The Fourier coefficients of the McKay-Thompson series and the traces of CM values

The elliptic modular j function enjoys many beautiful properties. Its Fourier coefficients are related to the Monster group, and its CM values generate abelian extensions over imaginary quadratic fields. Kaneko gave an arithmetic formula for the Fourier coefficients expressed in terms of the traces of the CM values. In this article, we are concerned with analogues of Kaneko's result for the McKay-Thompson series of square-free level.

On the other hand, the Fourier coefficients of the j-function have a connection with the degrees of irreducible representations of the Monster group M, the largest of the sporadic simple groups. This is known as the monstrous moonshine. The first few observations are  [4] formulated the monstrous moonshine conjecture as follows.
• There exists a graded infinite-dimensional module It is called the monster module. • For each g ∈ M, we define the McKay-Thompson series Then there exists a genus 0 subgroup Γ g ⊂ SL 2 (R) such that T g (τ ) is a hauptmodul, that is, the generator of the modular function field A 0 (Γ g ).
In 1992, R. Borcherds [1] proved this conjecture. In this paper, we are concerned with the analogues of Kaneko's formula (1.2) for the McKay-Thompson series of level N such that N is a square-free integer and the genus of the congruence subgroup Γ 0 (N) is 0, that is, N = 2, 3, 5, 6, 7, 10, and 13. (Kaneko's result is the case of N = 1.) For these N, let Γ * 0 (N) be the Fricke group, which is generated by Γ 0 (N) and all Atkin-Lehner involutions W e for e such that e|N and (e, N/e) = 1. Here W e is a matrix of the form 1 √ e [ xe y zN we ] with detW e = 1 and x, y, z, w ∈ Z. Let d be a positive integer such that −d is congruent to a square modulo 4N. We denote by Q d, For genus zero groups Γ 0 (N) and Γ * 0 (N), the corresponding hauptmoduln j N (τ ) and j * N (τ ) can be described by means of the Dedekind η-function η(τ ) := q 1/24 ∞ n=1 (1 − q n ), 3,5,7,13), For a modular function f on Γ * 0 (N), we define two modular trace functions by Ohta [12] and the author and Osanai [11] obtained the analogues of Kaneko's formula (1.2) in the cases of N = p = 2, 3, and 5, first found experimentally, and then showed the coincidence of q-series by using Riemann-Roch theorem. In the present paper, we use the theory of Jacobi forms to generalize (1.2). Let c 1 (x) is 0, and we put σ  (d), (see [6], [7], and [13]), these formulas can be interpreted as the sum of t The outline of this paper is as follows. In Section 2 and 3, we give a review of the theory of Jacobi forms [5] and Bruinier and Funke's work [2]. In Section 4 we prove Theorem 1.1.
Acknowledgement. The author is grateful to his advisor Professor Masanobu Kaneko for carefully reading the manuscript and helpful comments.

The theory of Jacobi forms
In this section, we follow the expositions in [5]. Let k and m be integers. For a function φ : H × C → C, we define slash operators by A weak Jacobi form of weight k and index m is a holomorphic function φ : • φ has a Fourier expansion of the form where the coefficients c(n, r) depend only on the value of 4mn − r 2 and on the class of r (mod 2m), that is, we can write as c(n, r) = c r (4mn−r 2 ), and it holds c r ′ (D) = c r (D) (r ′ ≡ r (mod 2m)). This property gives us coefficients c µ (D) for all µ ∈ Z/2mZ and all integers D satisfying D ≡ −µ 2 (mod 4m), namely For D ≡ −µ 2 (mod 4m), we define c µ (D) = 0, and set In addition, we put the theta functions then φ has the following decomposition, According to [5,Section 5], h µ and ϑ m,µ satisfy the following transformation laws, Moreover we have, (2.1) gives an isomorphism between the space of weak Jacobi forms of weight k and index m and the space of vector valued modular forms (h µ ) µ (mod 2m) on SL 2 (Z) satisfying the above transformation laws and some cusp conditions.
Finally, we show an easy lemma for a proof of Theorem 1.1.
Lemma 2.2. Let φ(τ, z) be a weak Jacobi form of even weight k and index m. Then the map sends a weak Jacobi form to a weakly holomorphic modular form of weight k on Γ 0 (m).
Proof. First, for any [ a b c d ] ∈ Γ 0 (m), we can easily see that

Bruinier and Funke's work
In this section, we give a review of Bruinier and Funke's work [2] and Kim's result [9].

