and so C(s, χ) = L(s, χ)/L(s, χ_{pr}) = L(s, ϑ_{χ})/L(s, ϑ_{χ_{pr}}) = 1, which means that χ is primitive.

Remark 30 (a) It follows from (8) and (3) that if f ∈ Q_{D}, then its (Epstein) zetafunction Z_{f} of Remark 6(b) can be written as a sum of the L-functions L(s, χ) := L(s, ϑ_{χ}) and the Hecke L-functions L(s, χ_{pr}) = L(s, χ̃_{pr}) in the following way:

Z_{f}(s) = \frac{w_{D}}{h_{D}} ∑_{χ∈Cl(D)^{∗}} \overline{χ(f)}L(s, χ) = \frac{w_{D}}{h_{D}} ∑_{χ} \overline{χ(f)}C(s, χ)L(s, χ_{pr}); (41)

this generalizes the well-known relation (cf. [4]) for Z_{f} when D = d_{k} is a fundamental discriminant. However, the above relation and Corollary 29 show that if D ≠ d_{K} is not a fundamental discriminant, then Z_{f} is never a linear combination of the associated Hecke L-functions because in that case the factor C(s, 1) is not a constant since the trivial character χ = 1 is not primitive; cf. Example 35 below.

(b) Similarly (and equivalently), by combining (8) with (2), we obtain in view of (35) and (38) the following explicit expression of the theta series ϑ_{f} in terms of the (extended) Atkin-Lehner basis:

ϑ_{f}(z) = \frac{w_{D}}{h_{D}} ∑_{χ∈Cl(D)^{∗}} \overline{χ(f)}ϑ_{χ}(z) = \frac{w_{D}}{h_{D}} ∑_{χ} \overline{χ(f)} ∑_{n|\overline{f}_{χ}} ϑ_{χ_{pr}}(nz). (42)

As was mentioned in the introduction, Theorem 28 yields immediately a formula for the L-function L(s, χ) = L(s, ϑ_{χ}).

Corollary 31 If χ ∈ Cl(D)^{∗}, then the L-function L(s, χ) of ϑ_{χ} has the Euler product

L(s, χ) = ∏_{p} L_{p}(s, χ) (43)

where for p ∤ f̄_{χ} the p-Euler factor L_{p}(s, χ) is

L_{p}(s, χ) = (1 − a_{p}(χ)p^{−s} + ψ_{D}(p)p^{−2s})^{−1} = (1 − a_{p}(χ_{pr})p^{−s} + ψ_{D_{χ}}(p)p^{−2s})^{−1} , (44)

whereas for p | f̄_{χ} it is given by

L_{p}(s, χ) = \frac{1 − p^{(1−2s)ē_{p}}}{1 − p^{1−2s}} + \frac{(1 − \frac{1}{p}ψ_{D_{χ}}(p)) p^{(1−2s)ē_{p}}}{1 − a_{p}(χ_{pr})p^{−s} + ψ_{D_{χ}}(p)p^{−2s}} . (45)

Proof. Since the a_{n}(χ)’s are multiplicative (cf. Theorem 12), it follows that L(s, χ) has an Euler product (43). Since C_{p}(s, χ) = 1 when p ∤ f̄_{χ}, it follows from (3) that L_{p}(s, χ) = L_{p}(s, χ_{pr}), so in particular a_{p}(χ) = a_{p}(χ_{pr}). Moreover, since D = D_{χ}f̄_{χ}^{2}, we see that ψ_{D}(p) = ψ_{D_{χ}}(p), so the second equality of (44) holds, and hence the first equality follows from (34).

The formula (45) follows immediately from (33) and (36).

Remark 32 (a) The above Corollary 31 shows that the second identity of the equation on the bottom of p. 280 of Lang[15] is incorrect when χ is not a primitive character.

(b) An alternate way of writing formula (45) is follows: if pē_{p}||f̄_{χ}, then

a_{p^{k}}(χ) = \{ \begin{matrix}0 & if k ≡ 1 (mod 2) and k < 2ē_{p} \\ p^{k/2} & if k ≡ 0 (mod 2) and k < 2ē_{p} \\ p^{e_{p}} (1 − \frac{1}{p}ψ_{D_{χ}}(p)) a_{p^{k−2ēp}}(χ_{pr}) & if k ≥ 2e_{p} \end{matrix}

(c) Corollary 31 can be viewed as a partial generalization of some of the main results of the articles of Sun and Williams[22], [23]. Indeed, for a negative discriminant D < 0, the function F (A, n) of [22] is essentially the same as the function a_{n}(χ) above. More precisely, if we put χ_{A}(K) = e^{2πi[K,A]}, where [K, A] (which depends on a choice of a “basis” of Cl(D)) is as defined on p. 143 of [22], then a comparison of formula (6) with the formula for F (A, n) on p. 144 of [22] shows that

a_{n}(χ_{A}) = F (A, n), ∀A ∈ Cl(D), n ≥ 1.

In particular, Corollary 31 generalizes Theorem 5.3 of [23] (for D < 0) to the non-cyclic case, and gives a succinct general formula for it (and also for the many special cases discussed in §8 of [22]). In addition, Corollary 31 generalizes (for D < 0) Theorem 4.3 of [22] because a_{n}(χ_{D,0}) = N(n, D)/w(D) in the notation of Example 35(b) and that of [22].

As another application of Theorem 28, we determine when ϑ_{χ} is an eigenfunction under the full Hecke algebra 𝕋_{|D|} = ⟨T_{n} : n ≥ 1⟩ of level |D|.

Corollary 33 If χ ∈ Cl(D)^{∗}, then ϑ_{χ} is a 𝕋_{|D|}-eigenfunction if and only if χ is primitive.

Proof. If χ = χ_{pr} is primitive, then by (34) the L-function of ϑ_{χ} has the Euler product

L(s, χ) = ∏_{p}(1 − a_{p}(χ)p^{−1} + ψ_{D}(p)p^{−2s})^{−1}. (46)

Thus, by a theorem of Hecke (cf. [16], Theorem 4.5.16), it follows that ϑ_{χ} is a 𝕋_{|D|}- eigenfunction.

