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## 保型関数の零点の配置、及び格子から得られる球面 デザインについての研究

重住, 淳一

九州大学大学院 数理学府: 学生(D3): 代数的組合せ論

https://doi.org/10.15017/458169

出版情報：Kyushu University, 2008, 博士（数理学）, 課程博士 バージョン：

権利関係：

### and

### Some Spherical Designs from Lattices

### Junichi Shigezumi

### A dissertation submitted in partial fulfillment of the requirements for the degree of

### Doctor of Philosophy in

### Mathematics

### Graduate School of Mathematics

### Kyushu University

## Acknowledgements

I would like to thank my supervisor, Professor Eiichi Bannai, for suggesting various topics for my doctor course project, and for insightful comments and helpful advice, which were essential to the success of the research in this thesis.

I would also like to thank Professor Masanobu Kaneko, Professor Masao Koike, and Professor Yuichiro Taguchi for their helpful comments regarding modular forms and Professor Etsuko Bannai for her com- ments and encouragement.

I am deeply indebted to the Japan Society for the Promotion of Science, which supported this research as well as my participation in several conferences.

The results presented in Chapter 1 were obtained through joint research with Dr. Hiroshi Nozaki and Dr. Tsuyoshi Miezaki. I am grateful for having had the opportunity to work with them.

Finally, I am grateful for the encouragement I received from my colleagues in the laboratory and from my family.

i

## Preface

The *location of the zeros of certain modular functions, particularly the Eisenstein series, Hecke type*
Faber polynomials, and certain Poincar´e series for certain genus zero Fuchsian groups of the first kind,
is investigated first.

Let*k*>4 be an even integer. For*z∈*H:=*{z∈*C; *Im(z)>*0}, let
*E**k*(z) := 1

2 X

(c,d)=1

(cz+*d)** ^{−k}*
be the

*Eisenstein series*associated with SL2(Z). Moreover, let

F:=*{|z|*>1, *−1/2*6*Re(z)*60} ∪ {|z|*>*1, 06*Re(z)<*1/2}

be the*standard fundamental domain*for SL2(Z).

F. K. C. Rankin and H. P. F. Swinnerton-Dyer considered the problem of locating the zeros of*E**k*(z)
in F[RSD]. They proved that*n* zeros are on the arc*A*:=*{z* *∈*C; *|z|*= 1, π/2*< Arg(z)<*2π/3} for
*k*= 12n+*s*(s= 4,6,8,10,0, and 14). They also stated that, “This method can equally well be applied
to Eisenstein series associated with subgroup of the modular group.” However, it is unclear how widely
this claim holds.

Subsequently, T. Miezaki, H. Nozaki, and the present author considered the same problem for Fricke
groups Γ^{∗}_{0}(2) and Γ^{∗}_{0}(3), which are not subgroups of SL2(Z) but are commensurable with SL2(Z). For a
fixed prime*p, we have the following:*

Γ^{∗}_{0}(p) := Γ0(p)*∪*Γ0(p)*W**p**,* (1)

where

Γ0(p) =©¡_{a b}

*c d*

¢*∈*SL2(Z) ; *c≡*0 (mod*p)*ª

*,* *W**p*:=

³ 0 *−1/**√*

*√* *p*

*p* 0

´
*.*

Then, all of the zeros of the Eisenstein series*E*_{k,p}* ^{∗}* (z) in a certain fundamental domain were proved to lie
on a circle having a radius equal to 1/

*√*

*p,* *p*= 2,3 (Chapter 1). Furthermore, for the case of*p*= 5,7, it
was proved that most of the zeros of the Eisenstein series in a certain fundamental domain lie on circles
having radii equal to 1/*√*

*p*or 1/(2*√*

*p) (Chapter 2).*

H. Hahn described the location of the zeros of the Eisenstein series [H]. Let Γ*⊆*SL2(R) be a genus
zero Fuchsian group of the first kind having *∞* as a cusp, and let *E*^{Γ}_{2k} be the holomorphic Eisenstein
series associated with Γ for the*∞*cusp that does not vanish at*∞*but vanishes at all other cusps. Under
the assumptions on Γ, and on a certain fundamental domain*F*, H. Hahn proved that all but at most
*c(Γ,F) (a constant) of the zeros of* *E*_{2k}^{Γ} lie on a certain subset (its lower arcs and the imaginary axis) of
*{z∈*H : *j*Γ(z)*∈*R}.

However, based on the numerical calculations in*Chapter 3, all of the zeros inF* lie on only its lower
arcs, not on the imaginary axis. Then, under the same assumption and considering a small generalization
of her result, most of the zeros of*E*_{2k}^{Γ} in *F* can be proved to lie on its lower arcs (Chapter 4). However,
if Γ does not satisfy the assumption (without any*acceptable*fundamental domain), then the zeros of*E*_{2k}^{Γ}
do not always lie on*{z∈*H : *j*Γ(z)*∈*R}(Chapter 3).

