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The Ramanujan Journal

, Volume 37, Issue 3, pp 461–467 | Cite as

On modular solutions of fractional weights for the Kaneko–Zagier equation for \(\varGamma ^*_0(2) \,\hbox {and}\, \varGamma ^*_0(3)\)

  • Yuichi Sakai
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Abstract

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.

Keywords

Modular form Differential equations The Heun series Supersingular \(j_N\)-polynomial 

Mathematics Subject Classification

Primary 11F11 Secondary 11F25 

1 Introduction and preliminaries

In [4], Kaneko–Koike studied various solutions for the so-called Kaneko–Zagier equation:
$$\begin{aligned} f^{\prime \prime }(\tau )-\frac{k+1}{6}E_2(\tau )f^\prime (\tau ) +\frac{k(k+1)}{12}E^\prime _2(\tau )f(\tau )=0, \end{aligned}$$
where \(\prime =(2\pi i)^{-1}d/d\tau =qd/dq\), \(q=e^{2\pi i\tau }\), \(\tau \) a variable in the Poincaré upper-half plane, \(k\) a fixed rational number, and \(E_2(\tau )\) is the (quasimodular) Eisenstein series of weight \(2\) for \(\mathrm {SL}_2(\mathbb {Z})\) defined by
$$\begin{aligned} E_2(\tau )=1-24\sum _{n=1}^\infty \Bigl (\sum _{d | n}d\Bigr )q^n. \end{aligned}$$
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 \(\varGamma _0^*(N) (N=2,3)\)
$$\begin{aligned} (\sharp )^{(N)}_k \quad f^{\prime \prime }(\tau )-\frac{k+1}{6-N}E_{NA}(\tau )f^\prime (\tau ) +\frac{k(k+1)}{2(6-N)}E_{NA}^\prime (\tau )f(\tau )=0 \end{aligned}$$
also has modular/quasimodular solutions similar to the case for \(\mathrm {SL}_2(\mathbb {Z})\), where the Fricke group of level \(N\) \((N=2,3)\) is defined by
$$\begin{aligned}&\varGamma _0^*(N)=\varGamma _0(N)\cup \varGamma _0(N)W_N,\qquad W_N= \left( {\begin{array}{ll} 0&{}-1/\sqrt{N}\\ \sqrt{N}&{}0 \end{array},}\right)&\\&\varGamma _0(N)=\biggl \{ \left( {\begin{array}{ll} a&{}b\\ c&{}d \end{array}}\right) \in \mathrm {SL}_2(\mathbb {Z})\ \biggl |\biggr .\ c\equiv 0\pmod {N}\biggr \},&\end{aligned}$$
and \(E_{NA}(\tau )\), the (quasimodular) Eisenstein series of weight \(2\) for \(\varGamma _0^*(N)\), is defined by
$$\begin{aligned} E_{NA}(\tau )=\frac{NE_2(N\tau )+E_2(\tau )}{N+1}. \end{aligned}$$