Conversely, suppose that ϑ_{χ} is a 𝕋_{|D|}-eigenfunction. Then by Hecke’s Theorem again, we know that its L-function L(s, χ) has an Euler product (46). If χ is not primitive, then f̄_{χ} = f_{D}/f_{χ} > 1, so there exists a prime p|f̄_{χ}. Then by (38) we know that a_{p}(χ) = 0, so the p-Euler factor of ϑ_{χ} is trivial, i.e. L_{p}(s, χ) = 1. (Recall that p|f̄_{χ}|D.) Thus by (33) we see that C_{p}(s, χ) = 1 − a_{p}(χ_{pr})p^{−1} + ψ_{D_{χ}}(p)p^{−2s}, i.e. that

c_{p}(χ) = −a_{p}(χ), c_{p^{2}}(χ) = ψ_{D_{χ}}(p) and c_{p^{k}}(χ) = 0, ∀k > 2.

Comparing this to (37) shows that ē_{p} ≤ 1. But if ē_{p} = 1, then by (37) c_{p^{2}}(χ) = p ≠ ψ_{D_{χ}}(p), contradiction. Thus f̄_{χ} = 1, so χ is primitive.

The above corollary allows us to determine when the spaces Θ_{D}, Θ_{D}^{E} and Θ_{D}^{S} are 𝕋_{|D|}-modules.

Proposition 34 (a) The space Θ_{D} is a 𝕋_{|D|}-module if and only if its Eisenstein space Θ_{D}^{E} is a 𝕋_{|D|}-module if and only if D is a fundamental discriminant.

(b) The space Θ_{D}^{S} of cusp forms is a 𝕋_{|D|}-module if and only if every proper discriminental divisor D′ of D idoneal, i.e. iff D satisfies the following condition:

(47) g_{D/p^{2}} = h_{D/p^{2}}, for all primes p such that p^{2}|D and D/p^{2} ≡ 0, 1 (mod 4).

Proof. (a) If D is a fundamental discriminant, then every χ ∈ Cl(D)^{∗} is primitive, and hence by Corollary 33 (and Proposition 9) Θ_{D} has a basis consisting of 𝕋_{|D|}-eigenforms and hence is a 𝕋_{|D|}-module.

Next, suppose that Θ_{D} is a 𝕋_{|D|}-module. Then we have:

(48) ϑ_{χ} is a 𝕋_{|D|}-eigenfunction, ∀χ ∈ Cl(D)^{∗}

To see this, note first f := ϑ_{χ} is a 𝕋(D)-eigenfunction by Theorem 12. Consider n ≥ 1 and put f_{n} := f|T_{n}. By hypothesis, f_{n} ∈ Θ_{D}. Since T_{n} commutes with the operators in 𝕋(D), f_{n} is the same 𝕋(D)-eigenspace as f. By multiplicity 1 (cf. Theorem 1), it follows that f_{n} = c_{n}f, for some c_{n} ∈ ℂ. Thus f = ϑ_{χ} is a 𝕋_{|D|}-eigenfunction, as claimed.

Thus, if Θ_{D} is a 𝕋_{|D|}-eigenspace, then it follows from (48) that Θ_{D}^{E} = ∑_{χ^{2}=1} ℂϑ_{χ} is also a 𝕋_{|D|}-eigenspace.

Now assume that Θ_{D}^{E} is a 𝕋_{|D|}-eigenspace. Then the above proof of (48) yields that for each quadratic χ, the series ϑ_{χ} is a 𝕋_{|D|}-eigenfunction, and hence by Corollary 33 we obtain that all quadratic characters are primitive. In particular, the the trivial character χ = 1 is primitive, which can happen only when D is fundamental; cf. Example 35 below.

(b) By the same argument as in part (a) we see that Θ_{D}^{S} is a 𝕋_{|D|}-module if and only if every non-quadratic χ ∈ Cl(D)^{∗} is primitive. But this means that for each prime p as in (47) we have that Cl(D/p^{2})^{∗} has only quadratic characters, and so Cl(D/p^{2}) is an elementary abelian 2-group, i.e. g_{D/p^{2}} = h_{D/p^{2}}. Thus (47) holds (and conversely).

We now illustrate the main results of this paper by working out some special cases. The first example examines ϑ_{χ} when χ = 1 ∈ Cl(D)^{∗} is the trivial character.

Example 35 (a) If D = d_{K} < 0 is a fundamental discriminant, then the theta function of associated to the trivial character χ_{d_{K},0} = 1 ∈ Cl(d_{K})^{∗} is

ϑ_{K}(z) := ϑ_{χ_{d_{K} ,0}}(z) = f_{1}(z; 1, ψ_{d_{K}}) with L(s, ϑ_{K}) = ζ(s)L(s, ψ_{d_{K}}) = ζ_{K}(s);

i.e. its associated L-function is just the Dedekind ζ-function of K = ℚ(\sqrt{D}). In particular, the T_{p}-eigenvalue of ϑ_{K} is a_{p}(ϑ_{K}) = 1 + (\frac{d_{K}}{p} ), for all primes p.

(b) If D = d_{K}f_{D}^{2} is any negative discriminant, then the trivial character χ_{D,0} = 1 on Cl(D) has conductor f_{χ_{D,0}} = 1, and its associated primitive character is (χ_{D,0})_{pr} = χ_{d_{K},0} = 1 ∈ Cl(d_{K})^{∗}. Thus, by (2), (3) and part (a) we have that

ϑ_{χ_{D,0}}(z) = ∑_{n|f_{D}^{2}} c_{n}(χ_{D,0})ϑ_{K}(nz) and L(s, ϑ_{χ_{D,0}}) = C(s, χ_{D,0})ζ_{K}(s),

where the coefficients c_{n}(χ_{D,0}) of the finite Dirichlet series C(s, χ_{D,0}) are given explicitly by Theorem 28.

We thus obtain an expression for a_{n}(χ_{0,D}) = #Id_{n}(𝔒_{D}) for all n ≥ 1. In particular, the formula of Remark 32(b) complements the expression obtained in the Appendix (cf. Proposition 48). Thus, if p^{e_{p}}||f_{D}, then we have

#I_{p^{k}}(O_{D}) = p^{e_{p}} (1 − \frac{1}{p}ψ_{d_{K}}(p)) #I_{p^{k−2ep}}(𝔒_{K}) = p^{e_{p}} (1 − \frac{1}{p}ψ_{d_{K}}(p)) ∑_{i=0}^{k−2e_{p}} ψ_{d_{K}}(p)^{k},

if k ≥ 2e_{p}. (For k < 2e_{p} we have the expressions #I_{p^{k}}(O_{D}) = 0 and #I_{p^{k}}(O_{D}) = p^{k/2}, for k odd and even, respectively, which are obtained in the Appendix §6.2.)

We now illustrate what happens when the class number h_{D} is small.

