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

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.


Introduction
Let D be a negative integer with D ≡ , (mod ) so that D is an imaginary quadratic discriminant. More explicitly, if we let : Q(x, y) = ax + bxy + cy → Q γ (x, y) = Q(γ x + γ y, γ x + γ y), where I denotes the × identity matrix. Then the action induces an equivalence relation ∼ on Q D as Q ∼ Q if and only if Q = Q γ for some γ ∈ Γ( ).
If we let Q D ⊂ Q D be the set of all primitive forms (i.e. gcd(a, b, c) = ), then the set of equivalence classes Q D /Γ( ) becomes a nite 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] = ax + bxy + cy ∈ Q D , let τ Q be the zero of Q(x, ) = in the complex upper half plane H = {z ∈ C | Im(z) > }, namely, The classical j-invariant on H is a Γ( )-modular function de ned by where τ ∈ H and q = e πiτ . Letting H D be the ring class eld of order O D over K, we have where the sum allows the classes of imprimitive forms and Γ( ) Q is the stabilizer of Q. Furthermore, Kaneko [2] found another description for t J (D) as where the rst 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 is the number of units in O d . Therefore, we can see that the modi ed trace of J is essentially a sum of usual Galois traces. Zagier proved that the generating series is a weakly holomorphic modular form of weight / for the Hecke subgroup Γ ( ). After Zagier's work, Bruinier and Funke [4] de ned 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 coe cients of the holomorphic part of a harmonic weak Maass form of weight / .
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 eld of an imaginary quadratic eld. 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 (τ D )) = K(j(τ D )), 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 coe cient 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 where η(τ) is the classical Dedekind's eta-function. They proved that the modi ed Galois traces of those invariants can be identi ed with the Fourier coe cients of weakly holomorphic modular forms of weight / .
In this paper, we shall construct real-valued class invariants by using the generalized Weber functions of level given by

Generalized Weber function of level
In this section, we shall brie y 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 πi/N be the primitive N-th root of unity and let F N be the eld of modular functions on the principal congruence group Γ(N) = γ ∈ Γ( ) | γ ≡ I (mod N) whose Fourier coe cients lie in the N-th cyclotomic eld Q(ζ N ). Then, it is well known that F N is a Galois extension over F = Q(j(τ)) with The group GL (Z/NZ)/{±I } can be decomposed into where Each element σu ∈ G N acts on the function f (τ) ∈ F N by where c σu n denotes the image of cn ∈ Q(ζ N ) via the automorphism of Q(ζ N ) de ned by σu : ζ N → ζ u N . Besides, the action of Γ(N) is given by where γ ∈ Γ( ) is a preimage of γ via the natural surjection Γ( ) → Γ(N) and τ → γτ is the fractional linear transformation with respect to γ. We write c = λ(c) · c with c odd and put c = λ(c) = if c = for convenience. Then, Here, ( a c ) is the Legendre symbol. In particular, we have Proof. See [5, §38] and [8, §4].
The generalized Weber functions are de ned by Then, these functions have the following modular properties. From now on, let us consider the case N = . For each n ∈ Z, let kn be an integer such that kn ≡ (mod ) and kn ≡ n (mod ).
We then de ne

Remark 2.4.
From the q-product of η(τ), one can easily see that By Proposition 2.1 and Remark 2.4, we obtain that S : (g∞, g , g , g , g , g ) → (g , g∞, g , g , g , g ), where σu ∈ G . Further by using (4), (5) and the following lemma, we can compute explicitly the Galois actions on the generalized Weber functions of level . Proof. See [7, §5].
Remark 2.6. We see that the Galois conjugates of gν(τ) for ν ∈ { , , , , } ∪ {∞} in F are given by On the other hand, the generalized Weber functions of level have the following algebraic relations with the j-invariant.

