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

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.

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 $\alpha \ \text {or}\ \beta \in -\mathbb {N}$.

Main result

Theorem 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) 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.

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 $(\sharp )_k^{(2)}$.

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.

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:

• (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.

• (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.