Example 36 (a) h_{D} = 1. In this case Cl(D) = {1_{D}} and Cl(D)^{∗} = {χ_{D,0}}. Thus, by (6) we have that ϑ_{1_{D}} = w_{D}ϑ_{χ_{D,0}}. If D = d_{K} is a fundamental discriminant (i.e. if −D = 3, 4, 7, 8, 11, 19, 43, 67, 163), then we have by Example 35(a) that

ϑ_{χ_{D,0}} = ϑ_{K} and L(s, χ_{D,0}) = ζ_{K}(s).

In the remaining cases (i.e. −D = 12, 16, 27, 28) this formula is no longer true. Indeed, since f = f_{D} ∈ {2, 3} is here a prime, we obtain from Theorem 28 that C(s, χ_{D,0}) = 1−a_{D}f^{−s}+f^{1−2s}, where a_{D} := a_{f}(χ_{d_{K},0}) = 1+(\frac{d_{K}}{f} ), (i.e. a_{D} = 0, 1, 1, 2, respectively), and so

ϑ_{χ_{D,0}}(z) = ϑ_{K}(z) − a_{D}ϑ_{K}(fz) + fϑ_{K}(f^{2}z), L(s, ϑ_{χ_{D,0}}) = (1 − a_{D}f^{−s} + f^{1−2s})ζ_{K}(s).

(b) h_{D} = 2. Here Cl(D) = {1_{D}, cl(f)} and Cl(D)^{∗} = {χ_{D,0}, χ}. Since D < −4, we have that w_{D} = 2 and hence it follows from (6) that

ϑ_{χ_{D,0}} = \frac{1}{2}(ϑ_{1_{D}} + ϑ_{f}) and ϑ_{χ} = \frac{1}{2}(ϑ_{1_{D}} − ϑ_{f}).

Since g_{D} = h_{D} = 2, we know by genus theory that D = D_{1}D_{2}c^{2}, where D_{1} < 0 and D_{2} > 0 are fundamental discriminants (with (D_{1}, D_{2}) = 1) and c ≥ 1, and that the associated Hecke character χ̃ satisfies χ̃(𝔭) = ψ_{D_{i}}(N_{K}(𝔭)), for i = 1, 2 and 𝔭 ∤ D. Thus, f_{χ} = D_{1}D_{2} and f̄_{χ} = c, and

ϑ_{χ_{pr}}(z) = f_{1}(z; ψ_{D_{1}}, ψ_{D_{2}}) and L(s, ϑ_{χ_{pr}}) = L(s, ψ_{D_{1}})L(s, ψ_{D_{2}}).

In particular, the T_{p}-eigenvalue of ϑ_{χ} is a_{p}(χ) = (\frac{D_{1}}{p} ) + (\frac{D_{2}}{p} ), for p ∤ D. Note that if χ is primitive, i.e. if c = 1, then ϑ_{χ} = ϑ_{χ_{pr}}. If c > 1, then a check of all 29 D’s with h_{D} = 2 shows that necessarily D = −60, and then ϑ_{χ} is worked out in Example 37(b).

(c) h_{D} = 3. Here Cl(D) = {1_{D}, cl(f), cl(f)^{−1}} and Cl(D)^{∗} = {χ_{D,0}, χ, χ^{−1}}. Since Re(χ(f)) = Re(\frac{−1±\sqrt{−3}}{2}) = −\frac{1}{2}, we have by (7) that

ϑ_{χ_{D,0}} = \frac{1}{2}(ϑ_{1_{D}} + 2ϑ_{f}) and ϑ_{χ} = ϑ_{χ^{−1}} = \frac{1}{2}(ϑ_{1_{D}} − ϑ_{f}).

By Remark 17(a) we know that Θ_{D}^{E} = ℂϑ_{χ_{D,0}} and that Θ_{D}^{S} = ℂϑ_{χ}; in particular, ϑ_{χ} is a cusp form and a 𝕋(D)-eigenfunction; cf. Theorem 1. Moreover, Corollary 29 shows that ϑ_{χ} is a newform of level |D| if and only if χ is primitive, i.e. if and only there is no c > 1 such that h_{D/c^{2}} = 3. This condition holds for 23 of the cases with h_{D} = 3 (i.e. for −D = 23, 31, 44, 59, 76, 83, 92, 107, 108, 124, 139, 172, 211, 243, 268, 283, 307, 331, 379, 499, 547, 643, 652, 883, 907) but fails for the two cases −D = 92 = 2^{2} · 23 and −D = 124 = 2^{2} · 31.

We complement the above discussion by working out two numerical examples.

Example 37 (a) D = −23. Here Cl(D) = {[1, 1, 6], [2, 1, 3], [2, −1, 3]}, where [a, b, c] denotes the equivalence class of the form ax^{2} + bxy + cy^{2}, and so h_{D} = 3. Since D is a fundamental discriminant, we have by Examples 35(a) and 36(c) that ϑ_{χ_{D,0}} = ϑ_{K} = f_{1}( ·; 1, ψ_{−23}) and ϑ_{χ} are primitive forms of level 23, and that ϑ_{χ} is a cusp form. (In fact, N = |D| = 23 is the smallest level for which Θ_{D}^{S} ≠ {0}.) Moreover,

ϑ_{χ_{D,0}} = ϑ_{K} = \frac{1}{2}(ϑ_{[1,1,6]} + 2ϑ_{[2,1,3]}) and ϑ_{χ} = \frac{1}{2}(ϑ_{[1,1,6]} − ϑ_{[2,1,3]}),

and from this (or from (8)) we obtain that

ϑ_{[1,1,6]} = \frac{2}{3}(ϑ_{K} + 2ϑ_{χ}) and ϑ_{[2,1,3]} = \frac{2}{3}(ϑ_{K} − ϑ_{χ}).

