On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

Ick Sun Eum; Ho Yun Jung

doi:10.1515/math-2019-0115

Article Open Access

On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

Ick Sun Eum and Ho Yun Jung

Published/Copyright: December 31, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

Keywords: modular forms; modular traces; Galois traces; class field theory

MSC 2010: Primary 11F37; Secondary 11F30; 11G15; 11R27; 11R37

1 Introduction

Let D be a negative integer with D ≡ 0, 1 (mod 4) so that D is an imaginary quadratic discriminant. More explicitly, if we let

τD=D2if D≡0(mod4),−1+D2if D≡1(mod4), (1)

then the ℤ-lattice 𝓞_D = [τ_D, 1] becomes a quadratic order of discriminant D = d_K ⋅ t² in the imaginary quadratic field K = ℚ(τ_D) where d_K is a fundamental discriminant of K and a positive integer t is the conductor of 𝓞_D.

Let 𝓠_D be the set of all positive definite integral binary quadratic forms of discriminant D, namely,

QD={ax2+bxy+cy2∈Z[x,y] | a>0, b2−4ac=D}.

The modular group Γ(1) = SL₂(ℤ)/{± I₂} acts on the set 𝓠_D from the right by the rule

γ=γ1γ2γ3γ4 : Q(x,y)=ax2+bxy+cy2↦Qγ(x,y)=Q(γ1x+γ2y,γ3x+γ4y), (2)

where I₂ denotes the 2 × 2 identity matrix. Then the action induces an equivalence relation ∼ on 𝓠_D as

Q1∼Q2if and only ifQ1=Q2γ ~for some γ∈Γ(1).

If we let QD0 ⊂ 𝓠_D be the set of all primitive forms (i.e. gcd(a, b, c) = 1), then the set of equivalence classes QD0 /Γ(1) becomes a finite abelian group under Dirichlet composition which is called the form class group of discriminant D and is denoted by C(D).

For each quadratic form Q = [a, b, c] = ax² + bxy + cy² ∈ 𝓠_D, let τ_Q be the zero of Q(x, 1) = 0 in the complex upper half plane ℍ = {z ∈ ℂ | Im(z) > 0}, namely,

τQ=−b+D2a. (3)

The classical j-invariant on ℍ is a Γ(1)-modular function defined by

j(τ)=1+240∑n=1∞∑m|nm3qn3q∏n=1∞(1−qn)24=q−1+744+196884q+21493760q2+⋯,

where τ ∈ ℍ and q = e^2πiτ. Letting H_D be the ring class field of order 𝓞_D over K, we have

HD=K(j(τD))

by the theory of complex multiplication. Furthermore, we have the classical isomorphism

Gal(HD/K)≅C(D)=QD0/Γ(1)

and the special values j(τ_Q) for all Q ∈ QD0 /Γ(1) become the Galois conjugates of j(τ_D) in H_D over K which are called the singular moduli.

Let J(τ) = j(τ)-744 be the normalized Hauptmodul on the modular group Γ(1). In [1], Zagier defined the modified Galois trace t_J(D) of index D as

tJ(D)=∑Q∈QD/Γ(1)J(τQ)Γ(1)Q,

where the sum allows the classes of imprimitive forms and Γ(1)_Q is the stabilizer of Q. Furthermore, Kaneko [2] found another description for t_J(D) as

tJ(D)=∑Od⊃OD2ωd⋅∑a∈Cl(Od)J(a),

where the first sum runs over all imaginary quadratic orders 𝓞_d ⊃ 𝓞_D, Cl(𝓞_d) denotes the 𝓞_d-ideal class group which is isomorphic to Gal(H_d/K) (see [3, §9]) and

ωd=6 if d=−3,4 if d=−4,2 otherwise,

is the number of units in 𝓞_d. Therefore, we can see that the modified trace of J is essentially a sum of usual Galois traces.

Zagier proved that the generating series

−q−1+2+∑D=1∞tJ(D)qD=−q−1+2−248q3+492q4−4119q7+7256q8+⋯

is a weakly holomorphic modular form of weight 3/2 for the Hecke subgroup Γ₀(4). After Zagier’s work, Bruinier and Funke [4] defined the modular traces of the CM values of modular functions for congruence subgroups of arbitrary genus and showed that modular traces of the values of an arbitrary modular function at Heegner points are Fourier coefficients of the holomorphic part of a harmonic weak Maass form of weight 3/2.

On the other hand, it is well known that the value of every modular function at an imaginary quadratic number lies in a ray class field of an imaginary quadratic field. In particular, we call the value f(τ_D) of a modular function f(τ) at τ = τ_D a class invariant if

K(f(τD))=K(j(τD)),

following Weber [5]. We can easily see that the modular trace of the CM value of J(τ) at a Heegner point is naturally its Galois trace. However, it is not obvious to see whether the Galois trace of a given algebraic integer is a modular trace and hence a Fourier coefficient of a certain automorphic form. In [6], the authors paid attention to real-valued class invariants given in terms of the singular values of the classical Weber functions

f∞=3η(3τ)η(τ),f0=η(τ3)η(τ),f1=η(τ+13)η(τ),f2=η(τ+23)η(τ),

where η(τ) is the classical Dedekind’s eta-function. They proved that the modified Galois traces of those invariants can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2.

In this paper, we shall construct real-valued class invariants by using the generalized Weber functions of level 5 given by

g∞(τ)=53⋅η5τη(τ)6,g0(τ)=ητ5η(τ)6,g1(τ)=ητ+965η(τ)6,g2(τ)=ητ+725η(τ)6,g3(τ)=ητ+485η(τ)6,g4(τ)=ητ+245η(τ)6

(Theorems 4.3 and 4.5) by extending the argument of [6, §6]. Furthermore, we shall prove that their Galois traces are the Fourier coefficients of holomorphic parts of weight 3/2 harmonic weak Maass forms (Theorem 6.4). To do this, we shall use the results on the Bruinier-Funke modular trace (Propositions 5.4 and 5.6) and Shimura’s reciprocity law (Proposition 3.3).

2 Generalized Weber function of level 5

In this section, we shall briefly introduce some arithmetic properties of generalized Weber functions (See [5] or [7, §4] for details). Throughout this paper, we let N be a positive integer.

Let ζ_N = e^2πi/N be the primitive N-th root of unity and let 𝓕_N be the field of modular functions on the principal congruence group Γ(N) = {γ ∈ Γ(1) | γ ≡ I₂ (mod N)} whose Fourier coefficients lie in the N-th cyclotomic field ℚ(ζ_N). Then, it is well known that 𝓕_N is a Galois extension over 𝓕₁ = ℚ(j(τ)) with

Gal(FN/F1)≃GL2(Z/NZ)/{±I2}.

