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# The case k ≡ 4 (mod 6)

## 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< α7,k< π/3.

We can use Lemma 2.5.1 (1) for the case α7,k < π/6, and we can use Lemma 2.5.1 (4) and (6) for the cases 0< α7,k< π/4 andπ/4< α7,k< π/3, respectively.

The case π/3< α7,k< π.

(i) “The case 3π/4< α7,k< π”

For 5π/6< α7,k < π (resp. 29π/36< α7,k <5π/6, 3π/4< α7,k <5π/6), we can use Lemma 2.5.1 (1) (resp. (3), (2)).

(ii) “The caseπ/3< α7,k<13π/36” We can use Lemma 2.5.1 (7).

(iii) “The case 2π/3< α7,k<3π/4”

We define cos(c00π) =−cos((x/180)π−(t/2)π). Then, we prove “For (x/180)π < α7,k<(y/180)π, we have|R7,1|<2 cos(c00π) forθ1=π/2 +α7−tπ/k.” for two cases, namely, (x, y, t) = (127.6,135,59/250), (120,127.6,1/4).

(iv) “The case 13π/36< α7,k<5π/9”

We define cos(c00π) = cos((y/180)π−π/3+(t/2)π). Then, we prove “For (x/180)π < α7,k<(y/180)π, we have |R7,2 |<2 cos(c00π) for θ2 =α7−π/6 +tπ/k.” for two cases, namely, (x, y, t) = (65,90,2/5), (90,100,2/5).

Now, we can write

Fk,7,11) = 2 cos (kθ1/2) + 2Re(3e−iθ1/2+

7e1/2)−k+R7,1 00, Fk,7,22) = 2 cos (kθ2/2) + 2k·2Re(3e−iθ2/2−√

7e2/2)−k+R7,200. (v) “The case 73π/120< α7,k<2π/3”

We have X1 = vk(2,1, θ1)−2/k > 1 + 2

3t(π/k) and X2 = vk(3,1, θ1)−2/k 6 e−33(t/2)π. Then, Sign{cos (kθ1/2)}=Sign{Re(3e−iθ1/2+

7e1/2)−k}, and we prove “For (x/180)π < α7,k<(y/180)π, we have|R7,100|<|2 cos (kθ1/2) + 2Re(3e−iθ1/2+

7e1/2)−k|forθ1=π/2 +α7−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 1/2 = k(π/2 +α7)/2(t/2)π. We can show Sign{cos (kθ1/2)} = Sign{Fk,7,11)} , and then we have at least one zero between the second to last integer point forA7,1 and the point1/2.

(vi) “The case 5π/9< α7,k<217π/360”

We haveX1= (1/4)vk(1,−1, θ2)−2/k>1+t(π/k) andX2= (1/4)vk(3,−1, θ2)−2/k 6e−(33/2)(t/2)π. Then, we have Sign{cos (kθ2/2)} =Sign{Re(3e−iθ2/2−√

7e2/2)−k}, and we prove “For (x/180)π <

α7,k < (y/180)π, we have |R7,200| < |2 cos (kθ2/2) + 2k ·2Re(3e−iθ2/2−√

7e2/2)−k| for θ2 = α7 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 FORΓ0(5)ANDΓ0(7) Proposition 2.5.5. Let k>4 be an integer which satisfiesk 4 (mod 6), and let α7,k [0, π] be the angle which satisfies α7,k k(π/2 +α7)/2 (modπ). If we have α7,k <217π/360 or 73π/120 < α7,k, then all of the zeros ofEk,7 (z)inF(7)lie on the arc A7. Otherwise, all but at most one zero ofEk,7 (z) inF(7) are on A7

7,k

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, whenkis large enough, we have the following observation.

The case “k2 (mod 6)and 266π/375< α7,k<3217π/4500”

Lett >0 be small enough, then we haveπ/2< α7,k−(t/2)π < π,π <2π/3+α7,k+d1,1(t/2)π < α7,k,10 <

2π/3 +α7,k+ (t/2)π <3π/2, and 2π <4π/3 +α7,k−tπ < α7,k,20 <4π/3 +α7,k−d1,2(t/2)π <5π/2.

Thus, we have

cos(α7,k(t/2)π)cos(2π/3 +α7,k+d1,1(t/2)π)·(1 + 2

3t(π/k))−k/2

cos(4π/3 +α7,k−d1,2(t/2)π)·e−(33/2)πt

>|cos(kθ1/2)|+

¯¯

¯¯Re

½³

2e1/2+

7e−iθ1/2

´−k¾¯¯

¯¯

¯¯

¯¯Re

½³

3e1/2+

7e−iθ1/2

´−k¾¯¯

¯¯

>−cos(α7,k(t/2)π)cos(2π/3 +α7,k+ (3t/2)π)·e3πt

cos(4π/3 +α7,k−tπ)·(1 + 3

3t(π/k))−k/2.

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

∂tA|t=0 = ∂tB|t=0 = 0. Second, let C = (5

3/2)π2(−cosα7,k)(tanα7,k+ 11/(5

3)), then we have

2

∂t2A|t=0=C+ 6π2(−cos(2π/3 +α7,k))/k and ∂t22B|t=0=C−(27/2)π2cos(4π/3 +α7,k)/k. Finally, we have B >0 ifα7,k>3π/27, and we haveA <0 ifα7,k <3π/27for large enough kand small enought.

Similarly, we consider the lower and the upper bounds of|cos(kθ2/2)|−|Re{2k·(e2/2−√

7e−iθ2/2)−k}|−

|Re{2k·(3e2/2−√

7e−iθ2/2)−k}|. The lower bound is positive if α7,k <3π/27, while the upper bound is negative ifα7,k>3π/27 for large enoughk and for small enought.

