On modular solutions of fractional weights for the Kaneko–Zagier equation for Γ ∗ 0 ( 2 ) and Γ ∗ 0 ( 3 )

For the full modular group, Kaneko gave a modular form solution of fractional weights for a holomorphic modular differential equation of second order. In this paper, we give modular form solutions of fractional weights for a modular differential equation on the Fricke group of levels 2 and 3.


Introduction and preliminaries
In [4], Kaneko-Koike studied various solutions for the so-called Kaneko-Zagier equation: where = (2πi) −1 d/dτ = qd/dq, q = e 2πiτ , τ a variable in the Poincaré upper-half plane, k a fixed rational number, and E 2 (τ ) is the (quasimodular) Eisenstein series of weight 2 for SL 2 (Z) defined by They gave modular forms expressed in terms of hypergeometric polynomials and quasimodular forms as its solution for weight k, where k is integer or half-integer. In particular, Kaneko studied in [3] the modular form as its solution for weight one-fifth, which is closely related to certain models in conformal field theory. From [5,6] and [11], we know that the Kaneko-Zagier equation for also has modular/quasimodular solutions similar to the case for SL 2 (Z), where the Fricke group of level N (N = 2, 3) is defined by and E N A (τ ), the (quasimodular) Eisenstein series of weight 2 for Γ * 0 (N ), is defined by In this paper, we give modular forms of a fractional weight as a solution of the Kaneko-Zagier equation for Γ * 0 (N )(N = 2, 3). Hereafter, N denotes the level 2 or 3. For any complex numbers v and s, we take −π < arg(v) ≤ π and put v s = |v| s e is arg(v) . Define φ (2)  for Γ 0 (6).

Remark 1
The following can be expressed in terms of theta series: acts on the space of solutions as follows: where F(X, Y ) is a homogenous polynomial of two variables, and | k [·] is a slash operator of weight k. Finally, Heun's local series Hl is defined by where the coefficients satisfy the recursion; c 0 = 1, c 1 = w aγ c 0 , and where γ + δ + ε = α + β + 1. This is a solution of Heun's equation, which is the canonical form of a second-order linear differential equation with four regular singularities (cf. [10]): In particular, Hl is a polynomial when α or β ∈ −N. Then Eq. ( ) (2) k has a two-dimensional space of solutions in C[φ (2) 1 , φ (2) 2 ] wt=k . Its generators are (2) Assume k = (3n + 1)/2 such that n = 0, 1, 2, . . . , n ≡ 1 (mod 2). Then Eq.
( ) 2 ] wt=k . Its generators are Remark 2 By the conditions for weight k, the Heun local series in the above theorem become polynomials.

Proof
We will prove the result only for the case of level 2. The case of level 3 can be treated in a similar manner. To prove this theorem, we need the following proposition. Hl(a, w; α, β, γ , δ; x) is a solution of the Heun differential equation (1), then x 1−γ Hl(a, w ; α , β , γ , δ; x) is also a solution of Eq. (1),

Proposition 1 ([10, p.18]) If
Putting f (τ )/φ (2) 1 (τ ) 3k = g(τ ) and X = 8φ 3 , Eq. ( ) (2) k can be transformed into Using the relation of derivatives between 2πiτ and X : we have Comparing this equation with Eq. (1), we can obtain Heun's solution. Using Proposition 1, we can obtain another solution, and the two solutions are polynomials, because α or β, α or β ∈ −N. Therefore, f = φ In this section, we present a certain conjecture about a reduction mod prime p of Heun polynomials. Let 2 , and further let , −n; X 2 and be the Heun polynomials of degree n(> 0).

Conjecture 1 Let p > 5 be a prime. Then T
where Y N runs through those values for which the corresponding Hauptmodul j N A (Y N ) is supersingular.

The function like characters
From [4], we know that some solutions of the Kaneko-Zagier equation for SL 2 (Z) are closely related to the character for two-dimensional conformal field theory. Precisely, we can get the character from the solution of weight k divided by η(τ ) 2k .
From numerical examination, for the Fricke group of levels 2 and 3, we can get something like the character from the solution f (τ ) of weight k divided by N A (τ ) k/(12−2N ) , where 2 A = η(τ 8 )η(2τ ) 8 and 3A = η(τ ) 6 η(3τ ) 6 are cusp forms for Γ 0 (N ). For example, in the case for Γ * 0 (2), we get the following: (a) For k = 1/3, the number of partitions of n in which no part appears more than twice and no two parts differ by 1.   For several other weights k, we observe that each coefficient of f (τ )/ N A (τ ) k/ (12−2N ) is a positive integer, where f (τ ) is a modular solution of weight k for ( ) (N ) k . But we do not know to which the function corresponds and what are the properties of these functions.