The group GL₂(ℤ/Nℤ)/{± I₂} can be decomposed into

GL2(Z/NZ)/{±I2}=GN⋅Γ(N)=Γ(N)⋅GN, (4)

where

GN=σu=100u u∈Z/NZ×.

Each element σ_u ∈ G_N acts on the function f(τ) ∈ 𝓕_N by

σu∈GN : f(τ)=∑n≥n0cnqn/N↦fσu(τ)=∑n≥n0cnσuqn/N,

where cnσu denotes the image of c_n ∈ ℚ(ζ_N) via the automorphism of ℚ(ζ_N) defined by σ_u : ζ_N ↦ ζNu . Besides, the action of Γ(N) is given by

γ∈Γ(N) : f(τ)↦fγ(τ)=f(γ~τ),

where γ͠ ∈ Γ(1) is a preimage of γ via the natural surjection Γ(1) → Γ(N) and τ ↦ γ͠τ is the fractional linear transformation with respect to γ͠.

Let η(τ)=q1/24∏n=1∞1−qn be the classical Dedekind eta-function and

S=0−110andT=1101

be generators of Γ(1). We have the following transformation formula of η(τ) under the composition of Γ(1).

Proposition 2.1

Let γ=abcd∈Γ(1). We may assume that

c≥0andd>0 if c=0.

We write c = 2^λ(c) ⋅ c₀ with c₀ odd and put c₀ = λ(c) = 1 if c = 0 for convenience. Then,

η(γτ)=ε(γ)⋅cτ+d⋅η(τ)with Recτ+d>0,

where

ε(γ)=ac0⋅ζ24ba+c(d(1−a2)−a)+3(a−1)c0+32λ(c)(a2−1).

Here, (ac0) is the Legendre symbol.

In particular, we have

η∘S(τ)=−iτ⋅η(τ)andη∘T(τ)=ζ24⋅η(τ).

Proof

See [5, §38] and [8, §4]. □

The generalized Weber functions are defined by

v∞,N(τ)=N⋅η(Nτ)η(τ)andvk,N(τ)=η(τ+kN)η(τ),(τ∈H, k∈Z).

Then, these functions have the following modular properties.

Proposition 2.2

For a positive integer N and an integer k, we have

v_∞,N and v_k,N belong to 𝓕_24N.
Let {r_n}_n|N be a set of integers indexed by the positive divisors of N. If gcd(N, 6) = 1 and k ≡ 0 (mod 24), then we have

∑n|N(n−1)rn≡0(mod24)if and only if∏n|N(vk,N)rn∈FN.

Proof

See [9, Theorem 3.2]. □

From now on, let us consider the case N = 5. For each n ∈ ℤ, let k_n be an integer such that

kn≡0(mod24)andkn≡n(mod5).

We then define

g∞(τ)=v∞,5(τ)6=53⋅η5τη(τ)6,gn(τ)=vkn,5(τ)6=ητ+kn5η(τ)6.

Lemma 2.3

Let n, n₁ and n₂ be integers. Then we have

𝔤_∞(τ) and 𝔤_n(τ) belong to 𝓕₅.
If n₁ ≡ n₂ (mod 5), then 𝔤_n₁(τ) = 𝔤_n₂(τ).

Proof

It is straightforward from Proposition 2.2 (ii) for 𝔤_n(τ), if we choose r₁ = 1, r₅ = 6. See [10, Theorem 1.64] for 𝔤_∞(τ).
For each i ∈ {1, 2}, we may write k_{n_i} = 5 ⋅ K_i + ν for some integers K_i and ν ∈ {0, 1, 2, 3, 4}. Then we have

ητ+kni5=ητ+ν5+Ki=ζ24Ki⋅ητ+ν5

by Proposition 2.1. Since k_n₁ − k_n₂ = p ⋅ (K₁ − K₂) ≡ 0 (mod 24) and gcd(5, 24) = 1, we get K₁ ≡ K₂ (mod 24) which implies that ζ24K1=ζ24K2. □

By the above lemma, the indices of the generalized Weber functions of level 5 can be chosen from ℤ/5ℤ ∪ {∞}, namely,

g∞(τ), g0(τ), g1(τ), g2(τ), g3(τ), g4(τ)⊂F5.

Remark 2.4

From the q-product of η(τ), one can easily see that

g∞(τ)=53⋅q⋅∏n=1∞1−q5n1−qn6,gν(τ)=ζ5−ν⋅q−1/5⋅∏n=1∞1−ζ5νnqn/51−qn6

for ν ∈ {0, 1, 2, 3, 4}.

By Proposition 2.1 and Remark 2.4, we obtain that

S : (g∞, g0, g1, g2, g3, g4)↦(g0, g∞, g4, g2, g3, g1),T : (g∞, g0, g1, g2, g3, g4)↦(g∞, g1, g2, g3, g4, g0),σu : (g∞, g0, g1, g2, g3, g4)↦(g∞, g0, gu, g2u, g3u, g4u), (5)

where σ_u ∈ G₅.

Further by using (4), (5) and the following lemma, we can compute explicitly the Galois actions on the generalized Weber functions of level 5.

Lemma 2.5

Let N = p^r be a power of a rational prime number. Let abcd ∈ SL₂(ℤ/Nℤ) so that either a or c is relatively prime to N. If gcd(c, N) = 1, let y ≡ (1 + a)c⁻¹ (mod N). Otherwise, let z ≡ (1 + c)a⁻¹ (mod N). Then we have

abcd≡TySTcSTdy−bif gcd(c,N)=1,ST−zST−aSTbz−dif gcd(a,N)=1,(modN).

Proof

See [7, § 5]. □

Remark 2.6

We see that the Galois conjugates of 𝔤_ν(τ) for ν ∈ {0, 1, 2, 3, 4} ∪ {∞} in 𝓕₅ are given by

gνσ(τ)=gν′(τ)for some ν′∈{0,1,2,3,4}∪{∞}

for any σ ∈ GL₂(ℤ/5ℤ)/{± I₂}.

On the other hand, the generalized Weber functions of level 5 have the following algebraic relations with the j-invariant.

Lemma 2.7

𝔤₀(τ), …, 𝔤₄(τ), and 𝔤_∞(τ) are the six distinct roots of

(X2+10X+5)3−j(τ)⋅X∈Z[j(τ)][X].

Proof

See [5, § 72]. □

3 The singular values of Weber functions

Let D ≡ 0, 1 (mod 4) be an imaginary quadratic discriminant. Then, the singular values of the generalized Weber functions of level 5 evaluated at τ_D lie in a finite abelian extension of an imaginary quadratic field by the theory of complex multiplication (See [11] and [3, §15]). In particular, there is a useful criterion for determining whether the values belonging to the ring class field H_D so that we can illustrate the Galois action of C(D) ≅ Gal(H_D/K) by Shimura’s reciprocity law.

