Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms

1 Introduction and statement of results

Let p be one or a prime and Γ 0 + ( p ) be the group generated by the congruence subgroup Γ 0 ( p ) and the Fricke involution W p = 0 − 1 / p p 0 . For any even integer k , let M k ! ( p ) be the space of weakly holomorphic (i.e., meromorphic with poles only at the cusps) modular forms f of weight k for Γ 0 ( p ) . For ε ∈ { ± 1 } , let M k ! , ε ( p ) be the subspace of M k ! ( p ) with f ∣ k W p = ε f . Each f ∈ M k ! , ε ( p ) has a Fourier development of the form

where the parameter q stands for exp ( 2 π i z ) , as usual. We set ord ∞ f = n 0 if a f ( n 0 ) ≠ 0 . Let S be the set consisting of values of p for which the genus of Γ 0 + ( p ) is zero, that is,

When p ∈ S , the space M k ! , ε ( p ) has a canonical basis: see [1,2, 3,4]. We define

When k > 2 , we note that m k ε is given by dim S k ε ( p ) . Indeed, for every integer m with − m ≤ m k ε , there exists a unique weakly holomorphic modular form f k , m ε ∈ M k ! , ε ( p ) with Fourier expansion of the form

and together they form a basis for M k ! , ε ( p ) . Indeed the basis element f k , m ε can be given explicitly in the form f k , m ε = f k , − m k ε ε F k , m + m k ε ( j p + ) , where j p + is the Hauptmodul for Γ 0 + ( p ) and F k , D ( x ) is a monic polynomial in x of degree D . Moreover, the Fourier coefficients a k ε ( m , n ) of q n in f k , m ε turn out to be rational integers. Throughout this paper, we simply write f k , m ≔ f k , m + .

As usual, we denote by S k + ( p ) the space of holomorphic cusp forms of weight k for Γ 0 + ( p ) . Let S k ! , + ( p ) be the subspace of M k ! , + ( p ) consisting of weakly holomorphic modular forms for Γ 0 + ( p ) with zero constant term in the Fourier expansion. We now consider Hecke operators on S k + ( p ) . For each positive integer n relatively prime to p , the usual Hecke operator T n on S k + ( p ) extends to S k ! , + ( p ) . In particular, for prime indices l ( ≠ p ) , the Hecke operators T ℓ on S k ! , + ( p ) are given as follows: for f ∈ S k ! , + ( p ) ,

Common eigenforms of all Hecke operators T n on S k + ( p ) with n coprime to p are called Hecke eigenforms. For later use, we let t = dim S k + ( p ) and

be a basis of S k + ( p ) consisting of normalized Hecke eigenforms. Following [5,6] we call f ∈ S k ! , + ( p ) a weakly holomorphic Hecke eigenform with respect to S k ! , + ( p ) / D k − 1 ( M 2 − k ! , + ( p ) ) if for every Hecke operator T n with ( n , p ) = 1 there is a complex number λ n for which

where D stands for the differential operator 1 2 π i d d z .

In the work of Bringmann et al. [5, Theorem 1.5] weakly holomorphic Hecke eigenforms are constructed in the level 1 case by making use of harmonic weak Maass forms which are preimages of Hecke eigenforms under the differential operator ξ k ≔ 2 i y k ∂ ∂ z ¯ ¯ originated from [7]. Thus, their construction is not explicit. In [6, Theorem 1.2], we extended it to higher level cases to the primes for which Γ 0 + ( p ) has genus zero (primes up to 71 excluding 37, 43, 61, and 67). The construction was given explicitly in terms of weakly holomorphic modular forms without relying on the theory of harmonic weak Maass forms, and we gave an explicit description of the “polar” eigenform h n in terms of a linear combination of cuspidal eigenforms and the dual form f n ∗ , as in (EF1)–(EF3). Here the duality is with respect to a certain pairing of functions introduced by Guerzhoy as follows.