In addition, T. Asai, M. Kaneko, and H. Ninomiya considered the problem of locating the zeros of
modular functions *F**m*(z) for SL2(Z), which correspond to the Hecke type Faber polynomial, *P**m*, that

ii

*m* *m* *m*

circle for each *m* >1. Then, E. Bannai, K. Kojima, and T. Miezaki considered the same problem for
the normalizers of congruence subgroups that correspond to the conjugacy classes of the Monster group
(see [BKM]). They observed the location of the zeros by numerical calculations, and most of the zeros
of the modular functions from the Hecke type Faber polynomial lie on the lower arcs when the group
satisfies the assumption of the theorem of H. Hahn. In particular, T. Miezaki proved that all but at most
one of the zeros of the modular functions from the Hecke type Faber polynomials for the Fricke group
Γ^{∗}_{0}(2) lie on the lower arcs of its fundamental domain.

Furthermore, R. A. Rankin considered the same problem for certain Poincar´e series associated with SL2(Z) [R]. He also applied the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer, and proved that all of the zeros of certain Poincar´e series inFalso lie on the unit circle. The distribution of the zeros of certain Poincar´e series is also interesting.

The distribution of the zeros of the Eisenstein series resembles a uniform distribution with the*argu-*
*ment*on the lower arc ofF. On the other hand, the distribution of the zeros of the modular functions from
the Hecke type Faber polynomials resembles a uniform distribution with the*real part*on the lower arc ofF.

Next, the Poincar´e series*G**k*(z;*t** ^{−m}*) is considered. Note that

*G*

*k*(z;

*t*

^{0}) =

*E*

*k*(z) and

*G*0(z;

*t*

*) =*

^{−m}*F*

*m*(z).

Furthermore, if*k* is sufficiently large compared with*m, then the distribution of zeros of* *G**k*(z;*t** ^{−m}*) re-
sembles that of

*E*

*k*(z). On the other hand, if

*m*is sufficiently large compared with

*k, then the distribution*resembles that of

*F*

*m*(z). Thus, the Poincar´e series

*G*

*k*(z;

*t*

*) “fills the space of two modular functions discretely” (Chapter 5).*

^{−m}In addition, it has been proved that all of the zeros of certain Poincar´e series for Γ^{∗}_{0}(p) in a certain
fundamental domain lie on a circle having a radius equal to 1/*√*

*p,p*= 2,3 (Chapter 5).

The second topic is *certain classifications of lattices and spherical designs.*

Let*S** ^{d−1}*:=

*{(x*1

*, . . . , x*

*d*)

*∈*R

*;*

^{d}*x*

^{2}

_{1}+

*· · ·*+

*x*

^{2}

*= 1} be the Euclidean sphere for*

_{d}*d*>1.

Definition(Spherical design [DGS]) Let*X* be a non-empty finite set on the Euclidean sphere*S** ^{d−1}*, and
let

*t*be a positive integer.

*X*is called a spherical

*t-design if*

1

*|S*^{d−1}*|*
Z

*S*^{d−1}

*f*(ξ)*dξ* = 1

*|X|*

X

*ξ∈X*

*f(ξ)* (2)

for every polynomial*f(x) =f(x*1*, . . . , x**d*) of degree at most*t.*

Here, the left-hand side of the above equation (2) represents the average on the sphere*S** ^{d−1}*, and the
right-hand side represents the average on the finite subset

*X*. Thus, if

*X*is a spherical design, then

*X*gives a certain approximation of the sphere

*S*

*.*

^{d−1}Let*L*be a Euclidean lattice, which is a discrete freeZ-module. The squared norm of a vector of the
lattice is called the*norm of the vector. Then, the sets**m*(L) of vectors of the lattice*L*that take the same
value*m*for their norm is called the*shell of the lattice, i.e.,s**m*(L) :=*{x∈L*; (x, x) =*m}. Moreover, the*
shell of*minimum*min* _{x∈L\{0}}*(x, x) of the lattice

*L*is called the

*minimal shell, which is denoted byS(L).*

If an integral lattice *L* is generated by the vectors of*s**q*(L), then*L* is referred to as a*q-lattice. For*
every nonempty shell *s**m*(L) of lattice *L, normalization* ^{√}^{1}_{m}*s**m*(L), which is a finite set on a Euclidean
sphere, is considered. Then, spherical designs from shells of lattices can be considered. A lattice having a
minimal shell that is a spherical 5-design is said to be*strongly perfect. Then, the following are obtained*
(Details of the definitions are omitted. Please refer to the references.):

Theorem (see [V]) A strongly perfect integral lattice of minimum 2 is isometric to one of root lattices
A_{1},A_{2},D_{4},E_{6},E_{7}, orE_{8}. Furthermore, the minimal shell of the lattice is a 7-design only for the case of
the latticeE8.