In conclusion, if k is large enough, then |R7,10| and |R7,20| is small enough, and then we have one more zero on the arc A7,1 when α7,k >3π/27, and we have one more zero on the arcA7,2 when α7,k<3π/27. However, ifkis 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 Γ0(5) and Γ0(7).

The case “k4 (mod 6)and 217π/360< α7,k<73π/120”

Lett >0 be small enough. Similar to Subsection 2.4.4, ifkis large enough, then we have one more zero on the arcA7,1 whenα7,k> π−α7, and we have one more zero on the arcA7,2 whenα7,k< π−α7. 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 Γ0(2) and Γ0(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 ofCours 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 forΓ0(5)andΓ0(7), arXiv:math.NT/0607247.

### 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 seriesEk(z) in the standard fundamental domainF (See [RSD]). They proved that all of the zeros of Ek(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 Γ0(p) (see [Kr], [Q]), and proved that all of the zeros of the Eisenstein series Ek,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/

por 1/(2

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

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

z∈H; −h/2< Re(z)< h/2, |cz+d|>1 for∀γa b

c d

¢Γ s.t.c6= 0ª

. (3.1)

We have a fundamental domainFΓ such thatF0,ΓFΓ F0,Γ. LetFΓ be such a fundamental domain.

For the modular group SL2(Z) and the Fricke groups Γ0(p) (p= 2,3), all the zeros of the Eisenstein series for the cusplie 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 withas a cusp whose hauptmodulJΓtakes real values on∂FΓ, and proved that almost all the zeros of the Eisenstein series for the cuspfor Γ lie on∂FΓ under some further assumptions (see [H]).

T. Asai, M. Kaneko, and H. Ninomiya further considered the problem of locating the zeros of modular functions Fm(z) for SL2(Z) that correspond to the Hecke type Faber polynomialsPm, that is, Fm(z) = Pm(J(z)) (See [AKN]). They proved that all of the zeros of Fm(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 Γ0(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 JΓ takes real values on∂FΓ, all of the zeros of the Eisenstein series for the cusp ∞forΓin FΓ lie on the arcs

∂FΓ\ {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 JΓ takes real values on∂FΓ, all but at most ch(Γ)of the zeros of modular function from the the Hecke type Faber polynomial of degree mforΓ inFΓ lie on the arcs

∂FΓ\ {z∈H; Re(z) =±h/2}

for all but a finite number ofmand for the constant ch(Γ), which does not depend onm.

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), Γ0(2), Γ0(2), Γ0(3), Γ0(3), Γ0(4), Γ0(4), Γ0(5), Γ0(6)+, Γ0(6), Γ0(6) + 3, Γ0(6), Γ0(7), Γ0(8), Γ0(8), Γ0(9), Γ0(10)+, Γ0(10), Γ0(10) + 5, Γ0(12)+, Γ0(12),

Γ0(12) + 4, and Γ0(12).

For case of Conjecture 3.1, the subgroups SL2(Z), Γ0(2), and Γ0(3) verify this conjecture. For the other subgroups, we can prove the conjecture by numerical calculation for Eisenstein series of weight k6500.

For the case of Conjecture 3.2, the subgroups SL2(Z) and Γ0(2) verify this conjecture for every degree m, where we have ch(Γ) = 0 for each case. Furthermore, for Γ0(2), Γ0(3), Γ0(3), Γ0(4), Γ0(4), Γ0(6)+, Γ0(6) + 3, Γ0(6), Γ0(8), Γ0(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 degreem6200 in each fundamental domain lie on the lower arcs.

On the other hand, for Γ0(5) and Γ0(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 havech(Γ) = 0 for each case. Whenm= 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 Γ0(6) and Γ0(8), we can prove the conjecture by numerical calculation for the modular functions from the the Hecke type Faber polynomials of every degreem6200 satisfyingm6≡0 (mod 2) andm6≡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,ch(Γ) = 1.

Finally, for Γ0(10) and Γ0(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) andm≡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), Γ0(2), Γ0(2), Γ0(3), Γ0(3),

Γ0(4), Γ0(4), Γ0(6)+, Γ0(6) + 3, Γ0(6), Γ0(8), Γ0(9), Γ0(10)+, Γ0(10) + 5, Γ0(12)+, Γ0(12) + 4, Γ0(12).

°

°

Γ0(5), Γ0(7) m= 2, h1i

Γ0(6) m: even, h1i

Γ0(8) m0 (mod 4), h1i

Γ0(10) m= 7,9,11,, m: even,h1i

Γ0(12) m= 3,6,12,13,15, m2,4 (mod 6),h1i

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

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

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

Table 3.1: Numerical calculation results.

If the hauptmodul JΓ does not take real values on ∂FΓ (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), Γ0(11), and Γ0(12) + 3.

Lower arcs of∂F6+2

-8 -6 -4 -2

-6 -4 -2 2 4 6

Figure 3.1: The image of the lower arcs byJ6+20(6) + 2).

For Γ0(5), Γ0(6)+2, Γ0(7), Γ0(10)+2, Γ0(10), and Γ0(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 ofEk,6+2 for 46k640

-8 -6 -4 -2

-6 -4 -2 2 4 6

The zeros ofE1000,6+2

-8 -6 -4 -2

-6 -4 -2 2 4 6

Figure 3.2: Image byJ6+20(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 ofFm,6+2

for 16m640

-8 -6 -4 -2

-6 -4 -2 2 4 6

The zeros ofF200,6+2

-8 -6 -4 -2

-6 -4 -2 2 4 6

Figure 3.3: Image byJ6+20(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 weightk6500 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 degreem6200 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.

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