Values of modular functions at real quadratics and conjectures of Kaneko

TL;DR

This paper proves some of Kaneko's conjectures regarding the values of modular functions at real quadratic irrationalities, which are defined via cycle integrals along geodesics, advancing understanding in this area.

Contribution

It generalizes Kaneko's conjectures to a broader class of modular functions, providing new proofs and insights into their values at real quadratic points.

Findings

Proved several of Kaneko's conjectures for general modular functions.

Established explicit relations between modular function values and cycle integrals.

Enhanced understanding of modular functions at real quadratic irrationalities.

Abstract

In 2008, M. Kaneko made several interesting observations about the values of the modular j invariant at real quadratic irrationalities. The values of modular functions at real quadratics are defined in terms of their cycle integrals along the associated geodesics. In this paper we prove some of the conjectures of M. Kaneko for a general modular function.

Equations

$$f(w):={\int }_{C_w}f(z)ds,$$

$$\mathrm{Re}(j^{nor}(w))\in [j^{nor}((1+\sqrt{5})/2),744]\text{mbox}and\mathrm{Im}(j^{nor}(w))\in (-1,1),$$

$$j^{nor}(w):=\frac{1}{2\log \epsilon }j(w).$$

$$(a_0,a_1,a_2,\dots )=a_0-\frac{1}{a_1-\frac{1}{a_2-\frac{1}{\ddots }}},$$

$$\mathrm{Re}(j^{nor}(w))\ge j^{nor}((1+\sqrt{5})/2).$$

$$\underset{N\to \infty }{\lim }{j}^{nor}((\overline{N}))=744.$$

$$\frac{\text{mbox}{Tr}_D(j)}{\text{mbox}{Tr}_D(1)}\to 720,$$

$$L:={\left\{\nu (x)\right\}}_{x\in \mathbb{R}}\subseteq \left[0,1/\sqrt{5}\right]\text{mbox}with\nu (x)=\underset{q\to \infty }{\mathrm{liminf}}q\Vert qx\Vert .$$

$$\frac{-3c+2k+\sqrt{9c^2-4}}{2c},$$

$$a^2+b^2+c^2=3abc.$$

$$a{x}^2+bx+c=0(a,b,c\in \mathbb{Z},(a,b,c)=1)$$

$$a|z{|}^2+b\mathrm{Re}(z)+c=0(z\in \mathcal{H}).$$

$$A_w=\left(\begin{array}{cc}\frac{1}{2}(t-bu)& -cu\\ au& \frac{1}{2}(t+bu)\end{array}\right),$$

$$f(w):={\int }_{C_w}\frac{\sqrt{D}f(z)}{Q_w(z,1)}dz.$$

$${\int }_{C_w}\frac{\sqrt{D}}{Q_w(z,1)}dz=2\log \epsilon$$

$$f^{nor}(w):=\frac{f(w)}{2\log \epsilon }.$$

$$w_0=w-1,w_{i+1}=A_{i+1}(w_i)(i\ge 0),$$

$$A_{i+1}=\left\{\begin{array}[]{ll}T^{-1}&\mbox{if $w_{i}\geq 1$},\\ V^{-1}=\tiny{\begin{pmatrix}1&0\\ -1&1\end{pmatrix}}&\mbox{otherwise},\end{array}\right.$$

$$K(z,w)=\sum _{i=0}^{{\ell }_w-1}\frac{1}{z-w_i}-\frac{1}{z-{\tilde{w}}_i}.$$

$$f(w)={\int }_{\rho }^{{\rho }^2}f(z)K(z,w)dz,$$

$$f(w)=f(w-1)$$

$$=-\sqrt{D}\sum _{i=0}^{{\ell }_w-1}{\int }_{z_i}^{z_{i+1}}\frac{f(z)}{Q_{w-1}(z,1)}dz$$