(b) D = −60 = −15 · 2^{2}. Here Cl(D) = {[1, 0, 15], [3, 0, 5]}, so h_{D} = 2 and Cl(D)^{∗} = {χ_{D,0}, χ}. Since h_{d_{K}} = h_{−15} = 2, we see that f_{χ_{D,0}} = f_{χ} = 1 and so both χ_{D,0} and χ are imprimitive with f̄_{χ_{D,0}} = f̄_{χ} = 2. The theta-functions of the associated primitive characters are ϑ_{K} = f_{1}( ·; 1, ψ_{−15}) and ϑ_{χ_{pr}} = f_{1}( ·; ψ_{−3}, ψ_{5}), respectively. Since a_{2}(ϑ_{K}) = 1 + (\frac{−15}{2}) = 2 and a_{2}(χ_{pr}) = (\frac{−3}{2}) + (\frac{5}{2}) = −2, it follows from Theorem 28 that C(s, χ_{D,0}) = 1 − 2 · 2^{−s} + 2 · 2^{−2s} and that C(s, χ) = 1 + 2 · 2^{−s} + 2 · 2^{−2s}. Thus,

ϑ_{χ_{D,0}}(z) = ϑ_{K}(z)−2ϑ_{K}(2z)+2ϑ_{K}(4z) and ϑ_{χ}(z) = ϑ_{χ_{pr}}(z)+2ϑ_{χ_{pr}}(2z)+2ϑ_{χ_{pr}}(4z),

and the associated L-functions are

L(s, χ_{D,0}) = (1 − 2^{1−s} + 2^{1−2s})ζ_{K}(s) and L(s, χ) = (1 + 2^{1−s} + 2^{1−2s})L(s, χ_{pr}),

where L(s, χ_{pr}) = L(s, ψ_{−3})L(s, ψ_{5}).

6 Appendix: Ideals of Quadratic Orders

The purpose of this appendix is to collect some well-known results about ideals in an order 𝔒_{D} of an imaginary quadratic field K, and to extend these to obtain the results which were used in this article.

6.1 Lattices in an imaginary quadratic field

Let K = ℚ(\sqrt{d_{K}}) be an imaginary quadratic field of discriminant d_{K} < 0, and let 𝔒_{K} = ℤ + ω_{K}ℤ denote its ring of integers, where ω_{K} = \frac{1}{2}(d_{K} + \sqrt{d_{K}}). For any f ≥ 1, put D = f^{2}d_{K} and

𝔒_{D} = ℤ + fω_{K}ℤ = ℤ + ω_{D}ℤ, where ω_{D} = \frac{1}{2}(D + \sqrt{D}).

It is immediate that 𝔒_{D} is a subring of 𝔒_{K} of index f_{𝔒_{D}} := [𝔒_{K} : 𝔒_{D}] = f. The ring 𝔒_{D} is called the order of discriminant D (or of conductor f), and it is well-known that every subring R of 𝔒_{K} with quotient field K is of this form, i.e. R = 𝔒_{D} with D = f_{R}^{2}d_{K}, where f_{R} = [𝔒_{K} : R]. (For such and other basic facts about orders see [26], §90-113, [1], §II.7, [15], §8.1.)

Let Lat(K) denote the set of all lattices in K, i.e. the set of all finitely generated ℤ-modules which contain some ℚ-basis of K.

If L ∈ Lat(K) is a lattice, then its associated order is 𝔒(L) = {λ ∈ K : λL ⊂ L}. It is easy to see that 𝔒(L) is an order of K, i.e. 𝔒(L) = 𝔒_{D}, for some D = f^{2}d_{K}.

The norm of the lattice L is by defined by N(L) = | det(T)|, where T ∈ GL_{2}(ℚ) is such that T (𝔒(L)) = L; cf. [1], §II.6. Note that if L ⊂ 𝔒_{K}, then

[𝔒_{K} : L] = [O_{K} : 𝔒(L)]N(L). (49)

If L_{1}, L_{2} ∈ Lat(K) are two lattices, then the product (module) L_{1}L_{2} is again a lattice. By [1], §II.7, Ex. 6 and 10, its order and norm are given by the formulae

𝔒(L_{1}L_{2}) = 𝔒(L_{1})𝔒(L_{2}) and N(L_{1}L_{2}) = N(L_{1})N(L_{2}). (50)

In particular, we see that the set I(𝔒_{D}) = {L ∈ Lat(K) : 𝔒(L) = 𝔒_{D}} is closed under multiplication. In fact, I(𝔒_{D}) is a group with unit 𝔒_{D}: it can be identified with the group of invertible 𝔒_{D}-submodules of K; cf. [15], p. 91. Thus, if P (𝔒_{D}) = {λ𝔒_{D} : λ ∈ K^{×}} denotes the subgroup of principal 𝔒_{D}-modules, then the quotient

Cl(𝔒_{D}) = Pic(𝔒_{D}) = I(𝔒_{D})/P (𝔒_{D})

is called the class group (or Picard group) of the order 𝔒_{D}, and its (group) order h_{D} := |Cl(𝔒_{D})| is called the class number of 𝔒_{D}.

If 𝔒_{D_{i}} are two orders of K with discriminants D_{i} = f_{D_{i}}^{2}d_{K}, then

(51) 𝔒_{D_{1}} ⊃ 𝔒_{D_{2}} if and only if f_{D_{1}} | f_{D_{2}},

and hence

(52) 𝔒_{D_{1}} ∩ 𝔒_{D_{2}} = 𝔒_{lcm(f_{D_{1}},f_{D_{2}})^{2}d_{K}} and 𝔒_{D_{1}}𝔒_{D_{2}} = 𝔒_{gcd(f_{D_{1}},f_{D_{2}})^{2}d_{K}}.

Proposition 38 If R and R′ are two orders of K with R ⊂ R′, then the rule L ↦ LR′ defines a surjective homomorphism π_{R,R′} : I(R) → I(R′) with kernel

Ker(π_{R,R′}) = {L ∈ I(R) : L ⊂ R′ and [R′ : L] = [R′ : R]}. (53)

Proof. If L ∈ I(R), then by (50) we have 𝔒(LR′) = 𝔒(L)𝔒(R′) = RR′ = R′, so LR′ ∈ I(R′). Thus, the rule L ↦ LR′ defines a map π = π_{R,R′} : I(R) → I(R′). Moreover, π is a homomorphism because π(L_{1})π(L_{2}) = L_{1}R′L_{2}R′ = L_{1}L_{2}R′R′ = L_{1}L_{2}R′ = π(L_{1}L_{2}).

To prove (53), let L ∈ Ker(π). Then LR′ = R′. Since R′ is an order, we have that N(R′) = [𝔒(R′) : R′] = [R′ : R′] = 1, and so by (50) we have that N(L) = N(L)N(R′) = N(LR′) = N(R′) = 1. Thus, since 𝔒(L) = R, we see that [R′ : R] = [R′ : R]N(L) = [R′ : R][R : L] = [R′ : L]. Moreover, since L = L · 1 ⊂ LR′ = R′ it follows that L ⊂ R′, and hence L ∈ K := {L ∈ I(R) : L ⊂ R′ and [R′ : L] = [R′ : R]}.