The singular values of Weber functions
Let D ≡ , (mod ) be an imaginary quadratic discriminant. Then, the singular values of the generalized Weber functions of level evaluated at τ D lie in a nite abelian extension of an imaginary quadratic eld 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 eld 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, Proposition 3.1. Let n be a positive integer prime to and k be an integer satisfying k ≡ (mod ) and F(−k) ≡ (mod n). If r is an even integer such that r · (n − ) ≡ (mod ), then we have Proof. See [12,Theorem 20].
From the above proposition, we obtain the following class invariants.

Lemma 3.2. For an imaginary quadratic discriminant D
are class invariants over K = Q(τ D ).
It is well known that the form class group C( (ii) for D ≡ (mod ), Note that for a given N ≥ , we can obtain a unique matrix M Q in GL (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 Proof. See [13, §6]. 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.
By Remark 2.4, for ν ∈ { , , , , }, we have One can see that the complex numbers appearing in the above product are of the form e (− ν−B)πi/ and e n( ν+B)πi/ for all n ≥ .

Real valued class invariants from the generalized Weber functions of level
In this section, we construct a real valued class invariants from the generalized Weber functions of level by using Shimura's reciprocity law and the lemmas on the absolute values of Galois conjugates. We shall assume that D ≡ (mod ) and gcd(D, ) = , i.e. splits completely in K = Q(τ D ). We start with the basic inequalities.
Proof. The proofs of (i) and (ii) are straightforward by basic calculus. Proof.
(i) We deduce that (ii) The proof is similar to the proof of (i).
(iii) We establish that Extending the arguments in [6, §6], we achieve the following theorems.
are real-valued class invariants over K = Q(τ D ).
Proof. We may assume that h D ≥ so that D ≤ − by Remark 3.4 (iii). Let Q = [a, b, c] ∈ Q D be a nonprincipal reduced form. By Proposition 3.3 and Remark 2.6, we have for some ν , ν ∈ { , . . . , } ∪ {∞}. Further by the above de nition of g prod and Lemma 3.5, we see that Therefore, it su ces to show that ≈ . and Then we deduce that for ν ∈ { , , , , } and a ≥ . We then deduce that 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 − ≤ D ≤ − . Therefore, we conclude that the only reduced form in Q D that xes 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.  Since by Proposition 3.3, it is enough to show that is a monotone increasing function for t ≥ and has the limit when t → ∞. Moreover, its value at t = is less than . Hence we get Note that the minimal polynomial of g prod (τ D ) is given by

are real-valued class invariants over K = Q(τ D ).
Proof. We prove the case D ≡ (mod ), D ≡ (mod ). The proofs for the other cases can be done similarly.
If h D = , there is nothing to prove. Therefore, we may assume that h D ≥ . Let Q = [a, b, c] ∈ Q D be a non-principal reduced form so that ≤ a ≤ −D/ by Remark 3.4 (ii). From the de nition of gsum(τ D ) and Lemma 3.5, we have Further by Remark 2.6 and Proposition 3.3, we see that Hence, it is enough to prove that Re(g (τ D )) > gν(τ Q ) for all ν ∈ { , , , , } ∪ {∞}. We estimate a lower bound of Re(g (τ D )) . Let us set q D = e πiτ D and r D = |q D | = e −π √ −D .
In fact, q D = r D for D ≡ (mod ). By Remark 2.4, we then have We put De ne which is an increasing function for D ≤ − . Then, by using the fact that x ≤ tan x for < x < π/ , we obtain that Thus, by using that sin x ≤ x for x > , we get We then arrive at Note that Therefore, we achieve by (12), (13) that On the other hand, from (8) Hence, we achieve that for D ≤ − , The nite remaining cases are given by We see that for ν ∈ { , , , , }, By evaluating (14) at D = − and the last formula at the actual values of a (see Table 1) of non-principal reduced forms Q = [a, b, c] ∈ Q D , we again achieve that gν(τ Q ) Re(g (τ D )) < for the remaining cases. Hence, we conclude from (11) that for any reduced forms Q representing non-identity classes in C(D) ∼ = Gal(H D /K). This completes the proof by Galois theory.