$$=-\sqrt{D}\sum _{i=0}^{{\ell }_w-1}{\int }_{{\rho }^2}^{A_{i+1}{\rho }^2}\frac{f(z)}{(Q_{w-1}|A_0^{-1}\dots A_i^{-1})(z,1)}dz,$$

$$=-\sum _{i=0}^{{\ell }_w-1}{\int }_{{\rho }^2}^{A_{i+1}{\rho }^2}f(z)\left(\frac{1}{z-w_i}-\frac{1}{z-{\tilde{w}}_i}\right)dz.$$

$$w_{r,k}=(k,\overline{a_{r+1},\dots ,a_n,a_1,\dots ,a_r}),$$

$${\tilde{w}}_{r,k}=-(a_r-k,\overline{a_{r-1},\dots ,a_1,a_n,\dots ,a_r}).$$

$$K(z,w)=\sum _{r=1}^n\sum _{k=1}^{a_r-1}\frac{1}{z-w_{r,k}}-\frac{1}{z-{\tilde{w}}_{r,k}}.$$

$$S_{w,v}(z,r):=\sum_{k=1}^{a_{r}-1}\Big{(}\dfrac{1}{z-w_{r,k}}-\dfrac{1}{z-\tilde{w}_{r,k}}\Big{)}-\sum_{k=1}^{b_{r}-1}\Big{(}\dfrac{1}{z-v_{r,k}}-\dfrac{1}{z-\tilde{v}_{r,k}}\Big{)}.$$

$$C_2(r)<\mathrm{Re}(S_{w,v}(e^{i\theta },r))<C_1(r)$$

$$C_1(r)$$

$$C_2(r)$$

$$-13.02<\mathrm{Im}(S_{w,v}(e^{i\theta },r))<15.$$

Full text

Bengoechea’s research is supported by SNF grant 173976.

Imamoglu’s research is supported by SNF grant 200021-185014.

1. Introduction

Let $\Gamma=\mathrm{SL}(2,\mathbb{Z})$, $\mathcal{H}$ be the upper complex half-plane and $j(z)=\frac{1}{q}+744+196884q+\cdots$ be the classical Klein modular invariant. The values of $j$ at imaginary quadratic irrationalities have a long and rich history going back to Kronecker and Weber and play an important role in the theory of complex muliplication. They have also seen considerable recent interest due to the beautiful results of Borcherds and Zagier, which relate their traces to the coefficients of half integral weight modular forms.

For a real quadratic irrationality $w\in\mathbb{Q}(\sqrt{D})$, where $D$ is a positive (not necessarily fundamental) discriminant, the “value" $f(w)$ of a holomorphic modular function $f$ at $w$ has been defined only recently in [8] and [7] using their periods along the closed geodesic associated to $w$. We define

[TABLE]

where $C_{w}$ is the closed geodesic associated to $w$ in $\Gamma\backslash\mathcal{H}$ with an orientation that will be defined later in the preliminaries (in p.3), and $ds$ is the hyperbolic arc length.

Even though the properties of these values and potential arithmetic applications remain inaccessible, Kaneko [8] studied the numerical values of $j(w)$ and made several remarkable observations. Among his many observations, we note the following boundedness conjecture.

Conjecture 1** (Kaneko).**

Let $w\in\mathbb{Q}(\sqrt{D})$ be a real quadratic irrationality and $\varepsilon>1$ be the smallest unit with positive norm in $\mathbb{Q}(\sqrt{D})$. Then

[TABLE]

where

[TABLE]

In this paper we look at the values $f(w)$ for any modular function $f$ which takes real values on the geodesic arc $\{e^{i\theta}\colon\pi/3\leq\theta\leq 2\pi/3\}.$

We denote by $(a_{0},a_{1},a_{2},\ldots)$ the ‘–’ continued fraction

[TABLE]

with $a_{i}\in\mathbb{Z}$ and $a_{i}\geq 2$ for $i\geq 1$. As a special case of our results we prove