Let F(X) denote the minimal polynomial of τ_D over ℚ, namely,

F(X)=X2−D/4if D≡0(mod4),X2+X+(1−D)/4if D≡1(mod4).

Proposition 3.1

Let n be a positive integer prime to 6 and k be an integer satisfying k ≡ 0 (mod 24) and F(−k) ≡ 0 (mod n). If r is an even integer such that r ⋅ (n − 1) ≡ 0 (mod 24), then we have

ητD+knη(τD)r∈HD .

Proof

See [12, Theorem 20]. □

From the above proposition, we obtain the following class invariants.

Lemma 3.2

For an imaginary quadratic discriminant D ≡ □ (mod 100) with gcd(D, 5) = 1, the values

g2(τD), g3(τD)if D≡0(mod4), D≡1(mod5)g1(τD), g4(τD)if D≡0(mod4), D≡4(mod5)g0(τD), g1(τD)if D≡1(mod4), D≡1(mod5)g2(τD), g4(τD)if D≡1(mod4), D≡4(mod5)

are class invariants over K = ℚ(τ_D).

Proof

Since j(τ) ∈ ℚ(𝔤_ν(τ)) by Lemma 2.7, we have

HD=K(j(τD))⊆K(gν(τD))

for each ν ∈ {0, …, 4} ∪ {∞}. Conversely, if we put n = 5, r = 6, and k = k_ν for each ν ∈ {0, …, 4} in Proposition 3.1, then we can determine the values of ν such that 𝔤_ν(τ_D) ∈ H_D. □

It is well known that the form class group C(D) = QD0 /Γ(1) is isomorphic to Gal(H_D/K) (See [3, Theorem 3.9]). Let Q = [a, b, c] ∈ QD0 be a primitive quadratic form. For each prime integer p, we define the matrix M_Q,p ∈ GL₂(ℤ/pℤ) as

for D ≡ 0 (mod 4),

MQ,p=ab/201if p∤a,−b/2−c10if p|a and p∤c,−a−b/2−c−b/21−1if p|a and p|c, (6)
for D ≡ 1 (mod 4),

MQ,p=a(b−1)/201if p∤a,−(b+1)/2−c10if p|a and p∤c,−a−(b+1)/2−c+(1−b)/21−1if p|a and p|c. (7)

Note that for a given N ≥ 2, we can obtain a unique matrix M_Q in GL₂(ℤ/Nℤ) satisfying M_Q ≡ M_Q,p (mod p^r) for all primes p with p^r||N by Chinese remainder theorem.

Then, Shimura’s reciprocity law tells us that

Proposition 3.3

For f ∈ 𝓕_N and Q ∈ C(D) ≅ Gal(H_D/K), we have

f(τD)Q−1=fMQ(τQ),

where Q⁻¹ denotes the inverse of Q in C(D).

Proof

See [13, §6]. □

Remark 3.4

The principal form

1, 0, −D/4if D≡0(mod4),1, 1, (1−D)/4if D≡1(mod4)

represents the identity class in C(D) ([3, Theorem 3.9]).
The form class group C(D) is usually represented by reduced quadratic forms Q = [a, b, c] ∈ QD0 characterized by the condition

(−a<b≤a<c or 0≤b≤a=c)andb2−4ac=D

([1, Theorem 2.8]). One can easily derive that if the class of Q is not the identity, then

2≤a≤−D/3.
Let h_D be the class number of an imaginary quadratic discriminant D. Then, it is well known that h_D = 1 if and only if

D=−3,−4,−7,−8,−11,−12,−16,−19,−27,−28,−43,−67,−163

([3, Theorem 7.30]).

We observe that the pair of class invariants appearing in Lemma 3.2 are not necessarily real numbers. However, it is guaranteed that their sums or products are real numbers for arbitrary discriminants D by the following lemma.

Lemma 3.5

We have

gν(τD)¯=g−ν(τD) if D≡0(mod4),gν(τD)¯=g1−ν(τD)if D≡1(mod4),

for each ν ∈ {0, 1, 2, 3, 4}. Here, the indices −ν and 1 − ν are integers in a complete set of residues {0, 1, 2, 3, 4} modulo 5 such that ν + (− ν) ≡ 0 (mod 5) and ν+(1 − ν) ≡ 1 (mod 5).

Proof

Let B = 0 if D ≡ 0 (mod 4) and B = − 1 if D ≡ 1 (mod 4) so that

τD=B+D2andqD=e2πiτD=eBπi⋅rD,whererD=|qD|=e−π−D.

By Remark 2.4, for ν ∈ {0, 1, 2, 3, 4}, we have

ζ5−ν⋅qD−1/5⋅∏n=1∞1−ζ5νnqDn/51−qDn6=ζ5−ν⋅e−Bπi/5⋅rD−1/5⋅∏n=1∞1−ζ5νn⋅enBπi/5⋅rDn/51−enBπi⋅rDn6=e(−2ν−B)πi/5⋅rD−1/5⋅∏n=1∞1−en(2ν+B)πi/5⋅rDn/51−enBπi⋅rDn6.

One can see that the complex numbers appearing in the above product are of the form

e(−2ν−B)πi/5anden(2ν+B)πi/5for all n≥1.

Then, we find that only ν′ ∈ {0, 1, 2, 3, 4} with B + ν + ν′ ≡ 0 (mod 5) satisfy

e(−2ν−B)πi/5⋅e(−2ν′−B)πi/5=1,en(2ν+B)πi/5⋅en(2ν′+B)πi/5=1for all n≥1.

This completes the proof. □

4 Real valued class invariants from the generalized Weber functions of level 5

In this section, we construct a real valued class invariants from the generalized Weber functions of level 5 by using Shimura’s reciprocity law and the lemmas on the absolute values of Galois conjugates. We shall assume that D ≡ □ (mod 100) and gcd(D, 5) = 1, i.e. 5 splits completely in K = ℚ(τ_D).

We start with the basic inequalities.

Lemma 4.1

We have

1 + X < e^X, for all X > 0.
If 0 < X ≤ 1/11, then

11−X≤1+1.1X.

Proof

The proofs of (i) and (ii) are straightforward by basic calculus. □

Lemma 4.2

Let x + y i ∈ ℍ and r = e^−2πy.

If 0 < r < 1/11, then g∞(x+yi)<53⋅r⋅e6r51−r5+6.6r1−r.
If 0 < r < 1/11, then gν(x+yi)<r−1/5⋅e6r1/51−r1/5+6.6r1−r for all ν ∈ {0, 1, 2, 3, 4}.
If 0 < r^1/5 < 1/11, then gν(x+yi)>r−1/5⋅e−6.6r1/51−r1/5−6.6r1−r for all ν ∈ {0, 1, 2, 3, 4}.