Theorem(B. B. Venkov [V], Theorem 7.4) The strongly perfect lattices that are integral and of minimum
3 are*O*1,*O*7,*O*16,*O*22, and*O*23. Furthermore, the minimal shell is a 7-design only for the case of lattice
*O*23.

iii

appears to be one method of searching for ‘good’ spherical designs. Thus, all of the irreducible 3-lattices of dimension at most seven were classified by numerical calculation. Furthermore, the spherical designs of norm-3 shells of all of the 3-lattices are calculated (Chapter 6).

As an expansion of the above theorem by B. B. Venkov, the classification of integral lattices having shells of norm 3 that are 5-designs is considered (Chapter 7).

### Chapter 1.

*On the zeros of Eisenstein series for*Γ

^{∗}_{0}(2)

*and*Γ

^{∗}_{0}(3)

This research was performed in collaboration with H. Nozaki and T. Miezaki. We considered the
locations of the zeros of the Eisenstein series for Γ^{∗}_{0}(2) and Γ^{∗}_{0}(3).

Let

*E*_{k,p}* ^{∗}* (z) := 1

*p*

*+ 1*

^{k/2}³

*p*^{k/2}*E**k*(pz) +*E**k*(z)

´

be the Eisenstein series associated with Γ^{∗}_{0}(p). The region
F* ^{∗}*(p) :=

*{|z|*>1/

*√*

*p,−1/2*6*Re(z)*60}[

*{|z|>*1/*√*

*p,* 06*Re(z)<*1/2}

is a fundamental domain for Γ^{∗}_{0}(p) when*p*= 2 or 3. Define*A*^{∗}* _{p}*:=F

*(p)*

^{∗}*∩ {z∈*C;

*|z|*= 1/

*√*

*p}.*

In Chapter 1, we apply the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (RSD Method)
to the Eisenstein series for Γ^{∗}_{0}(2) and Γ^{∗}_{0}(3) and prove the following theorems:

Theorem 1. *Let* *k*>4 *be an even integer. All of the zeros ofE*_{k,2}* ^{∗}* (z)

*in*F

*(2)*

^{∗}*are on the arcA*

^{∗}_{2}

*.*Theorem 2.

*Let*

*k*>4

*be an even integer. All of the zeros ofE*

_{k,3}*(z)*

^{∗}*in*F

*(3)*

^{∗}*are on the arcA*

^{∗}_{3}

*.*

This result was published in the Journal of the Mathematical Society of Japan [MNS].

### Chapter 2.

*On the zeros of Eisenstein series for*Γ

^{∗}_{0}(5)

*and*Γ

^{∗}_{0}(7)

We consider the location of the zeros of the Eisenstein series for Γ^{∗}_{0}(5) and Γ^{∗}_{0}(7).

Henceforth, we assume that*p*= 5 or 7. The region
F* ^{∗}*(p) :=

*{|z|*>1/

*√*

*p,* *|z|*>1/(2*√*

*p),* *−1/2*6*Re(z)*60}

[*{|z|>*1/*√*

*p,|z|>*1/(2*√*

*p),* 06*Re(z)<*1/2}

is a fundamental domain for Γ^{∗}_{0}(p). Define*A*^{∗}* _{p}* :=F

*(p)*

^{∗}*∩ {z∈*C;

*|z|*= 1/

*√*

*p*or*|z|*= 1/(2*√*
*p)}.*

We apply the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (RSD Method) to the
Eisenstein series associated with Γ^{∗}_{0}(5) and Γ^{∗}_{0}(7) and have the following conjectures:

Conjecture 3. *Letk*>4*be an even integer. Then all of the zeros ofE*_{k,5}* ^{∗}* (z)

*in*F

*(5)*

^{∗}*lie on the arcA*

^{∗}_{5}

*.*Conjecture 4.

*Letk*>4

*be an even integer. Then all of the zeros ofE*

_{k,7}*(z)*

^{∗}*in*F

*(7)*

^{∗}*lie on the arcA*

^{∗}_{7}

*.*First, we prove that all but at most 2 zeros of

*E*

_{k,p}*(z) inF*

^{∗}*(p) lie on the arc*

^{∗}*A*

^{∗}*. Second, if (24/(p+1))*

_{p}*|*

*k, we prove that all of the zeros ofE*

_{k,p}*(z) inF*

^{∗}*(p) lie on*

^{∗}*A*

^{∗}*. We can then prove that if (24/(p+ 1))-*

_{p}*k,*all but one of the zeros of

*E*

_{k,p}*(z) inF*

^{∗}*(p) lie on*

^{∗}*A*

^{∗}*. Furthermore, for most of the other cases, we prove Conjectures 3 and 4 (see Section 2.1 for details).*

_{p}This result was published in the Kyushu Journal of Mathematics [SJ2].