Theorem 1**.**

Let $j(z)$ be the classical modular invariant. Let $w$ be a real quadratic irrational and $(\overline{a_{1},\ldots,a_{n}})$ be its period in the negative continued fraction expansion. Then we have

1. (1)

$\mathrm{Re}(j^{nor}(w))\leq 744.$** 2. (2)

If all the partial quotients $a_{r}$ in the period of $w$ satisfy $a_{r}\geq 3M$ with $M=e^{55}$ then

[TABLE]

Theorem 1 proves the upper bound conjectured by Kaneko for real quadratic irrationalities while Theorem 2 below, our second main result, shows that in fact this bound is optimal.

Theorem 2**.**

Let $j(z)$ be the classical modular invariant. Let $w$ be a real quadratic irrationality and $(\overline{a_{1},\ldots,a_{n}})$ be its period in the negative continued fraction expansion. Then for any positive integer $N>2$, the value $j^{nor}((\overline{N}))$ for the quadratic irrationality $w=(\overline{N})$ is real and

[TABLE]

In[7], the values $j(w)$ were studied not individually but in ‘traces’. By analogy to the traces of the values of $j$ at CM points, the authors in [7] define $\mbox{Tr}_{D}j:=\sum j(w)$, where the sum is over classes of indefinite binary quadratic forms of fundamental discriminant $D$. They relate these traces to the coefficients of mock modular forms, generalizing the results of Borcherds and Zagier. It was conjectured in [7] and proved in [6] and [11] that

[TABLE]

as fundamental discriminants $D\rightarrow\infty.$ Theorem 2, when combined with the limiting behavior of the traces in equation (1), has the following corollary.

Corollary 1.1**.**

There are infinitely many discriminants of the form $D=N^{2}-4$ with class number greater than one and only finitely many such discriminants with class number one.

Corollary 1.1, was also observed in the master thesis of S. Päpcke [12] where another proof of Theorem 2 was also given. This corollary is in the spirit of the conjectures of Chowla and Yokoi. Chowla (see [5]) conjectured that for a positive integer $p$ the class number $h(4p^{2}+1)$ is greater than one if $p>13$. Whereas Yokoi’s conjecture says that $h(p^{2}+4)>1$ if $p>17$ (see [13]). These conjectures were proved effectively by A. Biró in [2] and [3], and some generalizations were proved by Biró and Lapkova in [4]. Even though our result, unlike the results of Biró and Biró-Lapkova, is not effective, as far as we know it provides the first direct application of cycle integrals of modular functions to a classical arithmetic question.

It is worth noting that Part (2) of Theorem 1 can be rephrased in terms of the diophantine properties of the quadratic irrationalities, more precisely in terms of the Lagrange spectrum. For any irrational number $x$, let $\|x\|$ denote the distance from $x$ to a closest integer. Then recall that the Lagrange spectrum $L$ is defined as

[TABLE]

It is known that if $x$ has a continued fraction $(a_{1},a_{2},\ldots)$ then $\nu(x)\leq\inf_{r\geq 1}a_{r}^{-1}$. Hence part (2) of Theorem 1 proves the lower bound conjectured by Kaneko for the quadratic irrationalities $w$ with constant of approximation $\nu(w)\in[0,1/3M]$.

For the values $j(w)$ with $\nu(w)\in\left(1/3,1/\sqrt{5}\right]$, Kaneko observed more specific phenomena that have been partially proved in [1]. Every such quadratic irrationality is $\mathrm{SL}(2,\mathbb{Z})$-equivalent to a Markov quadratic, i.e. a quadratic of the form

[TABLE]

where $0\leq k<c$, $k\equiv ab^{-1}\pmod{c}$ and $(a,b,c)\in\mathbb{N}^{3}$ is a solution to the Diophantine equation

[TABLE]

A very rich theory has been developed by Markov for this sort of quadratic irrationalities that the authors exploit in [1].