Proof

We deduce that

g∞(x+yi)≤53⋅r⋅∏n=1∞1+r5n1−rn6by Remark 2.4<53⋅r⋅∏n=1∞1+r5n61+1.1⋅rn6by Lemma 4.1 (ii)<53⋅r⋅∏n=1∞er5n6e1.1rn6by Lemma 4.1 (i)=53⋅r⋅e6⋅∑n=1∞r5n+6.6⋅∑n=1∞rn=53⋅r⋅e6r51−r5+6.6r1−r.
The proof is similar to the proof of (i).
We establish that

|gν(x+yi)|≥r−1/5⋅∏n=1∞1−rn/51+rn6by Remark 2.4>r−1/5⋅∏n=1∞1−rn/56⋅1−rn6since 11+X>1−X for all X>0>r−1/5⋅∏n=1∞e−1.1⋅rn/56e−1.1⋅rn6by Lemma 4.1 (i),(ii)=r−1/5⋅e−6.6⋅∑n=1∞rn/5−6.6⋅∑n=1∞rn=r−1/5⋅e−6.6r1/51−r1/5−6.6r1−r.

□

Extending the arguments in [6, § 6], we achieve the following theorems.

Theorem 4.3

For an imaginary quadratic discriminant D ≤ − 31, we assume that D ≡ □ (mod 100) and gcd(D, 5) = 1. Then the singular values

gprod(τD)=g2(τD)⋅g3(τD)if D≡0(mod4), D≡1(mod5),g1(τD)⋅g4(τD)if D≡0(mod4), D≡4(mod5),g0(τD)⋅g1(τD)if D≡1(mod4), D≡1(mod5),g2(τD)⋅g4(τD)if D≡1(mod4), D≡4(mod5)

are real-valued class invariants over K = ℚ(τ_D).

Proof

We may assume that h_D ≥ 2 so that D ≤ − 24 by Remark 3.4 (iii). Let Q = [a, b, c] ∈ QD0 be a non-principal reduced form. By Proposition 3.3 and Remark 2.6, we have

(gprod(τD))Q−1=gν1(τQ)⋅gν2(τQ)

for some ν₁, ν₂ ∈ {0, …, 4} ∪ {∞}. Further by the above definition of 𝔤_prod and Lemma 3.5, we see that

|gprod(τD)|=|gν(τD)|2for some ν∈{0,1,2}.

Therefore, it suffices to show that

|gν(τD)|>|gν′(τQ)|

for all ν′ ∈ {0, …, 4} ∪ {∞}.

As in the proof of Lemma 3.5, let

qD=e2πiτD, qQ=e2πiτQandrD=|qD|=e−π−D, rQ=|qQ|=e−π−D/a.

One can immediately see that for D ≤ − 24,

rD1/5=e−π−D/5<1/11andrD1/2≤rQ=rD1/a≤e−π3<1/11

since 2 ≤ a ≤ −D/3 from Remark 3.4 (ii). Then, we get

|g∞(τQ)|<53⋅rQ⋅e6rQ51−rQ5+6.6rQ1−rQby Lemma 4.2 (i)≤53⋅e−π3⋅e6e−5π31−e−5π3+6.6e−π31−e−π3because rQ≤e−π3≈0.55747 (8)

and

|gν′(τQ)|<rQ−1/5⋅e6rQ1/51−rQ1/5+6.6rQ1−rQby Lemma 4.2 (ii)≤rD−1/10⋅e6e−π3/51−e−π3/5+6.6e−π31−e−π3because rD1/2≤rQ≤e−π3≈21.66520⋅rD−1/10 (9)

for ν′ ∈ {0, 1, 2, 3, 4}. Further we have

|gν(τD)|>rD−1/5⋅e−6.6rD1/51−rD1/5−6.6rD1−rDby Lemma 4.2 (iii).

Then we deduce that

g∞(τQ)gν(τD)≤0.55747⋅rD1/5⋅e6.6rD1/51−rD1/5+6.6rD1−rD=0.55747⋅e−π−D/5⋅e6.6e−π−D/51−e−π−D/5+6.6e−π−D1−e−π−Dsince rD=e−π−D≤0.55747⋅e−π24/5⋅e6.6e−π24/51−e−π24/5+6.6e−π241−e−π24≈0.03530<1for D≤−24

and

gν′(τQ)gν(τD)≤21.66520⋅rD−1/10⋅rD1/5⋅e6.6rD1/51−rD1/5+6.6rD1−rD=21.66520⋅rD1/10⋅e6.6rD1/51−rD1/5+6.6rD1−rD=21.66520⋅e−π−D/10⋅e6.6e−π−D/51−e−π−D/5+6.6e−π−D1−e−π−Dsince rD=e−π−D≤21.66520⋅e−π99/10⋅e6.6e−π99/51−e−π99/5+6.6e−π991−e−π99≈0.96330<1for D≤−99

for all ν′ ∈ {0, 1, 2, 3, 4}. Hence we obtain the assertion for D ≤ − 99.

For the remaining finite cases where − 96 ≤ D ≤ − 31, we observe that

|gν′(τQ)|≤rQ−1/5⋅e6rQ1/51−rQ1/5+6.6rQ1−rQ=rD−1/5a⋅e6rD1/5a1−rD1/5a+6.6rD1/a1−rD1/abecause rQ=rD1/a

for ν′ ∈ {0, 1, 2, 3, 4} and a ≥ 2. We then deduce that

gν′(τQ)gν(τD)<rD1/5−1/5a⋅e6rD1/5a1−rD1/5a+6.6rD1/a1−rD1/a+6.6rD1/51−rD1/5+6.6rD1−rD=e−π−D⋅(1/5−1/5a)⋅e6e−π−D/5a1−e−π−D/5a+6.6e−π−D/a1−e−π−D/a+6.6e−π−D/51−e−π−D/5+6.6e−π−D1−e−π−D. (10)

By using the algorithm for counting reduced forms (see [14, Algorithm 5.3.5]), we can make the list of the actual values of a for each D (see Table 1 below). Evaluating (10) at those values, we attain the assertion for −96 ≤ D ≤ −31.

Table 1

The coefficients of x² of non-principal reduced form Q = [a, b, c] ∈ C(D) for D ≡ □ (mod 100) relatively prime to 5 for D > −200.