In this paper, we give modular forms of a fractional weight as a solution of the Kaneko–Zagier equation for \(\varGamma _0^*(N) (N=2,3)\). Hereafter, \(N\) denotes the level 2 or 3.
For any complex numbers \(v\) and \(s\), we take \(-\pi < \arg (v) \le \pi \) and put \(v^s= |v|^s e^{is\arg (v)}\). Define
$$\begin{aligned} \phi ^{(2)}_1(\tau )&=\Bigl (\frac{\eta (\tau )}{\eta (2\tau )^2}\Bigr )^{1/3}\frac{\eta (2\tau )\eta (3\tau )^2}{\eta (\tau )\eta (6\tau )}\\&=1 \!+\! \frac{2}{3}q \!+\! \frac{8}{9}q^2\! -\! \frac{50}{81}q^3\!+\! \frac{74}{243}q^4\!+\! \frac{320}{729}q^5\!+\!\frac{1232}{6561}q^6\!+\!\frac{7012}{19683} q^7\!+\!\cdots ,\\ \phi ^{(2)}_2(\tau )&=\Bigl (\frac{\eta (\tau )}{\eta (2\tau )^2}\Bigr )^{1/3}\frac{\eta (6\tau )^2}{\eta (3\tau )}\\&=q^{1/3}\Bigl (\!1 \!-\! \frac{1}{3}q\!+\! \frac{2}{9}q^2\!+\!\frac{40}{81}q^3\!+\!\frac{62}{243}q^4\!-\!\frac{307}{729}q^5\!+\!\frac{458}{6561} q^6\!-\!\frac{4136}{19683}q^7\!+\!\cdots \!\Bigr ),\\ \phi ^{(3)}_1(\tau )&=\Bigl (\frac{\eta (\tau )}{\eta (3\tau )^3}\Bigr )^{1/2}\frac{\eta (2\tau )^3\eta (3\tau )^2}{\eta (\tau )^2\eta (6\tau )}\\&=1 + \frac{3}{2}q+\frac{3}{8}q^2+\frac{15}{16}q^3+\frac{3}{128}q^4-\frac{99}{256}q^5 +\frac{1671}{1024}q^6+\frac{1383}{2048}q^7+\cdots ,\\ \phi ^{(3)}_2(\tau )&=\Bigl (\frac{\eta (\tau )}{\eta (3\tau )^3}\Bigr )^{1/2}\frac{\eta (6\tau )^3}{\eta (2\tau )}\\&=q^{1/2}\Bigl (\!1 - \frac{1}{2}q\!+\!\frac{3}{8}q^2\!+\!\frac{11}{16}q^3\!+\!\frac{35}{128}q^4\!-\!\frac{159}{256}q^5\!+\! \frac{359}{1024} q^6\!-\!\frac{573}{2048} q^7\!+\!\cdots \!\Bigr ), \end{aligned}$$
where \(\eta (\tau )=q^{1/24}\prod _{n=1}^{\infty }(1-q^n)\) is the Dedekind eta function. Then, we find that \(\phi ^{(N)}_1(\tau )\) and \(\phi ^{(N)}_2(\tau )\) are holomorphic modular forms of weight \(N/6\) for
$$\begin{aligned} \varGamma (6):=\Bigl \{ \left( {\begin{array}{ll} a&{}b\\ c&{}d \end{array}}\right) \in \varGamma _0(6) \Bigl | \Bigr .\ a\equiv d\equiv 1,\ b\equiv 0\pmod {6}\Bigr \} \end{aligned}$$
using properties of \(\eta (\tau )\) [7, §1.3 Theorem 1.7 and §2.3 Corollary 2.2]. Moreover, \((\phi ^{(N)}_1)^{6/N}\) and \((\phi ^{(N)}_2)^{6/N}\) are modular forms of weight \(1\) with the Legendre character \(\biggl (\dfrac{*}{3}\biggr )\) for \(\varGamma _0(6)\).