After giving some preliminaries in the next section, we will collect several technical results in Section 3 that will be needed in the sequel. In Section 4, we start by proving Theorem 2 for a holomorphic modular function which takes real values on the geodesic arc $\{e^{i\theta}:\pi/3\leq\theta\leq 2\pi/3\}$. The results from Section 3 are then used to prove Theorem  4.3, which compares the values of modular functions at different quadratic irrationalities by comparing their corresponding partial quotients. Theorem 4.3 is the main theorem of this paper and the results (1) and (2) in Theorem 1 follow as its simple corollaries.

Acknowledgement. We thank the referee for his/her many useful remarks which greatly improved the exposition of the paper.

2. Preliminaries

Let $w$ be a real quadratic irrationality and $\tilde{w}$ be its conjugate. $w$ and $\tilde{w}$ are the roots of a quadratic equation

[TABLE]

with discriminant $D=b^{2}-4ac>0$. We choose $a,b,c$ such that $w=\frac{-b+\sqrt{D}}{2a}$, $\tilde{w}=\frac{-b-\sqrt{D}}{2a}$. The geodesic $S_{w}$ in $\mathcal{H}$ joining $w$ and $\tilde{w}$ is given by the equation

[TABLE]

The stabilizer $\Gamma_{w}$ of $w$ preserves the quadratic form $Q_{w}=[a,b,c]$, and hence $S_{w}$. Let $A_{w}$ be the generator of the infinite cyclic group $\Gamma_{w}$ with

[TABLE]

where $(t,u)$ is the smallest positive solution to Pell’s equation $t^{2}-Du^{2}=4$. We denote by $\varepsilon=(t+u\sqrt{D})/2$ the smallest unit with positive norm.

For any modular function $f$, since the group $\Gamma_{w}$ preserves the expression $f(z)Q_{w}(z,1)^{-1}dz,$ one can define the cycle integral of $f$ along $C_{w}=\Gamma_{w}\backslash S_{w}$, also viewed as the “value" of $f$ at $w$, by the complex number

[TABLE]

The factor $\sqrt{D}$ is introduced here since $ds=\dfrac{\sqrt{D}f(z)}{Q_{w}(z,1)}dz$ on the geodesic $C_{w}$. The integral defining $f(w)$ is $\Gamma$-invariant and can in fact be taken along any path in $\mathcal{H}$ from $z_{0}$ to $A_{w}z_{0}$, where $z_{0}$ is any point in $\mathcal{H}$. Note that this gives an orientation on $S_{w}$ from $w$ to $\tilde{w}$, which is counterclockwise if $a>0$ and clockwise if $a<0$. We normalize the number $f(w)$ by the length of the geodesic $C_{w}$ which is given by

[TABLE]

and we define the normalized value as

[TABLE]

For a real quadratic irrationality $w$ with purely periodic continued fraction, it is known that the continued fraction expansion is given by setting $w_{0}=w$ and inductively $w_{i+1}=w_{i}-1$ if $w_{i}\geq 1$ and $w_{i+1}=-1/w_{i}$ if $0<w_{i}<1$. This algorithm is cyclic, which implies that the hyperbolic element $A_{w}$ is a word in negative powers of $T$ and $S$, where $T=\tiny{\begin{pmatrix}1&1\\ 0&1\end{pmatrix}}$, $S=\tiny{\begin{pmatrix}0&1\\ -1&0\end{pmatrix}}$ (for more detail we refer the reader to [14] and [9, Proposition 2.6]). It follows that the algorithm

[TABLE]

where

[TABLE]

is also cyclic, since $V^{-1}=T^{-1}ST^{-1}$, and hence comes back to $w_{0}=w-1$ after a finite number of steps. We denote by $\ell_{w}$ the length of the cycle, so that $A_{w-1}=A_{\ell_{w}}\cdots A_{1}$. The next lemma is crucial in our work and is inspired by Zagier’s argument in the proof of Theorem 7 in [10].