D	a	D	a	D	a	D	a
−24	2	−71	2, 3, 4	−116	2, 3, 5	−159	2, 3, 4, 5, 6
−31	2	−76	4	−119	2, 3, 4, 5, 6	−164	2, 3, 5, 6
−36	2	−79	2, 4	−124	5	−171	5, 7
−39	2, 3	−84	2, 3, 5	−131	3, 5	−176	3, 4, 5
−44	3	−91	5	−136	2, 5	−179	3, 5
−51	3	−96	3, 4, 5	−139	5	−184	2, 5
−56	2, 3	−99	5	−144	4, 5	−191	2, 3, 4, 5, 6
−59	3	−104	2, 3, 5	−151	2, 4, 5	−196	2, 5
−64	4	−111	2, 3, 4, 5	−156	3, 5	−199	2, 4, 5, 7

Therefore, we conclude that the only reduced form in QD0 that fixes 𝔤_prod(τ_D) is the principal form, which represents the identity in the group C(D) ≅ Gal(H_D/K).

This completes the proof of our theorem by Galois theory. □

Remark 4.4

In fact, we can see that Theorem 4.3 is still valid for D = −24. We have

C(−24)={Q0=[1, 0, 6], Q1=[2, 0, 3]}

so that the corresponding CM points are given by

τQ0=τD=−242=−6andτQ1=−244=−62.

Since

gprod(τD)Q1−1=g2(τD)⋅g3(τD)Q1−1=g1(τQ1)⋅g4(τQ1)

by Proposition 3.3, it is enough to show that

g2−6>g1−6/2.

From Lemma 4.2 (iii), a lower bound of g2−6 is given by

g2−6≥e2π6/5⋅e−6.6e−2π6/51−e−2π6/5−6.6e−2π61−e−2π6>15.79269.

On the other hand, for an upper bound of g1−6/2, we observe that

∏n=1∞1−ζ5nqQ1n/561−qQ1n6=∏t=0∞ ∏s=141−ζ55t+sqQ1(5t+s)/56=∏t=0∞ ∏s=141−ζ5sqQ1t+s/56

since

1−ζ5nqn/5=1−ζ55n0q5n0/5=1−qn0for n=5n0.

For each 1 ≤ s ≤ 4, we then obtain

1−ζ5sqQ1t+s/52=1−rQ1t+s/5cos2sπ52+rQ1t+s/5sin2sπ52since qQ1=rQ1=e−π6∈R=1−2rQ1t+s/5cos2sπ5+rQ12(t+s/5).

Then, it is routine to check that

∏s=141−2rQ1t+s/5cos2sπ5+rQ12(t+s/5)3=∏s=141−2e−π6⋅(t+s/5)cos2sπ5+e−2π6⋅(t+s/5)3

is a monotone increasing function for t ≥ 1 and has the limit 1 when t → ∞. Moreover, its value at t = 0 is less than 1. Hence we get

g1−6/2<rQ1−1/5=eπ6/5≤4.66021<15.79269<g2−6.

Note that the minimal polynomial of 𝔤_prod(τ_D) is given by

X−g2(−6)⋅g3(−6)X−g1(−6/2)⋅g4(−6/2)=X2−750X+15625.

Theorem 4.5

For an imaginary quadratic discriminant D, we assume that D ≡ □ (mod 100) and gcd(D, 5) = 1. Then the singular values defined by

gsum(τD)=g2(τD)+g3(τD)if D≡0(mod4), D≡1(mod5), D≤−44,g1(τD)+g4(τD)if D≡0(mod4), D≡4(mod5), D≤−56,g0(τD)+g1(τD)if D≡1(mod4), D≡1(mod5), D≤−59,g2(τD)+g4(τD)if D≡1(mod4), D≡4(mod5), D≤−71,

are real-valued class invariants over K = ℚ(τ_D).

Proof

We prove the case D ≡ 0 (mod 4), D ≡ 1 (mod 5). The proofs for the other cases can be done similarly.

If h_D = 1, there is nothing to prove. Therefore, we may assume that h_D ≥ 2. Let Q = [a, b, c] ∈ QD0 be a non-principal reduced form so that

2≤a≤−D/3

by Remark 3.4 (ii). From the definition of 𝔤_sum(τ_D) and Lemma 3.5, we have

gsum(τD)=g2(τD)+g3(τD)=2⋅Re(g2(τD)).

Further by Remark 2.6 and Proposition 3.3, we see that

gsum(τD)Q−1=gν1(τQ)+gν2(τQ)for some ν1,ν2∈{0,1,2,3,4}∪{∞}. (11)

Hence, it is enough to prove that

Re(g2(τD))>gν(τQ)

for all ν ∈ {0, 1, 2, 3, 4} ∪ {∞}.

We estimate a lower bound of |Re(𝔤₂(τ_D))|. Let us set

qD=e2πiτDandrD=|qD|=e−π−D.

In fact, q_D = r_D for D ≡ 0 (mod 4). By Remark 2.4, we then have

g2(τD)=ζ5−2⋅rD−1/5⋅∏n=1∞1−ζ52nrDn/51−rDn6.

We put

zD=ζ5−2⋅rD−1/5=cos4π5−isin4π5⋅rD−1/5andwD=∏n=1∞1−ζ52n⋅rDn/51−rDn6

so that 𝔤₂(τ_D) = z_D ⋅ w_D. Furthermore, let

L(D)=e−6.6e−π−D/51−e−π−D/5−6.6e−π−D1−e−π−DandU(D)=e6e−π−D/51−e−π−D/5+6.6e−π−D1−e−π−D.

One can check that 𝔏(D) and 𝔘(D) are decreasing and increasing functions for D ≤ −44, respectively. By substituting e−π−D for r in Lemma 4.2 (ii) and (iii), we see that

L(D)<wD<U(D)for D≤−44.

1−ζ52n⋅rDn/51−rDn=1−e4nπi/5⋅rDn/51−rDn

for each n. Since

1−rDn/5⋅e4nπi/5=1−rDn/5cos4nπ5−i rDn/5sin4nπ5,

we get

|tanθn|=rDn/5sin4nπ51−rDn/5cos4nπ5≤rDn/51−rDn/5≤rDn/5⋅(1+1.1⋅rDn/5)by Lemma 4.1 (ii).

Define

Θ(D)=6e−π−D/51−e−π−D/5+6.6e−2π−D/51−e−2π−D/5

which is an increasing function for D ≤ −44. Then, by using the fact that x ≤ tan x for 0 < x < π/2, we obtain that

|θ|=6⋅∑n=1∞θn≤6⋅∑n=1∞|tanθn|≤6⋅∑n=1∞rDn/5+1.1rD2n/5≤6rD1/51−rD1/5+6.6rD2/51−rD2/5=Θ(D)because rD=e−π−D.

Thus, by using that sin x ≤ x for x > 0, we get

|Im(wD)|=|wD|⋅|sinθ|≤|wD|⋅|θ|<U(D)⋅Θ(D). (12)

We then arrive at

|Re(wD)|=|wD|2−|Im(wD)|2≥L(D)2−U(D)2⋅Θ(D)2 . (13)

Note that

L(D)2−U(D)2⋅Θ(D)2>0for D≤−44.

