**2.3 The method of Rankin and Swinnerton-Dyer (RSD Method)**

**2.5.4 The case k ≡ 4 (mod 6)**

With the exception of some specific cases, we can prove this case in a similar way to the proof of Subsection 2.4.3 and the previous subsection.

*The case* 0*< α*7,k*< π/3.*

We can use Lemma 2.5.1 (1) for the case *α*7,k *< π/6, and we can use Lemma 2.5.1 (4) and (6) for the*
cases 0*< α*7,k*< π/4 andπ/4< α*7,k*< π/3, respectively.*

*The case* *π/3< α*7,k*< π.*

(i) “The case 3π/4*< α*7,k*< π”*

For 5π/6*< α*7,k *< π* (resp. 29π/36*< α*7,k *<*5π/6, 3π/4*< α*7,k *<*5π/6), we can use Lemma 2.5.1
(1) (resp. (3), (2)).

(ii) “The case*π/3< α*7,k*<*13π/36” We can use Lemma 2.5.1 (7).

(iii) “The case 2π/3*< α*7,k*<*3π/4”

We define cos(c0*0**π) =−*cos((x/180)π−(t/2)π). Then, we prove “For (x/180)π < α7,k*<*(y/180)π, we
have*|R*^{∗}_{7,1}*|<*2 cos(c0*0**π) forθ*1=*π/2 +α*7*−tπ/k.” for two cases, namely, (x, y, t) = (127.6,*135,59/250),
(120,127.6,1/4).

(iv) “The case 13π/36*< α*_{7,k}*<*5π/9”

We define cos(c0*0**π) = cos((y/180)π−π/3+(t/2)π). Then, we prove “For (x/180)π < α*7,k*<*(y/180)π,
we have *|R*_{7,2}^{∗}*|<*2 cos(c0*0**π) for* *θ*2 =*α*7*−π/6 +tπ/k.” for two cases, namely, (x, y, t) = (65,*90,2/5),
(90,100,2/5).

Now, we can write

*F*_{k,7,1}* ^{∗}* (θ1) = 2 cos (kθ1

*/2) + 2Re(3e*

^{−iθ}^{1}

*+*

^{/2}*√*

7e^{iθ}^{1}* ^{/2}*)

*+*

^{−k}*R*

_{7,1}

^{∗}

^{00}*,*

*F*

_{k,7,2}*(θ2) = 2 cos (kθ2*

^{∗}*/2) + 2*

^{k}*·*2Re(3e

^{−iθ}^{2}

^{/2}*−√*

7e^{iθ}^{2}* ^{/2}*)

*+*

^{−k}*R*

^{∗}_{7,2}

^{00}*.*(v) “The case 73π/120

*< α*

_{7,k}

*<*2π/3”

We have *X*1 = *v**k*(2,1, θ1)* ^{−2/k}* > 1 + 2

*√*

3t(π/k) and *X*2 = *v**k*(3,1, θ1)* ^{−2/k}* 6

*e*

^{−3}

^{√}^{3(t/2)π}. Then,

*Sign{cos (kθ*1

*/2)}*=

*Sign{Re(3e*

^{−iθ}^{1}

*+*

^{/2}*√*

7e^{iθ}^{1}* ^{/2}*)

^{−k}*}, and we prove “For (x/180)π < α*7,k

*<*(y/180)π, we have

*|R*

^{∗}_{7,1}

^{00}*|<|2 cos (kθ*1

*/2) + 2Re(3e*

^{−iθ}^{1}

*+*

^{/2}*√*

7e^{iθ}^{1}* ^{/2}*)

^{−k}*|*for

*θ*1=

*π/2 +α*7

*−tπ/k.” for four cases,*namely, (x, y) = (111.6,120,23/150), (110.1,111.6,1/10), (109.65,110.1,43/625), (109.5,109.65,21/400).