### Chapter 3.

*On the zeros of certain modular functions for the normalizers of congruence subgroups*We investigate the location of the zeros of the Eisenstein series and the 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.

iv

2

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
*.*

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

Then, we have the following conjectures:

Conjecture 5. *Let*Γ*be a genus zero Fuchsian group of the first kind with∞as a cusp. If the hauptmodul*
*J*Γ *takes a real value on∂F*Γ*, then 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 6. *Let*Γ*be a genus zero Fuchsian group of the first kind with∞as a cusp. If the hauptmodul*
*J*Γ *takes a real value on* *∂F*Γ*, then all but at most* *c**h*(Γ) *of the zeros of the modular function from the*
*Hecke type Faber polynomial of degree* *m* *for*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 number* *c**h*(Γ), which does not depend on*m.*

In Chapter 3, we consider normalizers of congruence subgroups that are of genus zero and of level
at most twelve. We investigate the general theory of each group, *i.e., fundamental domain, the valence*
formula, the Eisenstein series, the space of modular forms, and the hauptmodul. Then, we prove the
above conjectures for the Eisenstein series of weight *k* 6 500 and for the modular functions from the
Hecke type Faber polynomials of every degree*m*6200 by numerical calculations.

### Chapter 4.

*A note on zeros of Eisenstein series for genus zero Fuchsian groups*

We consider a small generalization of Hahn’s result “On zeros of Eisenstein series for genus zero
Fuchsian groups”, particularly on the domain on which the zeros of*E*_{2k}^{Γ} are located.

We denote by Γ a Fuchsian group of the first kind, which has*∞*as a cusp with width*h. LetF* be a
fundamental domain of Γ contained in*{z* : *−h/2*6*<(z)< h/2}, and letA* be the lower arcs of*F*. We
then define*y*0:= inf{y : *±h/2 +i y∈∂F}*and*a*0:=*j*Γ(−h/2 +*i y*0). We denote by*c(Γ,F) the number*
of equivalence classes under the action of Γ on the set of critical points of*F* at which _{(j} ^{y}^{0}^{(t)}

Γ*◦z**A*)* ^{0}*(t) changes
sign. Further details may be found in [H]. The main theorem of [H] is the following:

Theorem (H. Hahn [H])*Let* Γ *be a genus zero group that is good for the weight* 2k. Suppose that *F* *is*
*acceptable for* Γ, and that *j*Γ *has real Fourier coefficients. Then, all but possibly* *c(Γ,F)* *of the zeros of*
*P*(E^{Γ}_{2k}*, X)* *lie on*[a0*,∞), where* *a*0 *is the end point of the lower arcs ofF. Moreover, ifm* *denotes the*
*number of distinct zeros with odd multiplicity on*[a0*,∞), thenm*+*c(Γ,F)*>deg(P(E_{2k}^{Γ}*, X)).*

The terms *“good”* and *“acceptable”* are defined in [H, Section 1], and *P*(f, X) *∈* C[X] is the *divisor*
*polynomial*of a modular form*f* (see [H, Section 3]).

We denote by*s*^{1}_{2k}(Γ) the number of cusps other than 0 and*−1/2 at whichE*_{2k}^{Γ} has odd multiplicity.

Then, since*E*_{2k}^{Γ} vanishes at all of the cusps other than*∞, we can substitutec(Γ,F) forc(Γ,F)−s*^{1}_{2k}(Γ)
in the above theorem. Similarly to*y*0and*a*0, we define*y*1:= inf*{y* : *i y∈∂F}*and*a*1:=*j*Γ(i y1).

In Chapter 4, we prove the following theorem:

Theorem 7. *Let* Γ *be a genus zero group that is good for the weight* 2k. Suppose that *F* *is acceptable*
*for* Γ, and that *j*Γ *has real Fourier coefficients. Then, all but at most* *c(Γ,F)−s*^{1}_{2k}(Γ) *of the zeros of*
*P*(E^{Γ}_{2k}*, X)lie on* (−∞, a1]. Moreover, if*m* *denotes the number of distinct zeros with odd multiplicity on*
(−∞, a1], then*m*+*c(Γ,F)−s*^{1}_{2k}(Γ)>deg(P(E_{2k}^{Γ}*, X*)).