Following [1,5,6,8,9], for f , g ∈ M k ! , + ( p ) , we define a pairing { f , g } originated from Bruinier and Funke [7] by

It is antisymmetric (since k is even), bilinear, and Hecke equivariant. Specifically, for any prime ℓ ( ≠ p )

Moreover, D k − 1 ( M 2 − k ! , + ( p ) ) is perpendicular to all of S k ! , + ( p ) , they are the only weakly holomorphic modular forms with that property (by the Serre duality theorem), and two elements of S k + ( p ) are perpendicular to one another. Let

with μ ( m , n ) ∈ C be a linear combination of f k , 1 , … , f k , t , which is dual to f n with respect to the pairing, i.e., { f m ∗ , f n } = δ m n , where δ m n is the Kronecker delta function. Then such functions f n ∗ are unique. Moreover, it follows from [1,9] that T ℓ f n ∗ and λ ( n , ℓ ) f n ∗ represent the same coset in S k ! , + ( p ) / ( D k − 1 ( M 2 − k ! , + ( p ) ) ⊕ S k + ( p ) ) . Thus, we can write

for some g n , ℓ ∈ M 2 − k ! , + ( p ) and a j n ( ℓ ) ∈ C .

Let p ∈ S , t = dim S k + ( p ) , and ℓ be a prime different from p . We then obtain from [6, Theorem 1.2] the following assertions.

(EF1) Let i , n ∈ { 1 , … , t } with i ≠ n . Let r be a prime ( ≠ p ) such that λ ( i , r ) ≠ λ ( n , r ) and put

Then x i ( n ) is independent of the choice of r and the quantity a n i ( r ) is given by

(EF2) For each n with 1 ≤ n ≤ t , let

Then h n is a Hecke eigenform with respect to S k ! , + ( p ) / D k − 1 ( M 2 − k ! , + ( p ) ) having the same eigenvalues as those of f n . More explicitly one has

where g n , ℓ is the modular form defined in (5) and computed as

where μ ( ⋅ , n ) is the Fourier coefficient of f n ∗ , defined in (4).

(EF3) The set

forms a basis for S k ! , + ( p ) / D k − 1 ( M 2 − k ! , + ( p ) ) , where [ f ] stands for the class of f .

For Γ ∈ { Γ 0 ( p ) , Γ 0 + ( p ) } , let H 2 − k ( Γ ) be the space of harmonic weak Maass forms of weight 2 − k for Γ . Following [10,11], we say that F ( z ) ∈ H 2 − k ( Γ 0 ( p ) ) is “good” for the Hecke eigenform f c ( z ) ≔ f ( − z ¯ ) ¯ ∈ S k ( p ) if it satisfies the following:

1. The principal part of F at the cusp ∞ belongs to K f [ q − 1 ] . Here K f denotes the number field obtained by adjoining to Q the Fourier coefficients of f .

2. The principal part of F at the cusp 0 is constant.

3. We have ξ 2 − k F = f c ( f c , f c ) .

Let f , g ∈ M k ! ( p ) . We define a regularized inner product as follows. For T > 0 , we denote by ℱ T the truncated fundamental domain for SL 2 ( Z )

Moreover, we define the truncated fundamental domain for Γ 0 ( p ) by

where V is a fixed set of representatives of Γ 0 ( p ) \ SL 2 ( Z ) . Now we define the regularized inner product ( f , g ) reg as the constant term in the Laurent expansion at s = 0 of the function

As explained in [10] and [15], ( f , g ) reg exists if f or g is a holomorphic modular form. If both f and g are holomorphic modular forms such that f g is a cusp form, then ( f , g ) reg reduces to the Petersson inner product ( f , g ) .

In the next theorems, we give an explicit construction of D k − 1 F in terms of polar eigenforms and a canonical basis. Theorem 1.2 applies for p = 1 only, while Theorems 1.3 and 1.5 cover higher levels.

Let P k − 2 denote the space of all polynomials of degree at most k − 2 . For any meromorphic function f on the complex upper half plane H , we define the action of γ = a b c d ∈ G L 2 + ( R ) by

For p ∈ { 1 , 2 , 3 } , a subspace W k − 2 + ( p ) of P k − 2 is defined by

with T = 1 1 0 1 , U = T W p , and n p = 3 if p = 1 2 p if p = 2 , 3 . We call the elements of the space W k − 2 + ( p ) period polynomials. Period polynomials have been investigated in relation to Eichler integrals, cusp forms via Eichler-Shimura isomorphisms, and to special values of modular L -functions. Indeed, Eichler [16] discovered relations between periods of cusp forms, and Shimura [17] extended them. Later, Manin [18] made more explicit the connection of these relations with the Fourier coefficients, by using the Hecke operators and continued fractions. For more discussion on the classical theory of period polynomials, one is referred to [18,19,20, 21,22] and [23, Chapter 12]. The period polynomials are also related to weakly holomorphic modular forms as follows. For each f = ∑ n ≫ − ∞ a f ( n ) q n ∈ M k ! , + ( p ) we define the period polynomial for f by