For each case, we consider the point such that *kθ*1*/2 =* *k(π/2 +α*7)/2*−*(t/2)π. We can show
*Sign{cos (kθ*1*/2)}* = *Sign{F*_{k,7,1}* ^{∗}* (θ1)} , and then we have at least one zero between the second to last
integer point for

*A*

^{∗}_{7,1}and the point

*kθ*1

*/2.*

(vi) “The case 5π/9*< α*7,k*<*217π/360”

We have*X*1= (1/4)*v**k*(1,*−1, θ*2)* ^{−2/k}*>1+t(π/k) and

*X*2= (1/4)

*v*

*k*(3,

*−1, θ*2)

*6*

^{−2/k}*e*

^{−(3}

^{√}^{3/2)(t/2)π}. Then, we have

*Sign{cos (kθ*2

*/2)}*=

*Sign{Re(3e*

^{−iθ}^{2}

^{/2}*−√*

7e^{iθ}^{2}* ^{/2}*)

^{−k}*}, and we prove “For (x/180)π <*

*α*7,k *<* (y/180)π, we have *|R*^{∗}_{7,2}^{00}*|* *<* *|2 cos (kθ*2*/2) + 2*^{k}*·*2Re(3e^{−iθ}^{2}^{/2}*−√*

7e^{iθ}^{2}* ^{/2}*)

^{−k}*|*for

*θ*2 =

*α*7

*−*

*π/6 +tπ/k.” for five cases, namely, (x, y) = (100,*106,3/10), (106,107.7,11/50), (107.7,108.21,33/200), (108.21,108.42,2/15), (108.42,108.5,113/1000).

In conclusion, we have the following proposition:

28 *CHAPTER 2. ON THE ZEROS OF EISENSTEIN SERIES FOR*Γ_{0}(5)*AND*Γ_{0}(7)
Proposition 2.5.5. *Let* *k*>4 *be an integer which satisfiesk* *≡*4 (mod 6), and let *α*7,k *∈*[0, π] *be the*
*angle which satisfies* *α*7,k *≡* *k(π/2 +α*7)/2 (mod*π). If we have* *α*7,k *<*217π/360 *or* 73π/120 *< α*7,k*,*
*then all of the zeros ofE*_{k,7}* ^{∗}* (z)

*in*F

*(7)*

^{∗}*lie on the arc*

*A*

^{∗}_{7}

*. Otherwise, all but at most one zero ofE*

_{k,7}*(z)*

^{∗}*in*F

*(7)*

^{∗}*are on*

*A*

^{∗}_{7}

### 2.5.5 The remaining cases “k *≡* 2 (mod 6), 266π/375 *< α*

7,k *<* 3217π/4500” and

### “k *≡* 4 (mod 6), 217π/360 *< α*

7,k *<* 73π/120”

Similar the problem described in Subsection 2.4.4, it is difficult to prove Conjecture 2.1.2 for the remaining
cases. However, when*k*is large enough, we have the following observation.

The case “k*≡*2 (mod 6)and 266π/375*< α*7,k*<*3217π/4500”

Let*t >*0 be small enough, then we have*π/2< α*7,k*−(t/2)π < π,π <*2π/3+α7,k+d1,1(t/2)π < α7,k,1*0* *<*

2π/3 +*α*7,k+ (t/2)π <3π/2, and 2π <4π/3 +*α*7,k*−tπ < α*7,k,2*0* *<*4π/3 +*α*7,k*−d*1,2(t/2)π <5π/2.

Thus, we have

*−*cos(α7,k*−*(t/2)π)*−*cos(2π/3 +*α*7,k+*d*1,1(t/2)π)*·*(1 + 2*√*

3t(π/k))^{−k/2}

*−*cos(4π/3 +*α*7,k*−d*1,2(t/2)π)*·e*^{−(3}^{√}^{3/2)πt}

*>|*cos(kθ1*/2)|*+

¯¯

¯¯*Re*

½³

2e^{iθ}^{1}* ^{/2}*+

*√*

7e^{−iθ}^{1}^{/2}

´* _{−k}*¾¯¯

¯¯

*−*

¯¯

¯¯*Re*

½³

3e^{iθ}^{1}* ^{/2}*+

*√*

7e^{−iθ}^{1}^{/2}

´* _{−k}*¾¯¯

¯¯

*>−*cos(α7,k*−*(t/2)π)*−*cos(2π/3 +*α*7,k+ (3t/2)π)*·e*^{−}^{√}^{3πt}

*−*cos(4π/3 +*α*7,k*−tπ)·*(1 + 3*√*

3t(π/k))^{−k/2}*.*

We denote the upper bound by *A* and the lower bound by*B. First, we have* *A|**t=0* =*B|**t=0* = 0 and