### Chapter 5.

*On the zeros of certain Poincar´e series for*Γ

^{∗}_{0}(2)

*and*Γ

^{∗}_{0}(3)

The location of the zeros of certain Poincar´e series for Γ^{∗}_{0}(2) and Γ^{∗}_{0}(3) are considered.

Let

*G*^{∗}* _{k, p}*(z;

*R) :=*1 2

X

(c,d)=1
*p|c*

*R(e*^{2πiγz})
(cz+*d)** ^{k}* +1

2 X

(c,d)=1
*p|c*

*R(e*2πiγ(−1/(pz)))
(d*√*

*pz−c/√*
*p)*^{k}

be a Poincar´e series associated with Γ^{∗}_{0}(p), where*R(t) is a suitably chosen rational function oft.*

v

method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer and the method of R. A. Rankin are applied.

The dimension of cusp forms for Γ^{∗}_{0}(p) of weight*k* is denoted by*l. The following theorems are proved:*

Theorem 8. *Let* *k*>4 *be an even integer, and let* *m* *be a non-negative integer. Then, all of the zeros*
(i.e.,*k((p*+ 1)/24) +*mzeros)ofG*^{∗}* _{k,p}*(z;

*t*

*)*

^{−m}*in*F

*(p)*

^{∗}*lie on the arcA*

^{∗}

_{p}*forp*= 2,3.

Theorem 9. *Letk*>4 *be an even integer, and let* *m*6*l* *be a positive integer. Then,G*^{∗}* _{k,p}*(z;

*t*

*)*

^{m}*has at*

*leastk((p*+ 1)/24)

*−m*

*zeros on the arcA*

^{∗}

_{p}*and at least one zero at∞*

*forp*= 2,3.

### Chapter 6.

*On*3-lattices and spherical designs

Irreducible 3-lattices of dimension at most seven were classified by numerical calculations. The results
are presented in the appendix of Chapter 6. The following table lists the numbers of *d-dimensional*
irreducible 3-lattices of minimum 2 and minimum 3. (cf., [SJ7])

1 2 3 4 5 6 7

min. 2 *−* 1 4 18 90 591 5449

min. 3 1 1 3 9 39 247 2446

We then calculate the spherical designs of norm-3 shells of all of the 3-lattices, which are reducible and irreducible.

### Chapter 7.

*Spherical designs from the norm-3*

*shell of integral lattices*

B. B. Venkov classified strongly perfect lattices of minimum 3, the minimal shell of which is a spherical 5-design. As an expansion, we consider the classification of integral lattices having shells of norm 3 that are 5-designs.

In Chapter 7, the following theorem is proved:

Theorem 10. *Let* *L* *be an integral lattice. If its shell of norm* 3 *is a spherical* 5-design, then *L* *is*
*isometric to one of the following nine lattices:*

*1.* Z^{7}*, the minimum of which is equal to one.*

*2.* Λ16,2,1*,*Λ16,2,2*, and*Λ16,2,3*, the minima of which are equal to two.*

*3.* *O*1*,O*7*,O*16*,O*22*, andO*23*, the minima of which are equal to three.*

The definitions of the lattices in the above theorems are given in the chapter.

This result is to appear in the Asian-European Journal of Mathematics [SJ8].

### Bibliography

[AKN] T. Asai, M. Kaneko, and H. Ninomiya, *Zeros of certain modular functions and an application,*
Comment. Math. Univ. St. Paul. 46 (1997), 93–101.

[BKM] E. Bannai, K. Kojima, and T. Miezaki,*On the zeros of Hecke type Faber polynomial, Kyushu. J.*

Math. 62(2007), 15–61.

[H] H. Hahn, *On zeros of Eisenstein series for genus zero Fuchsian groups, Proc. Amer. Math. Soc.*

135(2007), No. 8, 2391–2401.

[M] Y. Mimura,*On*3-lattices(Japanese), The journal of Kobe Pharmaceutical University in humanities
and mathematics, 7(2006), 29–42.

[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.

vi

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

[SJ2] J. Shigezumi,*On the zeros of the Eisenstein series for*Γ^{∗}_{0}(5)*and*Γ^{∗}_{0}(7), Kyushu. J. Math. 61(2007),
527–549.

[SJ7] J. Shigezumi, *Classification of*3-lattices, published on WWW:

http://www2.math.kyushu-u.ac.jp/~j.shigezumi/L3/

[SJ8] J. Shigezumi, *Spherical designs from norm-3* *shell of integral lattices, to appear in Asian-Eur J.*

Math. (arXiv:0811.2653 [math.NT]).

