Cycle integrals and rational period functions for Γ₀⁺(2) and Γ₀⁺(3)

1 Introduction and statement of results

For a positive integer p , let Γ 0 ( p ) be the Hecke congruence subgroup consisting of all 2 × 2 matrices a b c d  with integer entries satisfying c ≡ 0 ( mod p ) . Further, let Γ 0 + ( p ) be the group generated by Γ 0 ( p ) and the Fricke involution W p = 0 − 1 ⁄ p p 0 . Let k be an even integer and ε C be the character on Γ 0 + ( p ) defined as

For a meromorphic function f on the complex upper half plane H , we define the action ∣ k , ε C for γ = a b c d ∈ Γ 0 + ( p ) as

Let T ≔ 1 1 0 1 and U = T W p = p − 1 ⁄ p p 0 . For p ∈ { 1 , 2 , 3 } , we consider a rational function q ( z ) satisfying

where n p = 3 if p = 1 , 2 p if p = 2 , 3 . Such a function q ( z ) is called a rational period function of weight k for Γ 0 + ( p ) and denote by RPF k , ε ( p ) the set of all such functions for C = ε 1 (see [1–4]). Here, we note that, when p = 1 , W 1 belongs to Γ 0 + ( 1 ) = SL 2 ( Z ) , and hence, we only need to consider the case C = + 1 .

For negative k , if the rational period functions are polynomials, we call them period polynomials of weight k for Γ 0 + ( p ) and denote by W − k ε ( p ) the set of all such period polynomials for C = ε 1 . That is,

We define a modular integral on Γ 0 + ( p ) of weight k to be a holomorphic function f on H that has a Fourier expansion at ∞ :

and satisfies

for a rational function q ( z ) , where C ∈ { + 1 , − 1 } . Then, q ( z ) ∈ RPF k , ε ( p ) (see [1–5]).

The study of period polynomials and rational period functions has been a significant area of research in the theory of modular forms. Knopp [1,2] initiated the study of rational period functions and their relation with modular integrals on SL 2 ( Z ) . Over the years, the theory of period polynomials and rational period functions has been significantly developed by many authors, and one of the significant advancements in the study of rational period functions was made by Duke et al. [6]. They provided an effective basis for the space of period polynomials W − k + ( 1 ) using weakly holomorphic modular forms on SL 2 ( Z ) and by using cycle integrals, explicitly constructed modular integrals for rational period functions for SL 2 ( Z ) arising from indefinite binary quadratic forms. Indeed, Knopp [1,2] already showed that rational period functions for SL 2 ( Z ) have modular integrals. But his construction is very difficult to compute. Choi and Kim [3,5] extended the results of Duke et al. to the space W − k + ( p ) and modular integrals on Γ 0 + ( p ) for p ∈ { 2 , 3 } and C = 1 . This article aims to further extend these results by studying the space W − k − ( p ) and rational period functions for Γ 0 + ( p ) for p ∈ { 2 , 3 } and C = − 1 .

For any even integer k and a prime p , let M k ( p ) (resp. S k ( p ) ) be the space of holomorphic modular forms (resp. cusp forms) of weight k on Γ 0 ( p ) . For ε ∈ { + , − } , we denote by M k ε ( p ) the subspace of M k ( p ) consisting of all eigenforms f of the Fricke involution ∣ k W p such that the eigenvalue of f is ε 1 , that is,

Similarly, we can also define two subspaces S k + ( p ) and S k − ( p ) of S k ( p ) as

Here ∣ k is the usual slash operator which is given by ( f ∣ k γ ) ( z ) = ( det γ ) k ⁄ 2 ( c z + d ) − k f ( γ z ) for γ = a b c d ∈ GL 2 ( R ) . Further, let M k ! ( p ) be the space of weakly holomorphic modular forms (i.e., meromorphic with poles only at the cusps) of weight k for Γ 0 ( p ) and let M k ! , ε ( p ) be the subspace of M k ! ( p ) consisting of all eigenforms f of the action ∣ k W p such that the eigenvalue of f is ε 1 . Then, it can be easily seen that

It is well-known that each f ∈ M k ! , ε ( p ) has a Fourier expansion at the cusp ∞ of the form

We set ord ∞ f = n 0 if n 0 is the smallest integer such that a f ( n 0 ) ≠ 0 . We additionally define the space S k ! , ε ( p ) by the subspace of M k ! , ε ( p ) consisting of weakly holomorphic modular forms with zero constant term in the Fourier expansion at the cusp ∞ . It is known [7,8] that when the genus of Γ 0 + ( p ) is zero, the space M k ! , ε ( p ) has a canonical basis: Let m k ε denote the maximal order of a nonzero f ∈ M k ! , ε ( p ) at ∞ . For every integer 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 at the cusp ∞ :

and together they form a basis for M k ! , ε ( p ) .