Lemma 2.1**.**

Let $w$ be a real quadratic irrationality with a purely periodic continued fraction expansion and let

[TABLE]

Then

[TABLE]

where $\rho=e^{\pi i/3}.$ Here $w_{i}$’s are defined in (2) and $\tilde{w_{i}}$ is the Galois conjugate of $w_{i}$.

Proof.

Let $A_{0}=\mathrm{Id}$ and $z_{i}=A_{0}^{-1}\ldots A_{i}^{-1}\rho^{2}$ for $i\geq 0$. We have

[TABLE]

Since $V(\rho^{2})=T(\rho^{2})=\rho$, we have proved the lemma. ∎

Let $w$ be a quadratic irrationality with purely periodic continued fraction and $\overline{a_{1},\ldots,a_{n}}$ be the period in its continued fraction expansion. Each $w_{i}$ from (2) has a continued fraction expansion of the form

[TABLE]

with $1\leq r\leq n$ and for each $r$, $1\leq k\leq a_{r}-1$. The Galois conjugate of $w_{r,k}$ is

[TABLE]

Hence

[TABLE]

If $r$ is fixed and there is no possible confusion, we will drop the dependence on $r$ and write $w_{k}:=w_{r,k}$ for simplicity.

3. Technical lemmas

Let $w=(\overline{a_{1},\ldots,a_{n}})$ and $v=(\overline{b_{1},\ldots,b_{m}})$ be two purely periodic quadratic irrationals with $n\geq m$. If $n>m$, then we define $b_{m+1},b_{m+2},\ldots,b_{n}$ by cycling back to $b_{1},b_{2},etc$. For a fixed $r,$ let

[TABLE]

Theorem 3.1**.**

Let $w=(\overline{a_{1},\ldots,a_{n}})$ and $v=(\overline{b_{1},\ldots,b_{m}})$ be two purely periodic quadratic irrationals. If $a_{r}\leq b_{r}$ for all $r=1,\ldots,n$, then we have for $\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$

[TABLE]

with

[TABLE]

and

[TABLE]

To prove Theorem 3.1, we start with the following two simple lemmas.

Lemma 3.2**.**

For $x,y\in\mathbb{R}$, $\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$, let

[TABLE]

As a function of $\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$, $F_{x,y}(\theta)$ satisfies the following properties:

1. (i)

$F_{x,y}(\theta)$* is decreasing for $x,y\in(2,\infty)$ and increasing for $x,y\in(-\infty,-2)$.* 2. (ii)

For $0\leq|x|\leq 1$ and $0\leq|y|\leq 1$, we have

[TABLE] 3. (iii)

For $1\leq|x|\leq 2$ and $1\leq|y|\leq 2$, we have

[TABLE] 4. (iv)

For $x\geq 2$ and $y\leq-2$,

[TABLE]

Proof.

A simple calculation gives

[TABLE]

The assertion (iv) is straightforward using (8). The other three assertions can be easily verified numerically. For example, for $x>2$ and $y>2$ the maximum of the derivative $\frac{dF}{d\theta}$ is $-0.00504971$ where as its minimum is $-0.19245$. This shows that for $x>2$ and $y>2,$ $\frac{dF}{d\theta}<0$ and hence $F_{x,y}(\theta)$ is decreasing.

∎

Similarly we have:

Lemma 3.3**.**

For $x,y\in\mathbb{R}$, $\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$, let

[TABLE]

As a function of $\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$, $G_{x,y}(\theta)$ satisfies the following properties:

1. (i)

$G_{x,y}(\theta)$* is decreasing for $x,y\in(1,\infty)$ and for $x,y\in(-\infty,-1)$.* 2. (ii)

$G_{x,y}(\theta)$* is increasing for $x\in(1,\infty)$ and $y\in(-\infty,-1)$.* 3. (iii)

For $0\leq|x|\leq 1$ and $0\leq|y|\leq 1$, we have

[TABLE]

with $G_{0,0}(e^{\pi i/3})=-\frac{\sqrt{3}}{2}$ and $G_{0,0}(e^{2\pi i/3})=\frac{\sqrt{3}}{2}$.