where c k = − Γ ( k − 1 ) ( 2 π i ) k − 1 and ℰ f denotes the Eichler integral

Let S k ! , + ( p ) be the subspace of M k ! , + ( p ) consisting of those f ∈ M k ! , + ( p ) with a f ( 0 ) = 0 . As described in [24, Section 4], r + gives a map from S k ! , + ( p ) to W k − 2 + ( p ) . For p ∈ { 1 , 2 , 3 } and even k > 2 , in virtue of [5, Theorem 1.6], [8, Theorem 1.1], [24, Theorem 1.2], and [25, Theorem 2], we have the following exact sequence:

Let p ∈ { 1 , 2 , 3 } , n be a positive integer relatively prime to p , and k be an even integer greater than 2. Following Knopp [19,20] we define a Hecke operator T ^ n on the space W k − 2 + ( p ) . Suppose that q ( z ) ∈ W k − 2 + ( p ) and F ( z ) is a meromorphic function on H satisfying

We also assume that F ( z ) is meromorphic in the local uniformizing variable at the cusp ∞ of a fundamental region for Γ 0 + ( p ) . In such a case, F ( z ) is called a modular integral for q ( z ) of weight 2 − k . We define T ^ n by

where

We then obtain that T ^ n ( q ( z ) ) ∈ W k − 2 + ( p ) and it follows from [6, Theorem 1.3] the following diagram

is commutative, i.e., r + is a Hecke equivariant homomorphism. We recall the Eisenstein series

and

where B k is the kth Bernoulli number and σ k − 1 denotes the usual divisor sum.

With the same notation as above, let p ∈ { 1 , 2 , 3 } and l be a prime different from p . We set p 0 ( z ) ≔ ( p z ) k − 2 − 1 . We then obtain from [6, Theorem 1.4] the following assertions.

(EP1) T ℓ ^ ( p 0 ) = ( 1 + ℓ 1 − k ) p 0 and T ℓ ^ ( r + ( f n ) ) = ℓ 1 − k λ ( n , ℓ ) r + ( f n ) for each n ∈ { 1 , … , t } .

(EP2) For each n with 1 ≤ n ≤ t we define

where c k is the constant defined in (7) and t ( m ) denotes the coefficient of q m in G k + ( z ) . Then we have

(EP3) The set

forms a basis for W k − 2 + ( p ) consisting of Hecke eigenpolynomials. As we see, the basis in (EP3) is constructed by using weakly holomorphic modular forms that are not holomorphic ones. Every period polynomial decomposes into an even and an odd part. By the Eichler-Shimura isomorphism, the space of even period polynomials consists of p 0 and those coming from cusp forms. Also the space of odd period polynomials can be constructed by using cusp forms. So it is natural to find a basis for the space W k − 2 + ( p ) consisting of odd and even period polynomials induced by only cusp forms that are Hecke eigenpolynomials. Although this result is well-known from the literature [18,21,22], we state it in the next theorem and reprove it by using the theory of harmonic weak forms because its proof will be used to prove our main results. For f ∈ S k + ( p ) , r − + ( f ) denotes r ( f , z ) − r ( f , − z ) 2 and r + + ( f ) stands for r ( f , z ) + r ( f , − z ) 2 i .

Since r + ( f n , − z ) ∈ W k − 2 + ( p ) , it follows from (9) that there exists g ∈ S k ! , + ( p ) such that

When p = 1 , by utilizing harmonic weak Maass forms, Bringmann et al. [5] constructed such g . So their construction is not explicit. In Theorem 1.8, we will provide an explicit construction of such g for all p ∈ S without using harmonic weak Maass forms.

We define a map r : S k + ( p ) × S k + ( p ) → W k − 2 + ( p ) by r ( f , g ) = r − + ( f ) + i r + + ( g ) and W 0 = im r . Then it is well known that W k − 2 + ( p ) = W 0 ⊕ ⟨ p 0 ⟩ and therefore we can consider a map P : W k − 2 + ( p ) → W 0 , which is the projection to the first component. When p = 1 , Bringmann et al. [5] suggested the following diagram of exact sequences

where a map Φ : S k ! , + ( p ) → S k + ( p ) × S k + ( p ) is not given explicitly. So they did not show that the diagram is commutative in the part.