[V] B. B. Venkov,*R´eseaux et “designs” sph´eriques*(French) [Lattices and spherical designs], in: R´eseaux
euclidiens, designs sph´eriques et formes modulaires, 10–86, Monogr. Enseign. Math.,37, Enseigne-
ment Math., Geneva, 2001. [Notes by J. Martinet]

vii

## Table of Contents

### Acknowledgements i

### Preface ii

Table of Contents ix

### I On the zeros of certain modular functions 1

1 On the zeros of Eisenstein series for Γ^{∗}_{0}(2) and Γ^{∗}_{0}(3) 3

(joint work with H. Nozaki and T. Miezaki) 3

2 On the zeros of Eisenstein series for Γ^{∗}_{0}(5) and Γ^{∗}_{0}(7) 15
3 On the zeros of certain modular functions for the normalizers ofΓ0(N) 31
4 A note on zeros of Eisenstein series for genus zero Fuchsian group 139
5 On the zeros of certain Poincar´e series for Γ^{∗}_{0}(2) and Γ^{∗}_{0}(3) 145

Bibliography 157

### II Certain classifications of lattices and spherical designs 159

6 On 3-lattices and spherical designs 161

7 Spherical designs from the norm-3 shell of integral lattices 275

Bibliography 287

ix

### Part I

## On the Zeros of

## Certain Modular Functions

1

### Chapter 1

## On the zeros of Eisenstein series for Γ ^{∗} _{0} (2) and Γ ^{∗} _{0} (3)

^{∗}

^{∗}

### Joint work with Hiroshi Nozaki and Tsuyoshi Miezaki

We locate all of the zeros of the Eisenstein series associated with the Fricke groups Γ^{∗}_{0}(2) and Γ^{∗}_{0}(3) in
their fundamental domains by applying and expanding the method of F. K. C. Rankin and H. P. F.

Swinnerton-Dyer (“On the zeros of Eisenstein series”, 1970).

### 1.1 Introduction

Let*k*>4 be an even integer. For*z∈*H:=*{z∈*C; *Im(z)>*0}, let
*E**k*(z) := 1

2 X

(c,d)=1

(cz+*d)** ^{−k}* (1.1)

be the*Eisenstein series*associated with SL2(Z). Moreover, let

F:=*{|z|*>1, *−1/2*6*Re(z)*60} ∪ {|z|*>*1, 06*Re(z)<*1/2}

be the*standaard fundamental domain*for SL2(Z).

F. K. C. Rankin and H. P. F. Swinnerton-Dyer considered the problem of locating the zeros of*E**k*(z)
in F[RSD]. They proved that*n* zeros are on the arc*A*:=*{z* *∈*C; *|z|*= 1, π/2*< Arg(z)<*2π/3} for
*k*= 12n+*s*(s= 4,6,8,10,0, and 14). They also said in the last part of the paper, “This method can
equally well be applied to Eisenstein series associated with subgroup of the modular group.” However, it
seems unclear how widely this claim holds.

Here, we consider the same problem for Fricke groups Γ^{∗}_{0}(2) and Γ^{∗}_{0}(3) (See [Kr], [Q]), which are
commensurable with SL2(Z). For a fixed prime*p, we define the following:*

Γ^{∗}_{0}(p) := Γ0(p)*∪*Γ0(p)*W**p**,* (1.2)
where

Γ0(p) :=©¡_{a b}

*c d*

¢*∈*SL2(Z) ; *c≡*0 (mod*p)*ª

*,* *W**p*:=³

0 *−1/**√*

*√* *p*

*p* 0

´

*.* (1.3)

Let

*E*_{k,p}* ^{∗}* (z) := 1

*p*

*+ 1*

^{k/2}³

*p*^{k/2}*E** _{k}*(pz) +

*E*

*(z)´*

_{k}(1.4)
be the Eisenstein series associated with Γ^{∗}_{0}(p). The regions

F* ^{∗}*(2) :=n

*|z|*>1/*√*

2, *−1/2*6*Re(z)*60o [ n

*|z|>*1/*√*

2, 06*Re(z)<*1/2o
*,*
F* ^{∗}*(3) :=n

*|z|*>1/*√*

3, *−1/2*6*Re(z)*60o [ n

*|z|>*1/*√*

3, 06*Re(z)<*1/2o
3

4 *CHAPTER 1. ON THE ZEROS OF EISENSTEIN SERIES FOR*Γ_{0}(2)*AND*Γ_{0}(3)
are fundamental domains for Γ^{∗}_{0}(2) and Γ^{∗}_{0}(3), respectively.