Proof.

Once again a simple calculation gives

[TABLE]

and the proof of the lemma follows in a similar way to Lemma 3.2. ∎

3.1. Proof of Theorem 3.1

We start by grouping some of the terms from $S_{w,v}$ into two sums $S_{1}$ and $S_{2}$ ($S_{1}$ corresponding to terms $w_{k},v_{k}$ and $S_{2}$ to conjugates $\tilde{w}_{k},\tilde{v}_{k}$) whose contribution, as we will see, will be minor. Since $r$ is fixed in the whole proof, we drop the dependence on $r$ in the notation for all these sums. Define

[TABLE]

[TABLE]

We group the remaining terms from $S_{w,v}$ in the sum $S_{3}$, which will be the major contribution:

[TABLE]

Hence,

[TABLE]

3.1.1. Proof of inequality (6).

The real part of $S_{1}(e^{i\theta})$ satisfies

[TABLE]

For $3\leq k\leq a_{r}-1$, since $\lfloor w_{k}\rfloor=\lfloor v_{k}\rfloor=k-1>2,$ by Lemma 3.2 (i),

[TABLE]

and

[TABLE]

Since $-1<w_{0}-v_{0}<0$ and using Lemma 3.2 (ii)-(iii), we have

[TABLE]

and similarly

[TABLE]

The real part of $S_{2}(e^{i\theta})$ satisfies

[TABLE]

For $1\leq k\leq a_{r}-3$, $\lfloor\tilde{w}_{k}\rfloor=\lfloor\tilde{v}_{b_{r}-a_{r}+k}\rfloor=-a_{r}+k<-2$ and hence by Lemma 3.2 (i), we have

[TABLE]

and

[TABLE]

Since $\lfloor\tilde{w}_{a_{r}}\rfloor=\lfloor\tilde{v}_{b_{r}}\rfloor=0$ and $b_{r}\geq a_{r}$, $0<\tilde{w}_{a_{r}}-\tilde{v}_{b_{r}}<1$ and using Lemma 3.2 (ii)-(iii),

[TABLE]

and

[TABLE]

The real part of $S_{3}(e^{i\theta})$ satisfies

[TABLE]

Since $-\lfloor\tilde{v}_{b_{r}-k}\rfloor=\lfloor v_{k}\rfloor=k,$ for $k\geq 3$, by Lemma 3.2 (iv),

[TABLE]

Therefore, similarly to above,

[TABLE]

with $k_{0}=1$ if $a_{r}=2$ or $k_{0}=0$ otherwise. We also have

[TABLE]

with

[TABLE]

By (10), (11), (14) and (16) we have that

[TABLE]

By (12), (13) and (15),

[TABLE]

3.1.2. Proof of the inequality (7)

The imaginary part of $S_{1}(e^{i\theta})$ satisfies

[TABLE]

For $k\geq 2$, by Lemma 3.3 (i),

[TABLE]

and

[TABLE]

Since $-1<w_{0}-v_{0}<0$ and using Lemma 3.3 (iii), we have

[TABLE]

The imaginary part of $S_{2}(e^{i\theta})$ satisfies

[TABLE]

For $1\leq k\leq a_{r}-2$, by Lemma 3.3 (i),

[TABLE]

and

[TABLE]

Since $0<\tilde{w}_{a_{r}}-\tilde{v}_{b_{r}}<1$ and using Lemma 3.3 (iii), we have

[TABLE]

The imaginary part of $S_{3}(e^{i\theta})$ satisfies

[TABLE]

For $k\geq 2$, by Lemma 3.3 (ii),

[TABLE]

and

[TABLE]

Therefore, similarly to previous cases we get

[TABLE]

By (10), (19), (20) and (21), we have

[TABLE]

The following corollary of Theorem 3.1 is crucial for the next section.