Therefore, we achieve by (12), (13) that

Re(g2(τD))=|Re(zD)⋅Re(wD)−Im(zD)⋅Im(wD)|≥||Re(zD)|⋅|Re(wD)|−|Im(zD)|⋅|Im(wD)||=|Re(wD)|⋅cos4π5−|Im(wD)|⋅sin4π5⋅rD−1/5>L(D)2−U(D)2⋅Θ(D)2⋅cosπ5−U(D)⋅Θ(D)⋅sinπ5⋅eπ−D/5. (14)

On the other hand, from (8), we have

|g∞(τQ)|<0.55747

for any reduced form Q ∈ QD0 . By evaluating 𝔏(D), 𝔘(D) and Θ(D) at D = −44, we obtain from (14) that

Re(g2(τD))>0.66224⋅eπ44/5≈42.76270>|g∞(τQ)|for D≤−44.

Furthermore, by (9), we have

|gν(τQ)|≤21.66520⋅rD−1/10=21.66520⋅eπ−D/10

for ν ∈ {0, 1, 2, 3, 4}. Then, by specializing 𝔏(D), 𝔘(D) and Θ(D) at D = −124, we get from (14) that

Re(g2(τD))>0.80087⋅eπ−D/5for D≤−124.

Hence, we achieve that for D ≤ −124,

gν(τQ)Re(g2(τD))<21.665200.80087⋅e−π124/10≈0.81827<1.

The finite remaining cases are given by

D=−104,−84,−64,−44.

We see that for ν ∈ {0, 1, 2, 3, 4},

|gν(τQ)|≤rQ−1/5⋅e6rQ1/51−rQ1/5+7.2rQ1−rQby Lemma 4.2 (ii)=rD−1/5a⋅e6rD1/5a1−rD1/5a+7.2rD1/a1−rD1/abecause rQ=rD1/a=eπ−D/5a⋅e6e−π−D/5a1−e−π−D/5a+7.2e−π−D/a1−e−π−D/a.

By evaluating (14) at D = −44 and the last formula at the actual values of a (see Table 1) of non-principal reduced forms Q = [a, b, c] ∈ QD0 , we again achieve that

gν(τQ)Re(g2(τD))<1

for the remaining cases.

Hence, we conclude from (11) that

gsum(τD)=2⋅Re(g2(τD))>gν1(τD)+gν2(τD)≥|(gsum(τD))Q−1|

for any reduced forms Q representing non-identity classes in C(D) ≅ Gal(H_D/K). This completes the proof by Galois theory. □

Remark 4.6

The finite exceptional cases of D with h_D ≥ 2 in the above theorem are given by D = -24, -31, -36, -39, −51. Using Proposition 3.3, we can directly compute the minimal polynomials of

gsum(τD)=g2(τD)+g3(τD)if D=−24,g1(τD)+g4(τD)if D=−36,g0(τD)+g1(τD)if D=−39,g2(τD)+g4(τD)if D=−31,−51

over ℚ, namely,

X2+56X+392if D=−24, h−24=2,X3+57X2+991X+6383if D=−31, h−31=3,X2−16X+16if D=−36, h−36=2,X4−29X3−2321X2−37041X−187867if D=−39, h−39=4,X2+68X+68if D=−51, h−51=2,

which are irreducible over ℚ. Thus, we can establish Theorem 4.3 again.
In fact, for D = −24, −36, −39, −51, we can apply the same argument as in Remark 4.4. However, we shall not repeat the same computations.

Example 4.7

Let D = −96 and K = Q(−6) . Then we have

C(−96)={Q0=[1, 0, 24], Q1=[3, 0, 8], Q2=[4, 4, 7], Q3=[5, 2, 5]}

with

τ−96=τQ0=2−6, τQ1=2−63, τQ2=−1+−162, τQ3=−2+2−65.

By Proposition 3.3, the class polynomials of

gprod(τ−96)=g1(τ−96)⋅g4(τ−96)andgsum(τ−96)=g1(τ−96)+g4(τ−96)

are given by

min(gprod(τ−96),K)=(X−g1(τQ0)⋅g4(τQ0))⋅(X−g2(τQ1)⋅g3(τQ1))⋅(X−g2(τQ2)⋅g4(τQ2))⋅(X−g0(τQ3)⋅g∞(τQ3))=X4−221000X3+60281250X2−3453125000X+244140625

and

min(gsum(τ−96),K)=(X−(g1(τQ0)+g4(τQ0)))⋅(X−(g2(τQ1)+g3(τQ1)))⋅(X−(g2(τQ2)+g4(τQ2)))⋅(X−(g0(τQ3)+g∞(τQ3)))=X4−236X3−11712X2−125528X+20164,

respectively.

5 Modular trace of a weakly holomorphic modular function

Throughout this section, we shall assume that an imaginary quadratic discriminant D = d_K ⋅ t² is congruent to a square modulo 4N² and relatively prime to N.

For each positive integer N, let

Γ=Γ00(N)=abcd∈Γ(1) b≡c≡0(modN)

which is a congruence subgroup of level N. We denote

QD,(N)=[a, b, c]∈QD | a≡c≡0(modN).

Then, the elements of 𝓠_D,(N) can be written as Q = [Na, b, Nc]. From (2), one can check that Γ acts on 𝓠_D,(N) and the action preserves the value of b (mod 2N²). Thus we obtain the following decomposition

QD,(N)/Γ=⋃β∈Z/2N2ZQD,(N),β/Γ,

where 𝓠_D,(N),β = {[Na, b, Nc] ∈ 𝓠_D |b ≡ β (mod 2N²)} for each β ∈ ℤ/2N²ℤ.

Remark 5.1

The values β with 𝓠_D,(N),β ≠ ∅ can be determined by the congruence equation β² ≡ D(mod 4N²). If N has ℓ distinct prime divisors, then the number of such β is equal to 2^ℓ by Chinese remainder theorem.
Let 𝓞_d be a quadratic order containing 𝓞_D in K = ℚ(τ_D). Then we can write d = d_K ⋅ (t/t′)² for some positive divisor t’ of t. By assigning

Q=[Na, b, Nc] (with gcd(Na,b,Nc)=t′)⟼Q~=1t′[Na, b, Nc]

for each t′|t, we then obtain the decomposition

QD,(N),β=⋃d|D t′⋅Qd,(N),βt′−10.

Moreover, we can easily see that τ_Q = τ_Q͠ and Γ_Q = Γ_Q͠.