Conversely, if L ∈ K, then N(L) = [R : L] = [R : L]/[R′ : R] = 1, and hence N(LR′) = N(L)N(R′) = 1 · 1 = 1. Now since 𝔒(LR′) = 𝔒(L)𝔒(R′) = RR′ = R′, we have that 1 = N(LR′) = [R′ : LR′]. But since L ⊂ R′, we have LR′ ⊂ R′ and so this forces LR′ = R′. Thus L ∈ Ker(π), which proves (53).

It remains to show that π is surjective. Let L′ ∈ I(R′). Then by [15], Theorem 5 (p. 93), there exists λ ∈ K^{×} such that λL′ + fR′ = R′, where f = [𝔒_{K} : R]. We then have that L := λL′ ∩ R ∈ I(R) and that LR′ = λL′ by applying [15], Theorem 4 (p. 92), to R′ and R. (See also Proposition 48 below.) Thus π(λ^{−1}L) = (λ^{−1}L)R′ = L′, and so π is surjective.

Remark 39 Since π_{R,R′}(λR) = λR′, for all λ ∈ K^{×}, it is clear that π_{R,R′} induces a surjective homomorphism

\overline{π}_{R,R′} : Cl(R) = I(R)/P (R) → Cl(R′) = I(R′)/P (R′). (54)

Moreover, it is immediate that

Ker(\overline{π}_{R,R′}) = Ker(π_{R,R′})P (R)/P (R) (55)

because if LP (R) ∈ Ker(\overline{π}_{R,R′}), then L = λR′, for some λ ∈ K^{×}, and then LP (R) = (λ^{−1}L)(λR)P (R) ∈ Ker(π_{R,R′})P (R)/P (R), so (55) follows.

Corollary 40 In the above situation put f̄ = [R′ : R]. Then the map λ ↦ L_{λ} := ℤλ + f̄R′ induces an exact sequence

0 ⟶ (ℤ/f̄ℤ)^{×} ⟶ (R′/f̄R′)^{×} ⟶^{L_{R′,f̄}} Ker(π_{R,R′}) ⟶ 0. (56)

Thus, if D′ denotes the discriminant of R′ = 𝔒_{D′} and if ψ_{R′}(p) = (\frac{D′}{p} ) denotes the associated Legendre-Kronecker symbol, then

|Ker(π_{R,R′})| = f̄∏_{p|f̄} (1 − \frac{1}{p}ψ_{R′}(p)) , (57)

Proof. First note that if λ ∈ R′, then L_{λ}/f̄R′ ≤ R′/f̄R′ is the cyclic subgroup generated by λ̄ = λ + f̄R′, so L_{λ} = Lλ̄ depends only on λ̄. Moreover, since λf̄R′ = f̄R′ if λ̄ ∈ (R′/f̄R′)^{×}, it is clear that L_{λ}L_{λ′} = L_{λλ′}, and so λ ↦ L_{λ} defines a homomorphism L_{R′,f} : (R′/f̄R′)^{×} → Lat(K). To see that Im(L_{R′,f}) ⊂ Ker(π_{R,R′}), note first that L_{λ} is clearly an R-module because R = ℤ + f̄R′. Moreover, since λλ′ ≡ 1 (mod f̄R′), for some λ′ ∈ R′, we see that L_{λ}L_{λ′} = L_{1} = R, and so L_{λ} ∈ I(R). In addition, L_{λ}R′ = R′ because L_{λ}R′ is an R′-ideal which contains 1, and so L_{λ} ∈ Ker(π_{R,R′}).

To show that L_{R′,f̄} is surjective, let L ∈ Ker(π_{R,R′}). Then by (53) we know that L ⊂ R′ and that [R′ : L] = f̄, so L ⊃ f̄R′ and [L : f̄R′] = \frac{[R′:f̄R′]}{[R′:L]} = \frac{f̄^{2}}{f̄} = f̄. Since LR′ = R′, we see that L ⊄ pR′, for any prime p|f̄, and hence L/f̄R′ is cyclic, i.e. L = ℤλ + f̄R′ for some λ ∈ R′. Since |L/f̄R′| = f̄, this means that λ has order f̄ in R′/f̄R′ ≃ ℤ/f̄ℤ × ℤ/f̄ℤ. Then λ = a + bf̄ω_{D′} with a, b ∈ ℤ and (a, b, f̄) = 1, where D′ is the discriminant of R′ = 𝔒_{D′}, and so it follows easily that λ ∈ (R′/f̄R′)^{×}. Thus L = L_{λ} = L_{R′,f̄}(λ), and hence L_{R,f̄} is surjective.

Next we observe that λ̄ ∈ Ker(L_{R′,f̄}) ⇔ Lλ̄ = R = ℤ + f̄R′ ⇔ λ̄ = n + f̄R′, for some n ∈ ℤ with (n, f̄) = 1. Thus, since the map ℤ/f̄ℤ → R′/f̄R′ is injective, it follows that the sequence (56) is exact.

It remains to verify (57). Applying (56) with R′ = 𝔒_{K} (and hence f̄ = f := f_{R}) yields that

|Ker(π_{R,𝔒_{K}})| = \frac{ϕ(f𝔒_{K})}{ϕ(f)} = f ∏_{p|f} (1 − \frac{1}{p}ψ_{𝔒_{K}}(p)) ,

where the second formula is deduced as in [15], p. 95. (Here, as in [15], ϕ(f𝔒_{K}) = |(𝔒_{K}/f𝔒_{K})^{×}|.) From this (and the surjectivity of π_{R,R′}) it follows that for any R′ ⊃ R we have

|Ker(π_{R,R′})| = \frac{|Ker(π_{R,𝔒_{K}})|}{|Ker(π_{R′,𝔒_{K}})|} = f̄ ∏_{p|f_{R},p∤f_{R′}} (1 − \frac{1}{p}ψ_{d_{K}}(p)) = f̄∏_{p|f̄} (1 − \frac{1}{p}ψ_{R′}(p)) .

Corollary 41 If R ⊂ R_{1} ∩ R_{2} and R_{3} = R_{1}R_{2}, then

Ker(π_{R,R_{3}}) = Ker(π_{R,R_{1}}) · Ker(π_{R,R_{2}}) and Ker(\overline{π}_{R,R_{3}}) = Ker(\overline{π}_{R,R_{1}}) · Ker(\overline{π}_{R,R_{2}}).