Remark 1

The following can be expressed in terms of theta series:
$$\begin{aligned} \frac{\eta (2\tau )\eta (3\tau )^2}{\eta (\tau )\eta (6\tau )}&=\sum _{n\in \mathbb {Z}}q^{(6n+1)^2/24},&\quad \frac{\eta (6\tau )^2}{\eta (3\tau )} \quad&=\sum _{n\in \mathbb {Z}}q^{3(4n+1)^2/8}. \end{aligned}$$
Note that if you find a solution \(F(\phi ^{(N)}_1,\ \phi ^{(N)}_2)\) of weight \(k\) for Eq. \((\sharp )^{(N)}_k\), you can get another solution \(F(\phi ^{(N)}_2,\ -\phi ^{(N)}_1/N)\) immediately because the group \(\varGamma _0^*(N)\) acts on the space of solutions as follows:
$$\begin{aligned} \left( {\begin{array}{c}\phi ^{(2)}_1\\ \phi ^{(2)}_2\end{array}}\right) \Biggl |_{\frac{1}{3}}\Biggr . \Biggl [ \begin{array}{ll} 2\sqrt{2}&{} \quad -3/\sqrt{2}\\ 3\sqrt{2}&{} \quad -2\sqrt{2} \end{array} \Biggr ]&= \sqrt{2}e^{-\frac{2}{3}\pi i} \left( {\begin{array}{c}\phi ^{(2)}_2\\ -\frac{1}{2}\phi ^{(2)}_1\end{array}}\right) ,\\ \left( {\begin{array}{c}\phi ^{(3)}_1\\ \phi ^{(3)}_2\end{array}}\right) \Biggl |_{\frac{1}{2}}\Biggr . \Biggl [ \begin{array}{ll} -\sqrt{3}&{} \quad -4/\sqrt{3}\\ 2\sqrt{3}&{} \quad -3\sqrt{3} \end{array} \Biggr ]&= \sqrt{3}e^{-\frac{1}{4}\pi i} \left( {\begin{array}{c}\phi ^{(3)}_2\\ -\frac{1}{3}\phi ^{(3)}_1\end{array}}\right) , \end{aligned}$$
where \(F(X,Y)\) is a homogenous polynomial of two variables, and \(|_k[\cdot ]\) is a slash operator of weight \(k\).
Finally, Heun’s local series \(Hl\) is defined by
$$\begin{aligned} Hl(a,w;\alpha ,\beta ,\gamma ,\delta ;x)={\displaystyle \sum _{n=0}^\infty c_n x^n}, \end{aligned}$$
where the coefficients satisfy the recursion; \(c_0=1,\ \ c_1=\frac{w}{a\gamma }c_0,\) and
$$\begin{aligned} c_{n\!+\!1}\!=\!\frac{(n[(n\!-\!1\!+\!\gamma )(1\!+\!a)\!+\!a\delta \!+\!\varepsilon ]\!+\!w)}{(n\!+\!1)(n\!+\!\gamma )a}c_n \!-\!\frac{(n\!-\!1\!+\!\alpha )(n\!-\!1\!+\!\beta )}{(n\!+\!1)(n\!+\!\gamma )a}c_{n\!-\!1}\qquad (n\ge 1), \end{aligned}$$
where \(\gamma +\delta +\varepsilon =\alpha +\beta +1\). 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]):
$$\begin{aligned} \frac{d^2y}{dx^2}+\biggl (\frac{\gamma }{x}+\frac{\delta }{x-1}+\frac{\varepsilon }{x-a}\biggr ) \frac{dy}{dx}+\frac{\alpha \beta x-w}{x(x-1)(x-a)}y=0. \end{aligned}$$
(1)
In particular, \(Hl\) is a polynomial when \(\alpha \ \text {or}\ \beta \in -\mathbb {N}\).

2 Main result

Theorem 1

  1. (1)
    Assume \(k=(4n+1)/3\ \text {such that}\ n=0,1,2,\ldots ,\ n\not \equiv 2\pmod {3}\). Then Eq. \((\sharp )^{(2)}_k\) has a two-dimensional space of solutions in \(\mathbb {C}[\phi ^{(2)}_1,\phi ^{(2)}_2]_{wt=k}\). Its generators are
    $$\begin{aligned} \phi ^{(2)}_1(\tau )^{3k} Hl\Bigl (-8, \frac{k(1\!-\!3k)}{2}; \!-\!k, \frac{1\!-\!3k}{4}, \frac{3\!-\!k}{4}, \frac{1\!-\!3k}{4}; 8\frac{\phi ^{(2)}_2(\tau )^3}{\phi ^{(2)}_1(\tau )^3}\Bigr )\\ \!=\! 1\!+\!O(q),\\ \phi ^{(2)}_2(\tau )^{\frac{k\!+\!1}{4}}\phi ^{(2)}_1(\tau )^\frac{11k-1}{4} Hl\Bigl (-8, \frac{(k\!-\!7)(1\!-\!3k)}{16}; \frac{1\!-\!3k}{4}, \frac{1\!-\!k}{2},\frac{k\!+\!5}{4}, \frac{1\!-\!3k}{4};\\ 8\frac{\phi ^{(2)}_2(\tau )^3}{\phi ^{(2)}_1(\tau )^3}\Bigr )\\ \ \!=\!q^\frac{k\!+\!1}{12}\!+\!O(q^\frac{k\!+\!13}{12}). \end{aligned}$$
     