Corollary 3.4**.**

Let $w=(\overline{a_{1},\ldots,a_{n}})$ and $v=(\overline{b_{1},\ldots,b_{m}})$ be two purely periodic quadratic irrationals. If $b_{r}\geq Ma_{r}$ for every $r$, then there exist constants $K_{1}(M)$ and $K_{2}(M)$ such that

[TABLE]

Moreover if $M\geq e^{55}$, then $K_{1}(M)$ and $K_{2}(M)$ are positive.

Proof.

This follows easily from the bounds (17), (18), (22) together with the simple observation that for $\pi/3\leq\theta\leq 2\pi/3$, $-1/2\leq\cos\theta\leq 1/2$ and $\sqrt{3}/2\leq\sin\theta\leq 1$. ∎

4. Values of Modular functions

We start this section by looking at the sequence of values $f^{nor}((\overline{N}))$, $N>2$ for a modular function which is real on the geodesic arc $\{e^{i\theta}\colon\pi/3\leq\theta\leq 2\pi/3\}.$ We have

Theorem 4.1**.**

Let $f(z)=\sum_{n\geq 0}c(n)q^{n}$ be a modular function which is real valued on the geodesic arc $\{e^{i\theta}\colon\pi/3\leq\theta\leq 2\pi/3\}$. For any positive integer $N>2$, the value $f^{nor}((\overline{N}))$ for the quadratic irrational $w=(\overline{N})$ is real and

[TABLE]

In particular we have

[TABLE]

For $w=(\overline{N})$, the function $K_{N}(z):=K(z,w)$ defined by (3) satisfies the following simple lemma.

Lemma 4.2**.**

[TABLE]

Proof.

A simple calculation shows

[TABLE]

and

[TABLE]

These two equalities imply the lemma. ∎

Proof.

(Theorem 4.1):

Let $w=(\overline{N})$. We have that $w=\frac{N+\sqrt{N^{2}-4}}{2}$ and $Q_{w}=[1,-N,1]$. Hence Pell’s equation becomes

[TABLE]

with $(N,1)$ being the smallest positive solution. Thus $\varepsilon_{w}=\frac{N+\sqrt{N^{2}-4}}{2}$. Since $w_{k}=(k,\overline{N})=-\tilde{w}_{N-k}$,

[TABLE]

with

[TABLE]

Now

[TABLE]

and

[TABLE]

Writing $\int_{\pi/3}^{2\pi/3}=\int_{\pi/3}^{\pi/2}+\int_{\pi/2}^{2\pi/3}$ and using Lemma 4.2, we have that

[TABLE]

Since $w_{k}=k-\frac{1}{a_{r+1}-\frac{1}{\cdots}},$ it follows from $\varepsilon_{w}=\frac{N+\sqrt{N^{2}-4}}{2},$ (23) and (24) that, for all $\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$,

[TABLE]

Thus

[TABLE]

To see the last equality we note that $f$ is real on the arc $\{e^{i\theta}\colon\pi/3\leq\theta\leq 2\pi/3\}$ and hence the integral is just the parametrization of $\int f(z)dz$ on this arc. On the other hand, due to the holomorphicity of $f$ the integral is independent of the path of integration and as such gives the constant Fourier coefficient of $f$.

∎

Our next result compares the values of a modular function at two different quadratic irrationalities by comparing their corresponding partial quotients. The results that were given in the introduction will then follow as corollaries of this main result. More precisely we have

Theorem 4.3**.**

Let $f(z)=\sum_{n\geq 0}c(n)q^{n}$ be a modular function which is real valued on the geodesic arc $\{e^{i\theta}\colon\pi/3\leq\theta\leq 2\pi/3\}$. Suppose that

[TABLE]

Then the following holds: Let $w$ and $v$ be two quadratic irrationals with respective periods $\overline{a_{1},\ldots,a_{n}}$ and $\overline{b_{1},\ldots,b_{m}}$ such that $m$ divides $n$. If $M=e^{55}$ and $b_{r}\geq Ma_{r}$ for all $r=1,\ldots,n$, then

[TABLE]

Remark 4.4**.**

As the proof of Theorem 4.3 will show, the condition (25) can be replaced by the condition that $\mathrm{Re}(f^{nor}(\alpha))<c(0)$ for some quadratic irrational $\alpha$.