*∂*

*∂t**A|**t=0* = _{∂t}^{∂}*B|**t=0* = 0. Second, let *C* = (5*√*

3/2)π^{2}(−cos*α*7,k)(tan*α*7,k+ 11/(5*√*

3)), then we have

*∂*^{2}

*∂t*^{2}*A|**t=0*=*C*+ 6π^{2}(−cos(2π/3 +*α*7,k))/k and _{∂t}^{∂}^{2}2*B|**t=0*=*C−*(27/2)π^{2}cos(4π/3 +*α*7,k)/k. Finally, we
have *B >*0 if*α*7,k*>*3π/2*−*2α7, and we have*A <*0 if*α*7,k *<*3π/2*−*2α7for large enough *k*and small
enough*t.*

Similarly, we consider the lower and the upper bounds of*|*cos(kθ2*/2)|−|Re{2*^{k}*·(e*^{iθ}^{2}^{/2}*−√*

7e^{−iθ}^{2}* ^{/2}*)

^{−k}*}|−*

*|Re{2*^{k}*·*(3e^{iθ}^{2}^{/2}*−√*

7e^{−iθ}^{2}* ^{/2}*)

^{−k}*}|. The lower bound is positive if*

*α*7,k

*<*3π/2

*−*2α7, while the upper bound is negative if

*α*7,k

*>*3π/2

*−*2α7 for large enough

*k*and for small enough

*t.*

In conclusion, if *k* is large enough, then *|R*^{∗}_{7,1}^{0}*|* and *|R*^{∗}_{7,2}^{0}*|* is small enough, and then we have one
more zero on the arc *A*^{∗}_{7,1} when *α*7,k *>*3π/2*−*2α7, and we have one more zero on the arc*A*^{∗}_{7,2} when
*α*7,k*<*3π/2*−*2α7. However, if*k*is small, a method of proving the conjecture for this case is not clear.

Remark 2.5.1. Getz considered a similar problem for the zeros of extremal modular forms of SL2(Z)
(see [G]). It seems that similar results do not hold for extremal modular forms of Γ^{∗}_{0}(5) and Γ^{∗}_{0}(7).

The case “k*≡*4 (mod 6)and 217π/360*< α*7,k*<*73π/120”

Let*t >*0 be small enough. Similar to Subsection 2.4.4, if*k*is large enough, then we have one more zero
on the arc*A*^{∗}_{7,1} when*α*7,k*> π−α*7, and we have one more zero on the arc*A*^{∗}_{7,2} when*α*7,k*< π−α*7.
*Acknowledgement.* I would like to thank Professor Eiichi Bannai for suggesting these problems.

### Bibliography

[Ge] J. Getz,*A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms.,*
Proc. Amer. Math. Soc.,132(2004), No. 8, 2221-2231.

[Kr] A. Krieg, *Modular Forms on the Fricke Group., Abh. Math. Sem. Univ. Hamburg,* 65(1995),
293-299.

[MNS] T. Miezaki, H. Nozaki, J. Shigezumi, *On the zeros of Eisenstein series for* Γ^{∗}_{0}(2) *and* Γ^{∗}_{0}(3), J.

Math. Soc. Japan,59(2007), 693–706.

[Q] H. -G. Quebbemann, *Atkin-Lehner eigenforms and strongly modular lattices, Enseign. Math. (2),*
43(1997), No. 1-2, 55-65.

[RSD] F. K. C. Rankin, H. P. F. Swinnerton-Dyer, *On the zeros of Eisenstein Series, Bull. London*
Math. Soc., 2(1970), 169-170.

[Se] J. -P. Serre,*A Corse in Arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New*
York-Heidelberg, 1973. (Translation of*Cours d’arithm´etique*(French), Presses Univ. France, Paris,
1970.)

[SG] G. Shimura, *Introduction to the arithmetic theory of automorphic functions, Kanˆo Memorial *
Lec-tures, No. 1. Publ. Math. Soc. Japan, No. 11. Iwanami Shoten Publishers, Tokyo; Princeton
Univ. Press, Princeton, 1971.