Preliminaries
Let N be a square-free positive integer and V a rational vector space of dimension 3 given by with a non-degenerate symmetric bilinear form (X, Y ) := −N · tr(XY ). We write q(X) := N · det(X) for the associated quadratic form. We fix an orientation for V once and for all. The action of G(Q) := Spin(V ) ≃ SL 2 (Q) on V is given as a conjugation, namely g.X := gXg −1 for X ∈ V and g ∈ G(Q). Let D be the orthogonal symmetric space defined by In particular, this is a bijective map and preserves G(Q)-action, that is, this map sends g.z := span g.[ x 1 x 2 −1 −x 1 ] to gτ for any g ∈ G(Q). Then the image of its inverse map span(X(τ )) is given by Let L ⊂ V (Q) be an even lattice of full rank and L # the dual lattice of L defined by L # := {X ∈ V (Q) | (X, Y ) ∈ Z for all Y ∈ L}. Let Γ be a congruence subgroup of Spin(L) which preserves L and acts trivially on the discriminant group L # /L. The set Iso(V ) of all isotropic lines in V (Q) corresponds to P 1 (Q) = Q ∪ {∞} via the bijection In particular, we put the isotropic line ℓ ∞ := ψ(∞) = span([ 0 1 0 0 ]). We orient all lines ℓ ∈ Iso(V ) by requiring that σ ℓ .[ 0 1 0 0 ] as a positively oriented basis vector of ℓ, where we pick σ ℓ ∈ SL 2 (Z) such that σ ℓ .ℓ ∞ = ℓ. For each ℓ ∈ Iso(V ), we define three positive rational numbers α ℓ , β ℓ , and ε ℓ . First, we pick α ℓ ∈ Q >0 as the width of the cusp ℓ, that is, where Γ ℓ is the stabilizer of the line ℓ. Next, we pick a positively oriented vector Y ∈ V (Q) such that ℓ = span(Y ) and Y is primitive in L. Then we define β ℓ ∈ Q >0 by σ −1 Finally, we put ε ℓ = α ℓ /β ℓ . Note that the quantities α ℓ , β ℓ , and ε ℓ depend only on the Γ-class of ℓ.
Let M := Γ\D be the modular curve. For X ∈ V (Q) with q(X) > 0, we define the Heegner point in M by D X := span(X) ∈ D, which corresponds to an imaginary quadratic irrational in H. For m ∈ Q >0 and h ∈ L # , Γ acts on L h,m := {X ∈ L + h | q(X) = m} with finitely many orbits. For a weakly holomorphic modular function f on Γ, we define the modular trace function by Next, we consider a vector X ∈ V (Q) with q(X) < 0. For such a vector X ∈ V (Q), we define a geodesic c X in D by and we put c(X) := Γ X \c X in M. If q(X) ∈ −N · (Q × ) 2 , then X is orthogonal to the two isotropic lines ℓ X = span(Y ) andl X = span(Ỹ ) such that Y andỸ positively oriented and the triple (X, Y,Ỹ ) is a positively oriented basis for V . We say ℓ X is the line associated to X, and write X ∼ ℓ X . We now define the modular trace function for negative index. For In particular, the geodesic c X is given in D ≃ H by A weakly holomorphic modular function f on Γ has a Fourier expansion at the cusp ℓ of the form By [2, Proposition 4.7], we can define the modular trace function for negative index by where r = Re(c(X)) for any X ∈ L h,−N k 2 ,ℓ and r ′ = Re(c(X)) for any X ∈ L −h,−N k 2 ,ℓ . If m ∈ Q <0 is not of the form m = −Nk 2 with k ∈ Q >0 , we put t f (h, m) = 0. In particular, t f (h, m) = 0 for m ≪ 0.
Finally, the modular trace function for zero index is defined by where δ h,0 is the Kronecker delta. By [2, Remark 4.9], we have

Modularity of the modular trace function
satisfies the following transformation laws.

(3.3)
A weakly holomorphic modular function f (τ ) = n a(n)q n on Γ * 0 (p 1 p 2 ) has a Fourier expansion of the form (3.1) at each cusp ℓ. By direct calculation, we have a ℓ∞ (n) = a(n), a ℓ 0 (n/p 1 p 2 ) = a(n), We apply Theorem 3.1 to the above case, then the function I h (τ, f ) satisfies By Theorem 2.1, we can obtain a weak Jacobi form of weight 2 and index p 1 p 2 . For further details, we compute the modular trace functions.
For a negative integer d, we define in the same way of Lemma 3.3 and Lemma 3.4 in [9]. Therefore, by Theorem 3.1 and Theorem 2.1, we obtain the following theorem.
Proof. We consider only the case of prime level N = p. We put the Atkin-Lehner involution  2 (τ, z), we obtain a weakly holomorphic modular form of weight 2 on Γ 0 (N) Since the weak Jacobi form g (N ) 2 (τ, z) has a theta decomposition (2.1) where h µ (τ ) is a partial generating function 2 (τ ) can be expressed as follows.
Note that we can easily see that By the modularities (2.2) of the functions h µ (τ ) and ϑ N,µ (τ, z), we have directlỹ Since the sum N −1 ℓ=0 e 2πi ℓ N µ is equal to N or 0 according as N|µ, we havẽ g (N )  Then we see that Thus we conclude that G In a similar way, we obtain Therefore we have Note that the value of the modular trace function t (N ) 2 (d) for negative index is zero except for d = −1, −4, and the partial generating functions h 0 (τ /N) and h N (τ /N) are given as Thus if N = 2, these functions have no pole at q = 0, that is, G (N * ) 2 (τ ) has no pole at τ = 0. If N = 2, the pole of h 2 (τ /2) at q = 0 is canceled out by the zero of ϑ 2,2 (τ /2, 0). For other cusps in the cases of N = 6, 10, we can check it in the same way.