Proof.

We have that

[TABLE]

with $S_{w,v}(z,r)$ defined as in (5), so

[TABLE]

By Theorem 3.1 and Corolary 3.4 we obtain

[TABLE]

with

[TABLE]

$K_{1}(M)$, $K_{2}(M)$ being the positive constants from Corollary 3.4. In particular, if $f=1$, then

[TABLE]

with

[TABLE]

By definition, the inequality

[TABLE]

holds if and only if

[TABLE]

Now (26) and (27) imply that (29) is equivalent to

[TABLE]

or also to

[TABLE]

Since the last inequality does not depend on $v$ and is equivalent to (30), the inequality (30) holds either for all or for no $v.$ We first show that $\dfrac{\mu(M)}{\lambda(M)}>c(0).$ Suppose on the contrary $\dfrac{\mu(M)}{\lambda(M)}\leq c(0).$ Let $v=(\overline{N}).$ For any $\epsilon>0$, for large enough $N>N_{0}$ using Theorem 4.1 we have $\mathrm{Re}(f^{nor}(v))>c(0)-\epsilon>\frac{\mu(M)}{\lambda(M)}$. But then, since (30) is equivalent to (28), for $N>\max{\{3M,N_{0}\}}$ and $w=\frac{1+\sqrt{5}}{2}$, we have

[TABLE]

Since this contradicts the assumption (25), we must indeed have $\dfrac{\mu(M)}{\lambda(M)}>c(0).$ Then with $w=\frac{1+\sqrt{5}}{2}$, we have $\mathrm{Re}(f^{nor}(w))=\mathrm{Re}(f^{nor}(\frac{1+\sqrt{5}}{2}))<c(0)<\dfrac{\mu(M)}{\lambda(M)}$. Hence for every $v=(\overline{b_{1},\ldots,b_{m}})$ with $b_{r}>3M$ we have that

[TABLE]

Now choose $v_{0}=(\overline{b_{1},\ldots,b_{m}})$ with $b_{r}>\max{\{3M,Ma_{r}\}}$. Then for this $v_{0}$ we have $\mathrm{Re}(f^{nor}(v_{0}))<\dfrac{\mu(M)}{\lambda(M)}$ and hence $\mathrm{Re}(f^{nor}(w))<\dfrac{\mu(M)}{\lambda(M)}.$ But this is equivalent to

[TABLE]

∎

We have the following immediate corollaries.

Corollary 4.5**.**

For any quadratic irrational $w$, and any modular function $f$ which satisfies the conditions of Theorem 4.3 we have

[TABLE]

Proof.

For any $w=(\overline{a_{1},\ldots,a_{n}})$, choose $N\in\mathbb{N}$ large enough, so that $N\geq e^{55}a_{r}$ for all $1\leq r\leq n$. Then by Theorem 4.3 we have

[TABLE]

Using Theorem 4.1, this gives

[TABLE]

∎

Corollary 4.6**.**

Let $f$ be a modular function which satisfies (25) and $v$ be a quadratic irrational. If all the partial quotients in the period of $v$ are greater than 3M, then

[TABLE]

Proof.

This follows from Theorem 4.3 applied to $w=\frac{1+\sqrt{5}}{2}=(2,\bar{3})$. ∎

Finally, these corollaries prove Theorem 1 since for the $j$ invariant the assumption in the statement of Theorem 4.3 is easily verified. Namely, we have

[TABLE]