[SH] H. Shimizu, *Hokei kansu. I-III.* (Japanese) [Automorphic functions. I-III], Iwanami Shoten Kiso
Sugaku [Iwanami Lectures on Fundamental Mathematics] 8, Iwanami Shoten Publishers, Tokyo,
1977-1978.

[SJ3] J. Shigezumi,*A detailed note on the zeros of Eisenstein series for*Γ^{∗}_{0}(5)*and*Γ^{∗}_{0}(7), arXiv:math.NT/0607247.

### Chapter 3

### On the zeros of certain modular functions for the normalizers of congruence subgroups

We consider the locations of the zeros of the Eisenstein series and modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups that are of genus zero and of level at most twelve.

### Introduction

The purpose of this study is to determine the locations of the zeros of modular functions. The Eisenstein series and the the Hecke type Faber polynomials are the most interesting and important modular forms.

F. K. C. Rankin and H. P. F. Swinnerton-Dyer considered the problem of locating the zeros of the
Eisenstein series*E**k*(z) in the standard fundamental domainF (See [RSD]). They proved that all of the
zeros of *E**k*(z) in F lie on the unit circle. They also stated towards the end of their study that “this
method can equally well be applied to Eisenstein series associated with subgroups of the modular group.”

However, it seems unclear how widely this claim holds.

Subsequently, T. Miezaki, H. Nozaki, and the present author considered the same problem for the
Fricke group Γ^{∗}_{0}(p) (see [Kr], [Q]), and proved that all of the zeros of the Eisenstein series *E*_{k,p}* ^{∗}* (z) in
a certain fundamental domain lie on a circle whose radius is equal to 1/

*√*

*p,* *p* = 2,3 (see [MNS]).

Furthermore, we also proved that almost all the zeros of the Eisenstein series in a certain fundamental
domain lie on circles whose radius are equal to 1/*√*

*p*or 1/(2*√*

*p),p*= 5,7 (see [SJ2]).

Let Γ be a discrete subgroup of SL2(R), and let*h*be the width of Γ, then we define
F0,Γ:=©

*z∈*H; *−h/2< Re(z)< h/2,* *|cz*+*d|>*1 for*∀γ*=¡_{a b}

*c d*

¢*∈*Γ s.t.*c6= 0*ª

*.* (3.1)

We have a fundamental domainFΓ such thatF0,Γ*⊂*FΓ *⊂*F0,Γ. LetFΓ be such a fundamental domain.

For the modular group SL2(Z) and the Fricke groups Γ^{∗}_{0}(p) (p= 2,3), all the zeros of the Eisenstein
series for the cusp*∞*lie on the arcs of the boundaries of certain fundamental domains.

H. Hahn considered the locations of the zeros of the Eisenstein series for the cusp *∞*for every genus
zero Fuchsian group Γ of the first kind with*∞*as a cusp whose hauptmodul*J*Γtakes real values on*∂F*Γ,
and proved that almost all the zeros of the Eisenstein series for the cusp*∞*for Γ lie on*∂F*Γ under some
further assumptions (see [H]).

T. Asai, M. Kaneko, and H. Ninomiya further considered the problem of locating the zeros of modular
functions *F**m*(z) for SL2(Z) that correspond to the Hecke type Faber polynomials*P**m*, that is, *F**m*(z) =
*P**m*(J(z)) (See [AKN]). They proved that all of the zeros of *F**m*(z) in F lie on the unit circle for each
*m*>1. Later, E. Bannai, K. Kojima, and T. Miezaki considered the same problem for the normalizers of
congruence subgroups that correspond to conjugacy classes of the Monster group (See [BKM]). In their
study, the locations of the zeros were numerically calculated, and it was found that almost all of the zeros
of the modular functions from the Hecke type Faber polynomial lie on the lower arcs when the group
satisfies the same assumptions as in the theorem of H. Hahn. In particular, T. Miezaki proved that all of
the zeros of the modular functions from the the Hecke type Faber polynomials for the Fricke group Γ^{∗}_{0}(2)
lie on the lower arcs of its fundamental domain.