From now on, we assume that 𝓠_D,(N),β ≠ ∅ for some suitable β ∈ ℤ/2N²ℤ. Let QD,(N),β0 ⊂ 𝓠_D,(N),β be the subset of primitive forms. Then, we have the following lemma.

Lemma 5.2

We have a canonical bijection between 𝓠_D/Γ(1) (resp. QD0 /Γ(1)) and 𝓠_D,(N),β/Γ (resp. QD,(N),β0 /Γ).

Proof

See [15, Proposition in § I.1] and [6, Lemma 5.1]. □

Let f be a modular function on Γ. We define the Zagier-type trace tf(β) (D) of index D as

tf(β)(D)=∑Q∈QD,(N),β/Γ1ΓQf(τQ),

where the weights of the summands are determined by the following lemma.

Lemma 5.3

For each Q ∈ 𝓠_D,(N),β, we have

|ΓQ|=2if D=−4⋅t2 and Q is Γ(1)−equivalent to [t, 0, t],3if D=−3⋅t2 and Q is Γ(1)−equivalent to [t, t, t],1otherwise.

In particular, if Q is primitive, then t should be 1.

Proof

It is a straighforward consequence from the fact that for each Q ∈ 𝓠_D,

|Γ(1)Q|=2 if Q is Γ(1)-equivalent to [t,0,t],3 if Q is Γ(1)-equivalent to [t,t,t],1 otherwise.

□

Now we briefly introduce the Bruinier-Funke modular trace of modular functions on Γ (see [4] for general statements). Let

V(Q)=X=bac−b a, b, c∈Q

be the vector space of dimension 3 over ℚ consisting of trace zero 2 × 2 matrices. It becomes a quadratic space of signature (1, 2) with the quadratic form q(X) = det(X) and the associated bilinear form (X, Y) = −tr(XY) for X, Y ∈ V(ℚ). One can see that the group SL₂(ℚ) acts on V by conjugation γ.X = γ X γ⁻¹ for X ∈ V(ℚ) and γ ∈ SL₂(ℚ).

Let 𝓓 be the space of positive lines in V(ℝ) = V(ℚ) ⊗ ℝ, namely,

D=z⊂V(R) | dim(z)=1, q|z>0.

We can identify 𝓓 with ℍ by assigning τ = x + yi ∈ ℍ to the line spanned by

Xτ=1y−12(τ+τ¯)ττ¯−112(τ+τ¯).

By direct computation, one can easily check that q(X_τ) = 1 and γ. X_τ = X(γτ) for γ ∈ SL₂(ℝ). Then, the CM points in ℍ can be viewed as positive lines ℝ X with the vectors X ∈ V(ℚ) of positive norms.

Let L be an even ℤ-lattice of V(ℚ) defined by

L=Nbca−Nb a, b, c ∈Z.

Then, the level of L is 4N² and the dual lattice is given by

L♯=Nbca−Nb a, c∈Z and b∈12N2Z. (15)

We then see that Γ acts on L by conjugation and acts trivially on the discriminant group L^♯/L. Furthermore, the group L^♯/L is isomorphic to a cyclic group ℤ/2N²ℤ. Therefore, each coset can be written in the form

L+h=X=Nb+h/2Nc−a−Nb−h/2N a, b, c∈Z (16)

for h ∈ {0, 1, …, 2N² − 1}.

Meanwhile, by using the fact that the stabilizer of each X ∈ V(ℝ) in SL₂(ℝ) ≅ SO(2) is compact, we get that Γ_X = (SL₂(ℝ))_X ∩ Γ is finite. Besides, if we let m be a positive rational number and h be a representative in L^♯/L ≅ ℤ/2N²ℤ, the group Γ acts on the set

Lh,m={X∈L+h | q(X)=m}

with the finite number of orbits. Then, the modular trace of a weakly holomorphic modular function f on Γ with respect to the lattice L for positive index m is defined by

MTfL(h,m)=∑X∈ΓN∖Lh,m1ΓXf(τX),

where τ_X is a CM point corresponding to the vector

(1/m)⋅X if a>0,(1/m)⋅(−X)if a<0.

The modular traces for zero or negative index are described by using a regularized integral or an infinite geodesic in ℍ (See [4, Definition 4.3]). Their explicit computations are given in [4, Proposition 4.7 and Remark 4.9]. We then have the following analytic property of modular traces.

Proposition 5.4

Let f be a weakly holomorphic modular function on Γ. Then the series

∑n≫−∞MTfL(h,n)qn

is the holomorphic part of a harmonic weak Maass form of weight 3/2 on Γ(4N²).

Proof

See [4, Theorem 4.5]. □

Remark 5.5

If h = 0, then the above series is the holomorphic part of a harmonic weak Maass form of weight 3/2 on a bigger group Γ₀(4N²) (See [4, §§3-4]).

Furthermore, the modular traces with respect to the lattice L can be related to the Zagier-type traces of modular functions.

Proposition 5.6

We have

MTfL(β,−D/4N2)=tf(β)(D)+tf(−β)(D).

Proof

See [6, Lemma 2.3]. □

6 Modular property of Galois traces of class invariants

Let us assume that N = 5 and D is an imaginary quadratic discriminant such that D ≡ □ (mod 100) and gcd(D, 5) = 1. In this section, we shall identify the Galois traces of real-valued class invariants defined in Theorems 4.3 and 4.5 with the Fourier coefficients of harmonic weak Maass forms of weight 3/2 by using the Bruinier-Funke modular traces and Shimura’s reciprocity law. For N = 5, we recall that Γ = Γ00 (5) and

L=5bca−5b a, b, c ∈Z.

Before we go further, we need some lemmas.

Lemma 6.1

𝔤₀ and 𝔤_∞ are Γ-modular functions.

Proof

By the definition of Γ = Γ00 (5), the only nontrivial transformation is given by the matrices γ ≡ [2003] (mod 5) in Γ. By Lemma 2.5, we have the decomposition

2003≡ST−3ST−2ST−3(mod5).

Using (5), we deduce that

2003 : g0 →S g∞ →T−3 g∞ →S g0 →T−2 g3 →S g3 →T−3 g0,g∞ →S g0 →T−3 g2 →S g2 →T−2 g0 →S g∞ →T−3 g∞.

This completes the proof. □

For a given discriminant D, we choose β ∈ ℤ/50ℤ satisfying β² ≡ D (mod 50) so that 𝓠_D,(5),β is nonempty.

Lemma 6.2

Let Q = [5a, b, 5c] ∈ QD,(5),β0 . Then we have

(gprod(τD))Q−1=g0(τQ)⋅g∞(τQ),(gsum(τD))Q−1=g0(τQ)+g∞(τQ).