Proof. By (55) it is enough to prove the first assertion. Moreover, since Ker(π_{R,R_{i}}) = π_{R,R_{1}∩R_{2}}^{−1}(Ker(π_{R_{1}∩R_{2},R_{i}})), for i = 1, 2, 3, it suffices to verify the formula for R = R_{1}∩R_{2}.

For this, write f = [𝔒_{K} : R], f_{i} = [𝔒_{K} : R_{i}] and f̄_{i} = \frac{f}{f_{i}} = [R_{i} : R], for i = 1, 2, 3. Then by (52) we have that f_{3} = (f_{1}, f_{2}). Since R = R_{1} ∩ R_{2}, we have that f = \frac{f_{1}f_{2}}{f_{3}} by (52), so f̄_{3} = f̄_{1}f̄_{2} and (f̄_{1}, f̄_{2}) = 1. It thus follows from (57) that |Ker(π_{R,R_{3}})| = |Ker(π_{R,R_{1}})| · |Ker(π_{R,R_{2}})|. Thus, the assertion follows once we have shown that Ker(π_{R,R_{1}}) ∩ Ker(π_{R,R_{2}}) = {R}. Now if L ∈ Ker(π_{R,R_{1}}) ∩ Ker(π_{R,R_{2}}), then L ⊂ R_{1} ∩ R_{2} = R. But since [R : L] = N(L) = 1 (the latter because LR_{i} = R_{i}), it follows that L = R, and hence Ker(π_{R,R_{1}}) ∩ Ker(π_{R,R_{2}}) = {R}, as claimed.

6.2 The ideal theory of 𝔒_{D}

We now study the ideals of the order R = 𝔒_{D} ⊂ K more closely. For this, we shall use the following general result which is applicable here because 𝔒_{D} is a noetherian domain of (Krull) dimension 1.

Proposition 42 Let A be a noetherian domain of dimension 1, and let 𝔞 ⊂ A be a non-zero ideal of A. If 𝔪 ∈ max(A) is any maximal ideal of A with 𝔞 ⊂ 𝔪, then 𝔞(𝔪) := 𝔞A_{𝔪} ∩ A is an 𝔪-primary ideal of A. In particular, if |A/𝔪| < ∞, then |A/𝔞(𝔪)| = |A/𝔪|^{s}, for some integer s = s_{𝔪}(𝔞) ≥ 1. Moreover, we have

𝔞 = ⋂_{𝔪}𝔞(𝔪) = ∏_{𝔪}𝔞(𝔪). (58)

Thus 𝔞 is an invertible ideal if and only 𝔞(𝔪) is invertible for all maximal ideals 𝔪.

Proof. Since A_{𝔪} is a local 1-dimensional noetherian domain, every proper ideal of A_{𝔪} is primary, and hence 𝔞(𝔪) is primary ideal of A with radical 𝔪. This proves the first assertion. From this the second assertion is an immediate consequence because by [2], p. 265, the A-module M = A/𝔞(𝔪) has a composition series {M_{i}} with M_{i}/M_{i+1} ≃ A/𝔪, and so the assertion follows (with s = length(M)).

The first identity of (58) is true in any domain; cf. [2], p. 89. Next, if 𝔪 ≠ 𝔪′ ∈ max(A) are two distinct maximal ideals, then 𝔞(𝔪) ⊄ 𝔪′ and hence 𝔞(𝔪)A_{𝔪′} = A_{𝔪′}. Thus, if we put 𝔞′ = ∏ 𝔞(𝔪), then 𝔞′A_{𝔪} = 𝔞(𝔪)A_{𝔪} = 𝔞A_{𝔪}, for every 𝔪 ∈ max(A), and so 𝔞′ = 𝔞, which proves the second equality of (58).

Finally, if 𝔞(𝔪) is invertible for all 𝔪 ∈ max(A), then 𝔞 is invertible by (58). Conversely, suppose that 𝔞 is invertible and fix 𝔪 ∈ max(A) with 𝔪 ⊃ 𝔞. Then 𝔞(𝔪)A_{𝔪} = 𝔞A_{𝔪} is principal, and for every 𝔪′ ≠ 𝔪 we have 𝔞(𝔪)A_{𝔪′} = A_{𝔪′}. Thus 𝔞(𝔪) is locally principal and hence invertible; cf. [2], p. 117.

Notation. Let Id(R) = {L ∈ I(R) : L ⊂ R} denote the set of invertible R-ideals. Moreover, for any integer n ≥ 1, let Id_{n}(R) = {𝔞 ∈ Id(R) : N(𝔞) = n} denote the set of invertible R-ideals of norm n, and let Id(R, n) = {𝔞 ∈ Id(R) : (N(𝔞), n) = 1} denote the set of invertible R-ideals whose norm is relatively prime to n.

Corollary 43 Let 𝔞 ∈ Id(R) be an invertible R-ideal, and 𝔪 ∈ max(R) be a maximal ideal, where R = 𝔒_{D}. Then 𝔞(𝔪) ∈ Id(R) is also invertible and N(𝔞(𝔪)) = p^{s}, for some s, where p|𝔪, i.e. p is the unique prime number with p ∈ 𝔪. Furthermore,

𝔞 = ∏_{𝔪}𝔞(𝔪) and hence N(𝔞) = ∏_{𝔪}N(𝔞(𝔪)).

In particular, if 𝔪 ∈ supp(𝔞) := {𝔪 ∈ max(R) : 𝔞 ⊂ 𝔪}, and p|𝔪, then p|N(𝔞).

Proof. If 𝔪 ∈ max(R), then R/𝔪 is a finite field of characteristic p, where p ∈ 𝔪. Thus, since R is a 1-dimensional noetherian ring and since N(𝔞) = [R : 𝔞], if 𝔞 ∈ Id(R), the corollary follows immediately from Proposition 42 (together with (50)).

We now want to study the set Id_{n}(R) of invertible ideals of norm n of R more closely. First we note:

Proposition 44 If (m, n) = 1, then the map (𝔞, 𝔟) ↦ 𝔞𝔟 induces a bijection

Id_{m}(R) × Id_{n}(R) →^{∼} Id_{mn}(R).