  2. (2)
    Assume \(k=(3n+1)/2\ \text {such that}\ n=0,1,2,\ldots ,\ n\not \equiv 1\pmod {2}\). Then Eq. \((\sharp )^{(3)}_k\) has a two-dimensional space of solutions in \(\mathbb {C}[\phi ^{(3)}_1,\phi ^{(3)}_2]_{wt=k}\). Its generators are
    $$\begin{aligned} \phi ^{(3)}_1(\tau )^{2k} Hl\Bigl (9, k(2k\!-\!1); \!-\!k, \frac{1\!-\!2k}{3}, \frac{2\!-\!k}{3}, \frac{1\!-\!2k}{3}; 9\frac{\phi ^{(3)}_2(\tau )^2}{\phi ^{(3)}_1(\tau )^2}\Bigr )\\ \!=\!1\!+\!O(q),\\ \phi ^{(3)}_2(\tau )^{\frac{k\!+\!1}{3}}\phi ^{(3)}_1(\tau )^\frac{5k\!-\!1}{3} Hl\Bigl (9, \frac{(k\!+\!10)(1\!-\!2k)}{9}; \frac{1\!-\!2k}{3}, \frac{2\!-\!k}{3},\frac{k\!+\!4}{3}, \frac{1\!-\!2k}{3};\\ 9\frac{\phi ^{(3)}_2(\tau )^3}{\phi ^{(3)}_1(\tau )^3}\Bigr )\\ \ \!=\!q^\frac{k\!+\!1}{6}\!+\!O(q^\frac{k\!+\!7}{6}). \end{aligned}$$
     

Remark 2

By the conditions for weight \(k\), the Heun local series in the above theorem become polynomials.

3 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.

Proposition 1

([10, p.18]) If \(Hl(a, w; \alpha , \beta , \gamma , \delta ; x)\) is a solution of the Heun differential equation (1), then \(x^{1-\gamma } Hl(a, w^\prime ; \alpha ^\prime , \beta ^\prime , \gamma ^\prime , \delta ; x)\) is also a solution of Eq. (1), where \(\alpha ^\prime =\alpha +1-\gamma \), \(\beta ^\prime =\beta +1-\gamma \), \(\gamma ^\prime =2-\gamma \), \(w^\prime =(a\delta +\varepsilon )(1-\gamma )+w\).

Putting \(f(\tau )/\phi ^{(2)}_1(\tau )^{3k}\!=\!g(\tau )\) and \(X\!=\!8\phi ^{(2)}_2(\tau )^3/\phi ^{(2)}_1(\tau )^3\), Eq. \((\sharp )_k^{(2)}\) can be transformed into
$$\begin{aligned} g^{\prime \prime }(\tau )&\!+\!\Bigl (\!-\!\frac{1}{4}E_{2A}(\tau )\!+\!\frac{k}{32}(X^2\!+\!28X\!-\!8) \phi ^{(2)}_1(\tau )^6\Bigr )g^\prime (\tau )\\&\!+\!\frac{k(3k\!-\!1)}{256}X(X\!-\!1)(X\!+\!2)(X\!+\!8)\phi ^{(2)}_1(\tau )^{12}g(\tau )\!=\!0. \end{aligned}$$
Using the relation of derivatives between \(2\pi i\tau \) and \(X\):
$$\begin{aligned} g^\prime (\tau )&\!=\!\!-\!\frac{1}{8}X(X\!-\!1)(X\!+\!8)\phi ^{(2)}_1(\tau )^6\frac{dg}{dX},\\ g^{\prime \prime }(\tau )&\!=\!\frac{1}{64}X^2(X\!-\!1)^2(X\!+\!8)^2\phi ^{(2)}_1(\tau )^{12}\frac{d^2g}{dX^2}\\&\quad \!+\!\frac{1}{256}X(X\!-\!1)(X\!+\!8)\phi ^{(2)}_1(\tau )^6\Bigl ((5X^2\!+\!28X\!-\!24) \phi ^{(2)}_1(\tau )^6-8E_{2A}(\tau ) \Bigr ) \frac{dg}{dX}, \end{aligned}$$
we have
$$\begin{aligned} \frac{d^2g}{dX^2}\!+\!\Bigl (\frac{1\!-\!3k}{4X}\!+\!\frac{3\!-\!k}{4(X\!-\!1)}\!+\!\frac{1\!-\!3k}{4(X\!+\!8)}\Bigr )\frac{dg}{dX} \!+\!\frac{k(3k\!-\!1)}{4}\cdot \frac{X\!+\!2}{X(X\!-\!1)(X\!+\!8)} g=0. \end{aligned}$$
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 \(\alpha \) or \(\beta \), \(\alpha ^\prime \) or \(\beta ^\prime \ \in -\mathbb {N}\). Therefore, \(f=\phi ^{(2)}_1(\tau )^{3k}\cdot g\) is a modular solution of \((\sharp )_k^{(2)}\).