Modular trace of a weakly holomorphic modular function
Throughout this section, we shall assume that an imaginary quadratic discriminant D = d K · t is congruent to a square modulo N and relatively prime to N. For each positive integer N, let which is a congruence subgroup of level N. We denote 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 N ). Thus we obtain the following decomposition for each t |t, we then obtain the decomposition Moreover, we can easily see that τ Q = τ Q and Γ Q = Γ Q .
From now on, we assume that Q D,(N),β ≠ ∅ for some suitable β ∈ Z/ N Z. Let Q D,(N),β ⊂ 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( ) (resp. Q D /Γ( )) and Q D,(N),β /Γ (resp. Q D,(N),β /Γ). Let f be a modular function on Γ. We de ne the Zagier-type trace t (β) f (D) of index D as where the weights of the summands are determined by the following lemma.
In particular, if Q is primitive, then t should be .
Proof. It is a straighforward consequence from the fact that for each Q ∈ Q D , otherwise.
Now we brie y introduce the Bruinier-Funke modular trace of modular functions on Γ (see [4] for general statements). Let be the vector space of dimension over Q consisting of trace zero × matrices. It becomes a quadratic space of signature ( , ) 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 SL (Q) acts on V by conjugation γ.X = γXγ − for X ∈ V(Q) and γ ∈ SL (Q). Let D be the space of positive lines in V(R) = V(Q) ⊗ R, namely, We can identify D with H by assigning τ = x + yi ∈ H to the line spanned by By direct computation, one can easily check that q(Xτ) = and γ.Xτ = X(γτ) for γ ∈ SL (R). Then, the CM points in H can be viewed as positive lines RX with the vectors X ∈ V(Q) of positive norms. Let L be an even Z-lattice of V(Q) de ned by Then, the level of L is N and the dual lattice is given by 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/ N Z. Therefore, each coset can be written in the form Meanwhile, by using the fact that the stabilizer of each X ∈ V(R) in SL (R) ∼ = SO( ) is compact, we get that Γ X = (SL (R)) X ∩ Γ is nite. Besides, if we let m be a positive rational number and h be a representative in L /L ∼ = Z/ N Z, the group Γ acts on the set with the nite 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 de ned by where τ X is a CM point corresponding to the vector Furthermore, the modular traces with respect to the lattice L can be related to the Zagier-type traces of modular functions.

Modular property of Galois traces of class invariants
Let us assume that N = and D is an imaginary quadratic discriminant such that D ≡ (mod ) and gcd(D, ) = . In this section, we shall identify the Galois traces of real-valued class invariants de ned in Theorems 4.3 and 4.5 with the Fourier coe cients of harmonic weak Maass forms of weight / by using the Bruinier-Funke modular traces and Shimura's reciprocity law. For N = , we recall that Γ = Γ ( ) and Before we go further, we need some lemmas. Lemma 6.1. g and g∞ are Γ-modular functions.
Proof. By the de nition of Γ = Γ ( ), the only nontrivial transformation is given by the matrices γ ≡ (mod ) in Γ. By Lemma 2.5, we have the decomposition Using (5), we deduce that This completes the proof.
For a given discriminant D, we choose β ∈ Z/ Z satisfying β ≡ D (mod ) so that Q D,( ),β is nonempty.
Proof. Since D = b − ac and gcd(D, ) = , we have D ≡ b (mod ) and gcd(b, ) = . From (6) and (7), the corresponding matrix M Q ∈ GL (Z/ Z)/{±I } is given by By Lemma 2.5, we obtain Since the computations for other cases are similar, we suppose that D ≡ (mod ) and D ≡ (mod ). Then, we get b = : This completes the proof by the de nitions of g prod and gsum. Proof. Since g · g∞ is Γ-modular function by Lemma 6.1, we deduce that Since GT is independent of the choice of β, we obtain the equalities on the right side.
For the second assertion, let h ∈ { , , . . . , } with gcd(h, ) = . Then, a vector X ∈ L + h is of the form (16). If it has a positive norm −D/ ∈ Q, then the corresponding point τ X is a root of a positive de nite form