Define *A*^{∗}_{2} :=*{z* *∈* C; *|z|* = 1/*√*

2, π/2*< Arg(z)<*3π/4}, and *A*^{∗}_{3} :=*{z* *∈*C; *|z|*= 1/*√*

3, π/2 *<*

*Arg(z)<*5π/6}. We then have*A*^{∗}_{2}=*A*^{∗}_{2}*∪ {i/√*

2, e^{i(3π/4)}*/√*

2}and*A*^{∗}_{3}=*A*^{∗}_{3}*∪ {i/√*

3, e^{i(5π/6)}*/√*
3}.

In the present paper, we will apply the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer
(RSD Method) to the Eisenstein series associated with Γ^{∗}_{0}(2) and Γ^{∗}_{0}(3). We will prove the following
theorems:

Theorem 1.1.1. *Letk*>4*be an even integer. All of the zeros of* *E*_{k,2}* ^{∗}* (z)

*in*F

*(2)*

^{∗}*are on the arcA*

^{∗}_{2}

*.*Theorem 1.1.2.

*Letk*>4

*be an even integer. All of the zeros of*

*E*

_{k,3}*(z)*

^{∗}*in*F

*(3)*

^{∗}*are on the arcA*

^{∗}_{3}

*.*

### 1.2 RSD Method

At the beginning of the proof in [RSD], F. K. C. Rankin and H. P. F. Swinnerton-Dyer considered the following:

*F**k*(θ) :=*e*^{ikθ/2}*E**k*

¡*e** ^{iθ}*¢

*,* (1.5)

which is real for all*θ∈*[0, π]. Considering the four terms with*c*^{2}+*d*^{2}= 1, they proved that

*F**k*(θ) = 2 cos(kθ/2) +*R*1*,* (1.6)

where*R*1is the rest of the series (i.e. *c*^{2}+*d*^{2}*>*1). Moreover they showed

*|R*1*|*61 +
µ1

2

¶* _{k/2}*
+ 4

µ2 5

¶* _{k/2}*
+20

*√*

2
*k−*3

µ9 2

¶_{(3−k)/2}

*.* (1.7)

They computed the value of the right-hand side of (1.7) at*k*= 12 to be approximately 1.03562, which is
monotonically decreasing in*k. Thus, they could show that|R*1*|<*2 for all*k*>12. If cos(kθ/2) is +1 or

*−1, thenF**k*(2mπ/k) is positive or negative, respectively.

In order to determine the location of all of the zeros of*E**k*(z) inF, we need the*valence formula:*

Proposition 1.2.1 (valence formula). *Let* *f* *be a modular function of weight* *k* *for*SL2(Z), which is
*not identically zero. We have*

*v**∞*(f) +1

2*v**i*(f) +1

3*v**ρ*(f) + X

*p∈SL*2(Z)\H
*p6=i, ρ*

*v**p*(f) = *Kr*

12*,* (1.8)

*wherev**p*(f)*is the order of* *f* *atp, andρ*:=*e** ^{i(2π/3)}* (See [Se]).

Write *m(k) :=* ¥_{Kr}

12 *−*_{4}* ^{t}*¦

, where *t* = 0 or 2, such that *t* *≡k* (mod 4). Then, *k* = 12m(k) +*s*(s =
4,6,8,10,0, and 14).

As F. K. C. Rankin and H. P. F. Swinnerton-Dyer observed, the fact that*E** _{k}*(z) has

*m(k) zeros on the*arc

*A, the valence formula, and Remark 1.2.1 below, imply that all of the zeros ofE*

*k*(z) in the standard fundamental domain for SL2(Z) are on

*A∪ {i, ρ}*for every even integer

*k*>4.

Remark 1.2.1. Let*k*>4 be an even integer. We have

*k* (mod 12) *v*_{i/}^{√}_{3}(E*k*) *v**ρ*3(E*k*) *k* (mod 12) *v*_{i/}^{√}_{3}(E*k*) *v**ρ*3(E*k*)

0 0 0 6 1 0

2 1 2 8 0 2

4 0 1 10 1 1

### 1.3 Γ

^{∗}_{0}

### (2) (Proof of Theorem 1.1.1)

### 1.3.1 Preliminaries

We define

*F*_{k,2}* ^{∗}* (θ) :=

*e*

^{ikθ/2}*E*

_{k,2}

^{∗}³
*e*^{iθ}*/√*

2

´

*.* (1.9)

Before proving Theorem 1.1.1, we consider an expansion of *F*_{k,2}* ^{∗}* (θ).