As previously mentioned, this article focuses on studying the rational period functions for Γ 0 + ( p ) and modular integrals on Γ 0 + ( p ) in the specific cases where p ∈ { 2 , 3 } and C = − 1 . In this case, the functional equation (1) can be written as

while the functional equation (2) for modular integrals becomes

using the usual slash operator.

The following is our first main result that gives the dimension of the space W k − 2 − ( p ) for p ∈ { 2 , 3 } and its relation to the minus space S k − ( p ) of cusp forms.

In addition to determining the dimension of the space W k − 2 − ( p ) of period polynomials, our second result introduces a method to construct an explicit basis for this space. For the basis construction, we utilize the Eichler integral of the canonical basis, where the Eichler integral is defined as follows: Suppose that k > 2 . For f ∈ ∑ a f ( n ) q n ∈ M k ! , ε ( p ) , we define the Eichler integral of f by

In addition, the period function for f is defined by

where c k = − Γ ( k − 1 ) ( 2 π i ) k − 1 . We note that for f ∈ S k ! , ε ( p ) , the function r ε ( f ; z ) is a polynomial in z of degree at most k − 2 with coefficients in C . Indeed, this fact follows from Bol’s identity. Specifically, for f ∈ M 2 − k ! , ε ( p ) and γ ∈ SL 2 ( R ) , we have

This identity implies that D k − 1 ( M 2 − k ! , ε ( p ) ) ⊆ S k ! , ε ( p ) .

To state our third result, consider a positive integer D congruent to a square modulo 4 p . In this context, Q D , p denotes the set of integral binary quadratic forms Q ( x , y ) = a x 2 + b x y + c y 2 = [ a , b , c ] , where p divides a , and the discriminant D = b 2 − 4 a c . The group Γ 0 ( p ) acts on Q D , p through the operation Q ↦ g Q , where g = α β γ δ ∈ Γ 0 ( p ) and Q ∈ Q D , p . This action is defined by

The set of classes Γ 0 ( p ) \ Q d , p is finite. Furthermore, if g = ± α β γ δ ∈ Γ 0 ( p ) , then g − 1 Q ( τ , 1 ) = Q ( g τ , 1 ) ( γ τ + δ ) 2 . From now on, assume that D is positive and not a perfect square. Note that Γ 0 ( p ) Q ⊆ Γ 0 + ( p ) Q , where Γ 0 ( p ) Q = { g ∈ Γ 0 ( p ) ∣ g Q = Q } and Γ 0 + ( p ) Q = { g ∈ Γ 0 + ( p ) ∣ g Q = Q } .

For Q = [ a , b , c ] ∈ Q D , p , let S Q be the oriented semi-circle defined by a ∣ τ ∣ 2 + ( Re ( τ ) ) b + c = 0 , with counterclockwise orientation for a > 0 and clockwise for a < 0 . Note that S g Q = g S Q for any g ∈ Γ 0 ( p ) . Let h Q be a generator of Γ 0 ( p ) Q . (For a detailed definition of h Q , see Remark 5.2.) For a weakly holomorphic modular form f ∈ M k ! ( p ) and Q ∈ Q d , p , we define

where C Q = C Q ( τ 0 ) is the directed arc on S Q from τ 0 ∈ S Q to h Q τ 0 , with the same orientation as S Q , and d τ Q ≔ Q ( τ , z ) k 2 − 1 d τ .

While the following result is parallel to [3, Theorem 1.5(i)(ii)], it has an application (see Corollary 1.5) which is specific to the case under consideration and has no direct analog in the setting adopted in [3].

The following corollary shows that for each m ≥ − m k − , we have r Q ( f k , m − ) = 0 under some condition.

This work extends the understanding of period polynomials with eigenvalue − 1 under the Fricke involution and modular integrals, building upon previous studies such as [6] and [3]. The proofs of Theorem 1.2 and 1.3 are based on the main ideas presented in [6], while our research specifically expands on the results of [3] to the case of eigenvalue − 1 under the Fricke involution. A key distinction of this article from previous work, particularly that of Choi and Kim [3], is our approach to calculating cycle integrals within Γ 0 ( p ) rather than Γ 0 + ( p ) . This shift requires a more refined computational method, enabling us to determine the exact values of cycle integrals. Our method allows for a more precise and comprehensive understanding of the relationship between cycle integrals and rational period functions in this setting.