Now, we have the following conjectures:

31

0

Conjecture 3.1. *Let* Γ *be a genus zero Fuchsian group of the first kind with* *∞* *as a cusp. If the*
*hauptmodul* *J*Γ *takes real values on∂F*Γ*, all of the zeros of the Eisenstein series for the cusp* *∞for*Γ*in*
FΓ *lie on the arcs*

*∂F*Γ*\ {z∈*H; *Re(z) =±h/2}.*

Conjecture 3.2. *Let* Γ *be a genus zero Fuchsian group of the first kind with* *∞* *as a cusp. If the*
*hauptmodul* *J*Γ *takes real values on∂F*Γ*, all but at most* *c**h*(Γ)*of the zeros of modular function from the*
*the Hecke type Faber polynomial of degree* *mfor*Γ *in*FΓ *lie on the arcs*

*∂F*Γ*\ {z∈*H; *Re(z) =±h/2}*

*for all but a finite number ofmand for the constant* *c**h*(Γ), which does not depend on*m.*

In this paper, as a first step towards the challenge of the above conjectures, we will observe the locations of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials for the normalizers of congruence subgroups.

The normalizers of congruence subgroups of level at most 12 which satisfy the assumptions of the above conjectures are

SL2(Z), Γ^{∗}_{0}(2), Γ0(2), Γ^{∗}_{0}(3), Γ0(3), Γ^{∗}_{0}(4), Γ0(4), Γ^{∗}_{0}(5), Γ0(6)+, Γ^{∗}_{0}(6), Γ0(6) + 3, Γ0(6),
Γ^{∗}_{0}(7), Γ^{∗}_{0}(8), Γ0(8), Γ^{∗}_{0}(9), Γ0(10)+, Γ^{∗}_{0}(10), Γ0(10) + 5, Γ0(12)+, Γ^{∗}_{0}(12),

Γ0(12) + 4, and Γ0(12).

For case of Conjecture 3.1, the subgroups SL2(Z), Γ^{∗}_{0}(2), and Γ^{∗}_{0}(3) verify this conjecture. For the
other subgroups, we can prove the conjecture by numerical calculation for Eisenstein series of weight
*k*6500.

For the case of Conjecture 3.2, the subgroups SL2(Z) and Γ^{∗}_{0}(2) verify this conjecture for every degree
*m, where we have* *c**h*(Γ) = 0 for each case. Furthermore, for Γ0(2), Γ^{∗}_{0}(3), Γ0(3), Γ^{∗}_{0}(4), Γ0(4), Γ0(6)+,
Γ0(6) + 3, Γ0(6), Γ0(8), Γ^{∗}_{0}(9), Γ0(10)+, Γ0(10) + 5, Γ0(12)+, Γ0(12) + 4, and Γ0(12), we can prove by
numerical calculation that all of the zeros of the modular functions from the Hecke type Faber polynomials
of every degree*m*6200 in each fundamental domain lie on the lower arcs.

On the other hand, for Γ^{∗}_{0}(5) and Γ^{∗}_{0}(7), we can prove the conjecture by numerical calculation for
modular functions from the Hecke type Faber polynomials of every degree *m* = 1 and 3 6 *m* 6 200,
where we have*c**h*(Γ) = 0 for each case. When*m*= 2, for each group there is just one zero which lies on
the boundary of its fundamental domain but not on the lower arcs.

For Γ^{∗}_{0}(6) and Γ^{∗}_{0}(8), we can prove the conjecture by numerical calculation for the modular functions
from the the Hecke type Faber polynomials of every degree*m*6200 satisfying*m6≡*0 (mod 2) and*m6≡*2
(mod 4), respectively. For the remaining degrees, there is just one zero which lies on the boundary of its
fundamental domain but not on the lower arcs for the each group, that is,*c**h*(Γ) = 1.

Finally, for Γ^{∗}_{0}(10) and Γ^{∗}_{0}(12), we have just two zeros which are not on the boundary of each
funda-mental domain for degrees *m*= 7,9,11 and *m*= 3,6,12,13,15, respectively. Furthermore, there is just
one zero which lies on the boundary of its fundamental domain but not on the lower arcs for the cases
*m≡*0 (mod 2) and*m≡*2,4 (mod 6). For the other cases, we can prove that all of the zeros are on the
lower arcs of each fundamental domain by numerical calculation.