Proof. If (𝔞, 𝔟) ∈ Id_{m}(R) × Id_{n}(R), then 𝔞𝔟 ∈ Id_{mn}(R) by (50). Suppose now that 𝔠 ∈ Id_{mn}(R), and put, for an integer k, supp_{k}(𝔠) := supp(𝔠 + kR) = {𝔪 ∈ supp(𝔠) : p|𝔪 ⇒ p|k} and 𝔠_{k} = ∏_{𝔪∈supp_{k}(𝔠)} 𝔞(𝔪). Then by Corollary 43 we have supp(𝔠) = supp(𝔠)_{m} ˙∪ supp(𝔠)_{n}, and so 𝔠 = 𝔠_{m}𝔠_{n}. Since (N(𝔠_{m}), n) = (N(𝔠_{n}), m) = 1, we see that (𝔠_{m}, 𝔠_{n}) ∈ Id_{m}(R) × Id_{n}(R), and so the map is surjective. It is injective because 𝔠 is uniquely determined by its local components 𝔠(𝔪).

Thus, by the above result, it is enough to study the sets Id_{n}(R) for prime powers n = p^{r}. If p is prime to the conductor f_{R} of R, i.e. if p ∤ f_{R} = [𝔒_{K} : R], then Id_{n}(R) is essentially the same as Id_{n}(𝔒_{K}), as the next (well-known) result shows.

Proposition 45 The rule 𝔞 ↦ 𝔞 ∩ R induces an injection φ̃_{R} : Id(𝔒_{K}, f_{R}) ↪ Id(R) with image Id(R, f_{R}), and we have that

φ̃_{R}(Id_{n}(𝔒_{K})) = Id_{n}(R), for all n ≥ 1, (n, f_{R}) = 1. (59)

Thus

#Id_{n}(R) = #Id_{n}(𝔒_{K}) = ∑_{d|n} ψ_{d_{K}}(d), if (n, f_{R}) = 1. (60)

Proof. The first assertion follows from [15], Theorem 4 (p. 92). Moreover, since that theorem also asserts that φ̃_{R}(𝔞)𝔒_{K} = 𝔞, if 𝔞 ∈ Id(𝔒_{K}, f_{R}), it follows from (50) that N(φ̃_{R}(𝔞)) = N(φ̃_{R}(𝔞))N(𝔒_{K}) = N(𝔞). Thus (59) holds, and hence the first equality of (60) follows. The second equality of (60) is well-known; cf. Weber[26], p. 345.

We now turn to study the ideals of R = 𝔒_{D} with norm dividing the conductor f_{R} = f_{D}. Here we first prove:

Proposition 46 If p|f_{D} is a prime divisor of the conductor f_{D} = [𝔒_{K} : 𝔒_{D}] of 𝔒_{D}, then 𝔪_{p} := p𝔒_{D/p^{2}} is the unique maximal ideal of 𝔒_{D} containing p. Moreover, |𝔒_{D}/𝔪_{p}| = p and 𝔪_{p} is an 𝔒_{D}-ideal which is not invertible, i.e. 𝔪_{p} ∉ Id(𝔒_{D}). In particular, Id_{p}(𝔒_{D}) = ∅.

Proof. Since 𝔪_{p} = p(ℤ + (f_{D}/p)ω_{K}ℤ) = pℤ + f_{D}ω_{K}ℤ, we see that 𝔪_{p} ⊂ 𝔒_{D} = ℤ + f_{D}ω_{K}ℤ and that [𝔒_{D} : 𝔪_{p}] = p. Clearly, 𝔪_{p} is a principal 𝔒_{D/p^{2}}-ideal, and hence a fortiori an 𝔒_{D}-ideal. Thus, 𝔪_{p} is a maximal ideal of 𝔒_{D} and 𝔒(𝔪_{p}) = 𝔒_{D/p^{2}}, so 𝔪_{p} cannot be invertible as an 𝔒_{D}-ideal.

Now suppose that 𝔪 ⊃ p𝔒_{D} is a maximal ideal. Then (by Cohen-Seidenberg) 𝔪 = 𝔪𝔒_{D/p^{2}} ∩ 𝔒_{D} ⊃ p𝔒_{D/p^{2}} ∩ 𝔒_{D} = 𝔪_{p}, and so 𝔪 = 𝔪_{p}. Thus 𝔪_{p} is the unique maximal ideal containing p.

Finally, if 𝔞 ∈ Id_{p}(𝔒_{D}), then [𝔒_{D} : 𝔞] = N(𝔞) = p, so 𝔞 is a maximal ideal of 𝔒_{D} with p ∈ 𝔞, and hence 𝔞 = 𝔪_{p}. But 𝔪 ∉ Id(𝔒_{D}), contradiction. Thus Id_{p}(𝔒_{D}) = ∅.

Corollary 47 Suppose that c|f_{D} = [𝔒_{K} : 𝔒_{D}]. If 𝔞 ∈ Id_{n}(𝔒_{D}) and c^{2}|n, then 𝔞𝔒_{D/c^{2}} = c𝔟, for some 𝔟 ∈ Id_{n/c^{2}}(𝔒_{D/c^{2}}).

Proof. Suppose first that c = p. Then by Corollary 43 and Proposition 46 we know that 𝔞 ⊂ 𝔪_{p} = p𝔒_{D/p^{2}}, and so 𝔟′ := 𝔞𝔒_{D/p^{2}} ⊂ p𝔒_{D/p^{2}}. Thus 𝔟 := \frac{1}{p}𝔟′ ⊂ 𝔒_{D/p^{2}}, and hence 𝔟 ∈ Id_{n/p^{2}}(𝔒_{D/p^{2}}) and 𝔞𝔒_{D/p^{2}} = p𝔟. Thus the assertion holds for c = p.

To prove the general case, induct on c. Since the assertion is vacuous for c = 1, we may assume that there is a prime p|c. Put D′ = D/c^{2}, D̄ = D/p^{2} and n̄ = n/p^{2}. Then, by what was just shown, 𝔞𝔒D̄ = p𝔞̄ with 𝔞̄ ∈ Idn̄(𝔒D̄). Put c̄ = c/p. Then c̄|\frac{f}{p} = [𝔒_{K} : 𝔒D̄] and c̄^{2}|n̄, so by induction 𝔞̄𝔒_{D′} = c̄𝔟 with 𝔟 ∈ Id_{n̄/c̄^{2}}(𝔒_{D}). Thus 𝔞𝔒_{D′} = (p𝔞̄)𝔒_{D′} = pc̄𝔟 = c𝔟, and so the assertion also holds for c.

As we shall now see, the above results show that when n|f_{D}^{2}, there is close connection between Id_{n}(𝔒_{D}) and Ker(π_{𝔒_{D},𝔒_{D/n}}).

Proposition 48 Suppose that n|f_{D}^{2} = D/d_{K}. If n is not a square, then Id_{n}(𝔒_{D}) = ∅, whereas for n = c^{2} we have that

Id_{c^{2}}(𝔒_{D}) = cKer(π_{𝔒_{D},𝔒_{D/c^{2}}}).

