Modular equations for congruence subgroups of genus zero (II)

Abstract

We present a result that the modular equation of a Hauptmodul for a certain congruence subgroup ${\mathrm{\Gamma }}_H(N,t)$ of genus zero satisfies Kronecker's congruence relation. This generalizes the author's previous result about ${\mathrm{\Gamma }}_1(m)\bigcap {\mathrm{\Gamma }}_0(mN)$. Furthermore we show that the similar result holds for a certain congruence subgroup Γ of genus zero with $[\mathrm{\Gamma }:{\mathrm{\Gamma }}_H(N,t)]=2$. Finally we prove a conjecture of Lee and Park, asserting that the modular equation of the continued fraction of order six satisfies a certain form of Kronecker's congruence relation.

Introduction

It is well known that the modular equation ${\mathrm{\Phi }}_n(x,y)$ of the elliptic modular function $j(\tau )$ for the full modular group ${\mathrm{SL}}_2(\mathbb{Z})$ satisfies the so-called Kronecker's congruence relation

$${\mathrm{\Phi }}_p(x,y)\equiv (x^p-y)(x-y^p)(\mathrm{mod}p\mathbb{Z}[x,y])$$

for any prime p (see [8, §2 in Chapter 5] or [6, Theorem 11.18]). After this classical result, Chen and Yui [2, Theorem 2.6], Cais and Conrad [1, Section 6], Cho, Koo, and Park [4], [5], and Cho [3, Theorem 2.1] showed that modular equations of other Hauptmoduln also satisfy certain forms of Kronecker's congruence relation. For more details, see the introduction in the author's article [3]. Instead, it is worth noting that every Kronecker's congruence relation that appeared in the articles above was of the form

$$(x^p-y)(x-y^p)$$

or

$$(x^p-y)((x+a)(y^p+b)+c)$$

modulo $p\mathbb{Z}[x,y]$.

Recently, Lee and Park [10] gave a systematic way to compute modular equations of the continued fraction $X(\tau )$ of order six and raised an interesting conjecture (see [10, Remark 3.10]) that its modular equation satisfies Kronecker's congruence relation of the form

$$F_p(x,y)\equiv \{\begin{array}{cc}(x^p-y)(x-y^p)(\mathrm{mod}p\mathbb{Z}[x,y])\hfill & \text{if }p\equiv \pm 1(\mathrm{mod}12)\hfill \\ (x^p-y)(x+y^p)(\mathrm{mod}p\mathbb{Z}[x,y])\hfill & \text{if }p\equiv \pm 5(\mathrm{mod}12).\hfill \end{array}$$

For explicit computations of $F_p(x,y)$, see [12, Section 2] and [10, Table 1]. Observe that a new form $(x^p-y)(x+y^p)$ of Kronecker's congruence relation has conjecturally appeared. Naturally, this observation motivated the author to deal with their conjecture.