The rest of the article is organized as follows. In Section 2, we provide examples illustrating our main results, including numerical calculations. Section 3 is dedicated to proving the dimension formula for W k − 2 − ( p ) stated in Theorem 1.1. In Section 4, we focus on constructing the basis for W k − 2 − ( p ) , as described in Theorem 1.2. Finally, the proofs of Theorem 1.3 and the proof of Corollary 1.5 are presented in Section 5.

2 Numerical examples

Let p = 2 and k = 10 . In this case, the space S 10 − ( 2 ) is one dimensional and spanned by

where η ( z ) = q 1 ⁄ 24 ∏ n = 1 ∞ ( 1 − q n ) denotes the Dedekind eta function and E 2 ( z ) = 1 − 24 ∑ n = 1 ∞ σ ( n ) q n is the normalized Eisenstein series of weight 2. Using [8], one can construct the first three basis elements of the space M 10 ! , − ( 2 ) as follows:

and their q -expansions are of the form

Here, j 2 + ( z ) denotes the Hauptmodul for Γ 0 + ( 2 ) described by

According to Theorem 1.1, the space W 8 − ( 2 ) is three-dimensional, and by Theorem 1.2, it is spanned by the polynomials 1 + 16 z 8 , ψ 10 , − 1 ( z ) , and ψ 10 , 1 ( z ) , where ψ 10 , m ( z ) is given by ψ 10 , m ( z ) = r − ( f 10 , m − ; z ) .

Using the n th Fourier coefficients of f 10 , m − for n ≤ 500 , one can explicitly compute the coefficients of the polynomials ψ 10 , m . The coefficients of each polynomial are given in Table 1. (The coefficients presented in this table are rounded off.)

For each f ∈ M 10 ! , + ( 2 ) and Q ∈ Q d , 2 , it follows from the definition of the cycle integral that

where C Q = C Q ( τ 0 ) is the directed arc on S Q from τ 0 ∈ S Q to g Q τ 0 .

We consider two cases.

Case (i) D = 8 and Q = [ 2 , 0 , − 1 ] ∈ Q 8 , 2 : In this case, we find that the smallest positive solution ( t 0 , u 0 ) of the Diophantine equation ( 2 t ) 2 − 8 u 2 = 8 is given by ( t 0 , u 0 ) = ( 2 , 1 ) . Thus, employing [3, Theorem 1.3 (i)], we see that the group Γ 0 + ( 2 ) Q ⁄ { ± 1 } is generated by 1 2 2 − 1 − 2 2 ∈ Γ 0 ( 2 ) W 2 , and hence, h Q in Remark 5.2 is given by

In (4), we take τ 0 = i and use the parametrization

Then, we can compute that the values of r Q ( f 10 , m − ) are equal to zero, as expected from Corollary 1.5.

Case (ii) D = 17 and Q = [ 2 , 1 , − 2 ] ∈ Q 17 , 2 : In this case, we find that the smallest positive solution ( t 1 , u 1 ) of the Pell equation t 2 − 17 u 2 = 4 is given by ( t 1 , u 1 ) = ( 66 , 16 ) . Thus, employing [3, Theorem 1.3 (ii)], we see that the group Γ 0 + ( 2 ) Q ⁄ { ± 1 } is generated by

In (4), we take τ 0 = − 1 4 + 17 4 i and use the parametrization

Then, one can estimate the values of r Q ( f 10 , m − ) which are listed in Table 2.

Here, each coefficient was rounded off to 30 digits. Meanwhile, one has

Thus, we find that

Thus, one should have an equality

which can be verified numerically from the values listed in Tables 1 and 2.

3 Proof of Theorem 1.1

We begin this section by noting that the matrix μ = − 1 0 0 1 satisfies the following properties:

• μ W p μ = − W p ,

• μ U ζ μ = W p U 2 p − ζ W p for every integer 0 < ζ < 2 p ,

• For a polynomial P ( z ) of degree at most k − 2 , the equality ( − 1 ) k ⁄ 2 P ( − z ) = − P ( z ) ∣ 2 − k μ = 0 holds if and only if P ( z ) = 0 .