Proof. Suppose first that n is not a square, i.e. suppose that there is a prime p such that p^{2r+1} ∥ n, for some r ≥ 0. If 𝔞 ∈ Id_{n}(𝔒_{D}), then 𝔞_{p} := 𝔞(𝔪_{p}) ∈ Id_{p^{2r+1}}(𝔒_{D}), and hence, putting D′ = D/p^{2r}, we have by Corollary 47 that 𝔞_{p}𝔒_{D′} = p^{r}𝔟, with 𝔟 ∈ Id_{p}(𝔒_{D′}). But since p|[𝔒_{K} : 𝔒_{D′}], this contradicts Proposition 46. Thus Id_{n}(𝔒_{D}) = ∅.

Now suppose that n = c^{2}, and put D′ = D/c^{2}. If 𝔞 ∈ Id_{n}(𝔒_{D}), then by Corollary 47 we have that 𝔞𝔒_{D′} = c𝔟 with 𝔟 ∈ Id_{1}(𝔒_{D′}), i.e. 𝔞𝔒_{D′} = c𝔒_{D′}, and hence 𝔞 = cL with L = \frac{1}{c}𝔞 ∈ Ker(π_{𝔒_{D},𝔒_{D′}}). Thus Id_{n}(𝔒_{D}) ⊂ cKer(π_{𝔒_{D},𝔒_{D′}}).

Conversely, if L ∈ Ker(π_{𝔒_{D},𝔒_{D′}}), then L ⊂ 𝔒_{D′} (cf. (53)), so cL ⊂ c𝔒_{D′} ⊂ 𝔒_{D}, the latter because c = [𝔒_{D′} : 𝔒_{D}]. Thus, cL ∈ Id_{n}(𝔒_{D}) because N(cL) = c^{2}N(L) = c^{2} = n, and hence Id_{n}(𝔒_{D}) = cKer(π_{D,D′}), as claimed.

Acknowledgments. I would like to thank Norm Hurt for his comments and notes on the first version of this paper, and for drawing my attention to the article of Sun and Williams.

References

[1] Z. Borevich, I. Shafarevich, Number Theory. Academic Press, New York, 1966.

[2] N. Bourbaki, Commutative Algebra, Ch. I-VII. Addison-Wesley, Reading, 1972.

[3] D. Cox, Primes of the Form x^{2} + ny^{2}: Fermat, Class Field Theory and Complex Multiplication. Wiley, New York, 1989.

[4] H. Davenport, H. Heilbronn, On the zeros of certain Dirichlet series I. J. London Math. Soc. 11 (1936), 181–185 = The Collected Papers of Hans Arnold Heilbronn, Wiley-Interscience, New York, 1988, pp. 272–276.

[5] G. Lejeune Dirichlet, Recherches sur diverses applications de l’analyse infinitésimale à la theorie des nombres. J. reine angew. Math. 19 (1939), 324–369; 21 (1940), 1–12, 134–155 = Werke I, Reimer, Berlin, 1889, pp. 413–496.

[6] C.F. Gauss, Untersuchungen über höhere Arithmetik. Translation (1889) of Disquisitiones Arithmeticae (1801) by H. Maser. Reprint: Chelsea Publ. Co., New York, 1981.

[7] G. Hardy, E. Wright, An Introduction to the Theory of Numbers (Fourth ed.) Oxford U. Press, Glasgow, 1968.

[8] E. Hecke, Zur Theorie der elliptischen Modulfunktionen. Math. Ann. 97 (1926), 210–242 = Math. Werke, Vandenhoeck & Ruprecht, Göttingen, 1983, pp. 428– 460.

[9] E. Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung I, II. Math. Ann. 114 (1937), 1–28, 316–351 = Math. Werke, Vandenhoeck & Ruprecht, Göttingen, 1983, pp. 644–707.

[10] H. Hijikata, A. Pizer, T. Schemanske, The basis problem for modular forms on Γ_{0}(N). Memoirs AMS 418 (1989).

[11] E. Kani, Idoneal numbers and some generalizations. Preprint, 34 pages. To appear in: Ann. Sci. Math. Quebec.

[12] E. Kani, Binary theta series and CM forms. Preprint, 16 pages.

[13] Y. Kitaoka, A note on Hecke operators and theta-series. Nagoya Math. J. 42 (1971), 189-195.

[14] L. Kronecker, Zur Theorie der elliptischen Funktionen. Sitzungsber. Köngl. Preuss. Akad. Wiss. 1883 – 1889 = Werke, vol. IV, Teubner, Leizig, 1897, pp. 347–495.

[15] S. Lang, Elliptic Functions. Addison-Wesley, Reading, MA, 1972.

[16] T. Miyake, Modular Forms. Springer-Verlag, Berlin, 1989.

[17] J. Piehler, Über Primzahldarstellungen durch binäre quadratische Formen. Math. Ann. 141 (1960), 239–241.

[18] B. Schoeneberg, Das Verhalten mehrfacher Thetareihen bei Modulsubstitutionen. Math. Ann. 116 (1939), 511–523.

[19] B. Schoeneberg, Elliptic Modular Functions. Springer-Verlag, Berlin, 1974.

[20] J.-P. Serre, Modular forms of weight 1 and Galois representations. In: Algebraic Number Fields (1977), pp. 193–268 = Œuvres/Collected Papers III, Springer-Verlag, Berlin, 1986, pp. 292–367.

[21] C.-L. Siegel, Über die analytische Theorie der quadratischen Formen. Ann. Math. 36 (1935), 527–606 = Gesam. Abh. I, Springer-Verlag, Berlin, 1966, pp. 322–405.

[22] Z.-H. Sun, K. S. Williams, On the number of representations of n by ax^{2} + bxy + cy^{2}. Acta Arith. 122 (2006), 101–171.

[23] Z.-H. Sun, K. S. Williams, Ramanujan identities and Euler products for a type of Dirichlet series. Acta Arith. 122 (2006), 349–393.

[24] H. Weber, Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist. Math. Ann. 20 (1882), 301– 329.

[25] H. Weber, Zahlentheoretische Untersuchungen aus dem Gebiete der elliptischen Funktionen III. Nachrichten Königl. Gesell. Wiss. Univ. Göttingen 7 (1893), 245– 264.

[26] H. Weber, Lehrbuch der Algebra III. Teubner, 1908. Chelsea Reprint, New York, 1961.