4 The relation to supersingular \(j_{NA}\)-polynomials

Koike defined in [9] (or see [2, 8, 11]) supersingular \(j_{NA}\)-polynomials. He proved that for the elliptic modular invariant \(j(\tau )\) over a finite field of characteristic \(p > 0\) to be supersingular is equivalent to the Hauptmodul \(j_{NA}(\tau )\) for \(\varGamma _0^*(N)\) being supersingular. From his result with respect to a supersingular elliptic curve, we get the following definition.

Definition 1

For a prime number \(p(\ge 5)\), we define the “supersingular \(j_{NA}\)-polynomials for \(\varGamma _0^*(N)\)” by
$$\begin{aligned} S^{(2A)}_p(X)&:= X^{\delta _2}(X-256)^{\varepsilon _2} {\left\{ \begin{array}{ll} X^{m_2} \mathbb {F}\bigl (\frac{1}{8}, \frac{3}{8}, 1, \frac{256}{X}\bigr ) &{} p\equiv 1,\ 3\pmod {8},\\ X^{m_2} \mathbb {F}\bigl (\frac{7}{8}, \frac{5}{8}, 1, \frac{256}{X}\bigr ) &{} p\equiv 5,\ 7\pmod {8}, \end{array}\right. }\\ S^{(3A)}_p(X)&:= X^{\delta _3}(X-108)^{\delta _3} {\left\{ \begin{array}{ll} X^{m_3} \mathbb {F}\bigl (\frac{1}{6}, \frac{1}{3}, 1, \frac{108}{X}\bigr ) &{} p\equiv 1\pmod {6},\\ X^{m_3} \mathbb {F}\bigl (\frac{2}{3}, \frac{5}{6}, 1, \frac{108}{X}\bigr ) &{} p\equiv 5\pmod {6}, \end{array}\right. } \end{aligned}$$
where \(m_2=[\frac{p}{8}]\), \(p-1=8m_2+2\delta _2+4\varepsilon _2\), \(m_3=[\frac{p}{6}]\), \(p-1=6m_3+4\delta _3\), \(\delta _2, \varepsilon _2, \delta _3 \in \{0,\ 1\}\), and \(\mathbb {F}(\alpha , \beta , \gamma ,x)\) is the hypergeometric series over a finite field of characteristic \(p>0\).

In this section, we present a certain conjecture about a reduction mod prime \(p\) of Heun polynomials.