Γ Eisenstein series (k6500) the Hecke type Faber polynomial (m6200)
SL2(Z), Γ^{∗}_{0}(2), Γ0(2), Γ^{∗}_{0}(3), Γ0(3),

Γ^{∗}_{0}(4), Γ0(4), Γ0(6)+, Γ0(6) + 3,
Γ0(6), Γ0(8), Γ^{∗}_{0}(9), Γ0(10)+, Γ0(10) + 5,
Γ0(12)+, Γ0(12) + 4, Γ0(12).

*°*

*°*

Γ^{∗}_{0}(5), Γ^{∗}_{0}(7) *m*= 2, *h1i*

Γ^{∗}_{0}(6) *m*: even, *h1i*

Γ^{∗}_{0}(8) *m**≡*0 (mod 4), *h1i*

Γ^{∗}_{0}(10) *m*= 7,9,11,[2], *m*: even,*h1i*

Γ^{∗}_{0}(12) *m*= 3,6,12,13,15,[2] *m**≡*2,4 (mod 6),*h1i*

‘°’: all of the zeros lie on lower arcs.

*h · i: the number of zeros that are on**∂F*but not on lower arcs.

[*·*] : the number of zeros that are not on*∂F.*

Table 3.1: Numerical calculation results.

If the hauptmodul *J*Γ does not take real values on *∂F*Γ (cf. Figure 3.1), the results are somewhat
different. The following cases have such hauptmoduls:

Γ0(5), Γ0(6) + 2, Γ0(7), Γ0(9), Γ0(10) + 2, Γ0(10), Γ^{∗}_{0}(11), and Γ0(12) + 3.

Lower arcs of*∂F*6+2

-8 -6 -4 -2

-6 -4 -2 2 4 6

Figure 3.1: The image of the lower arcs by*J*6+2 (Γ0(6) + 2).

For Γ0(5), Γ0(6)+2, Γ0(7), Γ0(10)+2, Γ0(10), and Γ^{∗}_{0}(11), we can observe from numerical calculations
that the zeros of the Eisenstein series for the *∞*cusp do not lie on the lower arcs of their fundamental
domains. However, when the weight of Eisenstein series increases, then the location of the zeros seems
to approach to lower arcs. (See Figure 3.2)

0

The zeros of*E*^{∞}* _{k,6+2}*
for 46

*k*640

-8 -6 -4 -2

-6 -4 -2 2 4 6

The zeros of*E*1000,6+2^{∞}

-8 -6 -4 -2

-6 -4 -2 2 4 6

Figure 3.2: Image by*J*6+2 (Γ0(6) + 2).

Furthermore, for the zeros of the modular functions from the the Hecke type Faber polynomials, we
can observe from numerical calculations that there are some zeros that do not lie on the lower arcs of
their fundamental domains. However, when the degree *m* increases, the location of the zeros seems to
approach the lower arcs. (See Figure 3.3)

The zeros of*F**m,6+2*

for 16*m*640

-8 -6 -4 -2

-6 -4 -2 2 4 6

The zeros of*F*200,6+2

-8 -6 -4 -2

-6 -4 -2 2 4 6

Figure 3.3: Image by*J*6+2 (Γ0(6) + 2).

In contrast, Γ0(9) and Γ0(12) + 3 seem to be special cases. We can prove that all of the zeros of
the Eisenstein series of weight*k*6500 lie on the lower arcs of their fundamental domains by numerical
calculation. We can also prove by numerical calculation that all of the zeros of the modular functions
from the the Hecke type Faber polynomials of degree*m*6200 lie on the lower arcs. However, they do
not satisfy the assumptions of Conjectures 3.1 and 3.2. However, the images of the lower arcs by their
hauptmoduls are interesting, as shown in Figure 3.4.

0

Γ0(9)

-6 -5 -4 -3 -2 -1 1

-3 -2 -1 1 2 3

Γ0(12) + 3

-4 -3 -2 -1

-4 -2 2 4

Figure 3.4: Image of the lower arcs of the fundamental domains by hauptmoduls.

We refer the reader to [MNS], [SJ1], and [SJ2] for definitions of some of the groups used in this study.

Note, however, that the definitions used in the present study do not always agree with those of these previous studies.