Proof

Since D = b²-100ac and gcd(D, 5) = 1, we have D ≡ b² (mod 5) and gcd(b, 5) = 1. From (6) and (7), the corresponding matrix M_Q ∈ GL₂(ℤ/5ℤ)/\{± I₂} is given by

−5a−b/2−5c−b/21−1=−b/2−b/21−1if D≡0(mod4),−5a−(b+1)/2−5c+(1−b)/21−1=−(b+1)/2(1−b)/21−1if D≡1(mod4).

By Lemma 2.5, we obtain

−b/2−b/21−1=100b⋅Tb(1+2b)STb−1STb−1if D≡0(mod4),−(b+1)/2(1−b)/21−1=100b⋅Tb(3+2b)STb−1STb−1if D≡1(mod4).

Since the computations for other cases are similar, we suppose that D ≡ 0 (mod 4) and D ≡ 1 (mod 5). Then, we get

b=1:g2 →1001 g2 →T3 g0 →S g∞ →T1 g∞ →S g0 →T0 g0,g3 →1001 g3 →T3 g1 →S g4 →T1 g0 →S g∞ →T0 g∞,

b=4:g2 →1004 g3 →T1 g4 →S g1 →T4 g0 →S g∞ →T3 g∞,g3 →1004 g2 →T1 g3 →S g3 →T4 g2 →S g2 →T3 g0.

This completes the proof by the definitions of 𝔤_prod and 𝔤_sum. □

Following Kaneko’s description, the modified Galois traces of 𝔤_prod and 𝔤_sum of index D are given by

GTgprod(D)=∑Od⊃OD2ωd⋅TrHd/K(gprod(τd)),GTgsum(D)=∑Od⊃OD2ωd⋅TrHd/K(gprod(τd)),

where Tr is the usual Galois trace.

Lemma 6.3

We have

GTgprod(D)=tg0⋅g∞(β)(D)=tg0⋅g∞(−β)(D),GTgsum(D)=tg0+g∞(β)(D)=tg0+g∞(−β)(D).

Proof

Since 𝔤₀ ⋅ 𝔤_∞ is Γ-modular function by Lemma 6.1, we deduce that

tg0⋅g∞(β)(D)=∑Q∈QD,(5),β/Γ1ΓQ⋅g0(τQ)⋅g∞(τQ)by definition=∑d|D∑Q~∈Qd,(5),βt′−10/Γ1ΓQ~⋅g0(τQ~)⋅g∞(τQ~)by Remark 5.1 (ii)=∑d|D 1ΓQ~⋅∑Q~∈Qd,(5),βt′−10/Γgprod(τd)Q~−1by Lemmas 5.3 and 6.2=∑d|D 1ΓQ~⋅TrHd/K(gprod(τd))by Lemma 5.2=∑d|D 2ωd⋅TrHd/K(gprod(τd))by Lemma 5.3=GTgprod(D),

where d = d_K ⋅ t′² runs over all discriminants of orders 𝓞_d ⊃ 𝓞_D in K = ℚ(τ_D). Similarly, we have

tg0+g∞(β)(D)=GTgsum(D).

Since GT is independent of the choice of β, we obtain the equalities on the right side. □

By combining the above lemmas, we deduce the following theorem.

Theorem 6.4

Let D be an imaginary quadratic discriminant congruent to a square modulo 100 and relatively prime to 5. Let β ∈ ℤ/50ℤ such that β² ≡ D (mod 50). Then we have

GTgprod(D)=12⋅MTg0⋅g∞L(β,−D/100)andGTgsum(D)=12⋅MTg0+g∞L(β,−D/100).

Moreover, there are finite principal parts 𝓐(τ) = ∑_m≤0 a(m)q^m and 𝓑(τ) = ∑_m≤0 b(m)q^m such that each of

A(τ)+∑D≡◻ (100)gcd(D,5)=1GTgprod(D)q−D/100

and

B(τ)+∑D≡◻ (100)gcd(D,5)=1GTgsum(D)q−D/100

is the holomorphic part of a harmonic weak Maass form of weight 3/2 on Γ(100).

Proof

The first assertion directly comes from Lemmas 5.6 and 6.3. Precisely, if D ≡ β² (mod 50) for some β ∈ ℤ/50ℤ, we deduce that

MTg0⋅g∞L(β,−D/100)=tg0⋅g∞(β)(D)+tg0⋅g∞(−β)(D)=2⋅GTgprod(D),MTg0+g∞L(β,−D/100)=tg0+g∞(β)(D)+tg0+g∞(−β)(D)=2⋅GTgsum(D).

For the second assertion, let h ∈ {0, 1, …, 49} with gcd(h, 5) = 1. Then, a vector X ∈ L + h is of the form

X=5b+h/10c−a−5b−h/10∈L+h

from (16). If it has a positive norm −D/100 ∈ ℚ, then the corresponding point τ_X is a root of a positive definite form

Q=5a, 50b+h, 5cif a>0,−5a,−50b−h,−5cif a<0,

whose discriminant is given by (50b + h)² − 100ac ≡ h² (mod 100). This implies that if gcd(h, 5) = 1, the generating series of MTg0⋅g∞L (h, −D/100) and MTg0+g∞L (h, -D/100) only allow the terms q^−D/100 with D ≡ □ (mod 100) and gcd(D, 5) = 1. This completes the proof by Proposition 5.4. □

Example 6.5

Let D = −96 and K = Q(−6) . If we choose β = 2, then we have

Q−96,(5),2/Γ=Q0=[25, 102, 105],Q1=[20, −48, 30],Q2=[20, 52, 35],Q3=[10, 52, 70],Q4=[15, −48, 40],Q5=[5, 52, 140]

with

τQ0=−102+−9650,τQ1=48+−9640,τQ2=−52+−9640,τQ3=−52+−9620,τQ4=48+−9630,τQ5=−52+−9610.

We obtain that

tg0⋅g∞(2)(−96)=∑k=05g0(τQk)⋅g∞(τQk)=221750,tg0+g∞(2)(−96)=∑k=05g0(τQk)+g∞(τQk)=180.

On the other hand, we have

GTgprod(D)=TrH−24/K(gprod(τ−24)+TrH−96/K(gprod(τ−96)=221000+750=221750by Remark 4.4 and Example 4.7

and

GTgsum(D)=TrH−24/K(gsum(τ−24)+TrH−96/K(gsum(τ−96)=−56+236=180by Remark 4.6 (i) and Example 4.7.

Acknowledgment

The authors would like to thank the referee for helpful and valuable comments. The first author was supported by the Dongguk University Research Fund of 2017 and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2017R1C1B5017567). The second (corresponding) author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2017R1C1B2010652) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1A6A1A11051177).

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Received: 2018-11-12

Accepted: 2019-10-10

Published Online: 2019-12-31

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0115

Keywords for this article

modular forms; modular traces; Galois traces; class field theory

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