Let
$$\begin{aligned} j_{2A}(X_2)&=\frac{(8-20X_2-X_2^2)^4}{X_2(1-X_2)^3(8+X_2)^3},&\quad j_{3A}(X_3)&=\frac{(X_3+3)^6}{X_3(1-X_3)^2(9-X_3)^2} \end{aligned}$$
be the Hauptmodul for \(\varGamma _0^*(N)\) expressed in terms of \(X_2=8\bigl (\phi ^{(2)}_2/\phi ^{(2)}_1\bigr )^3\) and \(X_3=9\bigl (\phi ^{(3)}_2/\phi ^{(3)}_1\bigr )^2\), and further let
$$\begin{aligned} T^{(2)}_n(X_2)&=Hl\Bigl (-8,-\frac{2n(4n+1)}{3}; -\frac{4n+1}{3}, -n, \frac{2-n}{3}, -n; X_2\Bigr )\, \hbox {and}\\ T^{(3)}_n(X_3)&=Hl\Bigl (9,\frac{3n(3n+1)}{2}; -\frac{3n+1}{2}, -n, \frac{1-n}{2}, -n; X_3\Bigr ) \end{aligned}$$
be the Heun polynomials of degree \(n(>0)\).

Conjecture 1

Let \(p>5\) be a prime. Then \(T^{(N)}_{p-1}(X_N)\mod p\) is a “supersingular \(X_N\)-polynomial,” i.e., it is equal to \(\prod _{Y_N\in \overline{\mathbb {F}}_p}(X_N-Y_N)\), where \(Y_N\) runs through those values for which the corresponding Hauptmodul \(j_{NA}(Y_N)\) is supersingular.

5 The function like characters

From [4], we know that some solutions of the Kaneko–Zagier equation for \(\mathrm {SL}_2(\mathbb {Z})\) 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 \(\eta (\tau )^{2k}\).

From numerical examination, for the Fricke group of levels \(2\) and \(3\), we can get something like the character from the solution \(f(\tau )\) of weight \(k\) divided by \(\Delta _{NA}(\tau )^{k/(12-2N)}\), where \(\Delta _{2A}=\eta (\tau ^8)\eta (2\tau )^8\) and \(\Delta _{3A}=\eta (\tau )^6\eta (3\tau )^6\) are cusp forms for \(\varGamma _0(N)\). For example, in the case for \(\varGamma ^*_0(2)\), we get the following:
  1. (a)
    For \(k=1/3\),
    $$\begin{aligned} \phi ^{(2)}_1/\Delta _{2A}^{1/24}=\frac{1}{q^\frac{1}{24}} + q^\frac{23}{24}+ 2 q^\frac{47}{24} + q^\frac{71}{24} + 3 q^\frac{95}{24} +3q^\frac{119}{24}+5 q^\frac{143}{24}+5 q^\frac{167}{24} + 8 q^\frac{191}{24}+\cdots ; \end{aligned}$$
    the number of partitions of n in which no part appears more than twice and no two parts differ by 1.
     
  2. (b)
    For \(k=2\),
    $$\begin{aligned} \frac{(\phi ^{(2)}_1)^6+20(\phi ^{(2)}_1 \phi ^{(2)}_2)^3-8(\phi ^{(2)}_2)^6}{\Delta _{2A}^{1/4}}= \frac{1}{q^\frac{1}{4}} + 26 q^\frac{3}{4} + 79 q^\frac{7}{4} + 326 q^\frac{11}{4} + 755 q^\frac{15}{4}\\ + 2106 q^\frac{19}{4} +\cdots , \end{aligned}$$
    the McKay–Thompson series of class 8C for the Monster. (cf. [1])
     
For several other weights \(k\), we observe that each coefficient of \(f(\tau )/\Delta _{NA}(\tau )^{k/(12\!-\!2N)}\) is a positive integer, where \(f(\tau )\) is a modular solution of weight \(k\) for \((\sharp )^{(N)}_k\). But we do not know to which the function corresponds and what are the properties of these functions.

Notes

Acknowledgments

The author would like to thank Professor Masanobu Kaneko and Professor Kiyokazu Nagatomo for giving this opportunity for conducting this study. Especially, he would like to thank Professor Masanobu Kaneko for helpful suggestions and support. Finally, he would like to thank the reviewer for helpful comments and suggestions.

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Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityNishi-kuJapan

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