By the definition of*E**k*(z), E_{k,2}* ^{∗}* (z) (cf. (1.1), (1.4)), we have
2(2

*+ 1)e*

^{k/2}

^{ikθ/2}*E*

_{k,2}*³*

^{∗}*e*^{iθ}*/√*
2´

= 2* ^{k/2}* X

(c,d)=1

(ce* ^{−iθ/2}*+

*√*

2de* ^{iθ/2}*)

*+ 2*

^{−k}*X*

^{k/2}(c,d)=1

(ce* ^{iθ/2}*+

*√*

2de* ^{−iθ/2}*)

^{−k}*.*

Now, (c, d) = 1 is split in two cases, namely,*c* is*odd*or *c*is*even. We consider the case in which* *c*is
*even. We have*

2* ^{k/2}* X

(c,d)=1
*c:even*

(ce* ^{−iθ/2}*+

*√*

2de* ^{iθ/2}*)

*= X*

^{−k}(c,d)=1
*d:odd*

(*√*

2c^{0}*e** ^{−iθ/2}*+

*de*

*)*

^{iθ/2}*(c= 2c*

^{−k}*)*

^{0}= X

(c,d)=1
*c:odd*

(ce* ^{iθ/2}*+

*√*

2de* ^{−iθ/2}*)

^{−k}*.*

Similarly,

2* ^{k/2}* X

(c,d)=1
*c:even*

(ce* ^{iθ/2}*+

*√*

2de* ^{−iθ/2}*)

*= X*

^{−k}(c,d)=1
*c:odd*

(ce* ^{−iθ/2}*+

*√*

2de* ^{iθ/2}*)

^{−k}*.*

Thus, we can write the following:

*F*_{k,2}* ^{∗}* (θ) =1
2

X

(c,d)=1
*c:odd*

(ce* ^{iθ/2}*+

*√*

2de* ^{−iθ/2}*)

*+1 2*

^{−k}X

(c,d)=1
*c:odd*

(ce* ^{−iθ/2}*+

*√*

2de* ^{iθ/2}*)

^{−k}*.*(1.10)

Hence, we use this expression as a definition.

In the last part of this section, we compare the two series in this expression. Note that for any pair
(c, d), (ce* ^{iθ/2}*+

*√*

2de* ^{−iθ/2}*)

*and (ce*

^{−k}*+*

^{−iθ/2}*√*

2de* ^{iθ/2}*)

*are conjugates of each other. Thus, we have the following lemma:*

^{−k}Lemma 1.3.1. *F*_{k,2}* ^{∗}* (θ)

*is real, for all*

*θ∈*[0, π].

### 1.3.2 Application of the RSD Method

We will apply the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (RSD Method) to the
Eisenstein series associated with Γ^{∗}_{0}(2). Note that *N* :=*c*^{2}+*d*^{2}.

First, we consider the case of *N* = 1. Because *c* is odd, there are two cases, (c, d) = (1,0) and
(c, d) = (−1,0). Then, we can write:

*F*_{k,2}* ^{∗}* (θ) = 2 cos(kθ/2) +

*R*

^{∗}_{2}

*,*(1.11) where

*R*

^{∗}_{2}denotes the remaining terms of the series.

Now,

*|R*^{∗}_{2}*|*6 X

(c,d)=1
*c:odd, N >1*

*|ce** ^{iθ/2}*+

*√*

2de^{−iθ/2}*|*^{−k}*.*

Let *v**Kr*(c, d, θ) := *|ce** ^{iθ/2}* +

*√*

2de^{−iθ/2}*|** ^{−k}*, then

*v*

*Kr*(c, d, θ) = 1/¡

*c*^{2}+ 2d^{2}+ 2*√*

2cdcos*θ*¢*k/2*

, and
*v**Kr*(c, d, θ) =*v**Kr*(−c,*−d, θ).*

Next, we will consider the following three cases, namely, *N* = 2,5, and *N* > 10. Considering *θ* *∈*
[π/2,3π/4], we have the following:

When*N* = 2, *v**k*(1,1, θ)61, *v**k*(1,*−1, θ)*6(1/3)^{k/2}*.*
When*N* = 5, *v**k*(1,2, θ)6(1/5)^{k/2}*,* *v**k*(1,*−2, θ)*6(1/3)^{Kr}*.*
When*N* >10,

6 *CHAPTER 1. ON THE ZEROS OF EISENSTEIN SERIES FOR*Γ_{0}(2)*AND*Γ_{0}(3)

*|ce*^{iθ/2}*±√*

2de^{−iθ/2}*|*^{2}>*c*^{2}+ 2d^{2}*−*2*√*

2|cd||cos*θ|*

>(c^{2}+*d*^{2})/N =*N/3,*

and the remaining problem concerns the number of terms with *c*^{2}+*d*^{2} =*N*. Because *c* is odd, *|c|* =
1,3, ...,2N^{0}*−*16*N*^{1/2}, so the number of*|c|*is not more than (N^{1/2}+ 1)/2. Thus, the number of terms
with*c*^{2}+*d*^{2}=*N* is not more than 2(N^{1/2}+ 1)63N^{1/2}, for*N* >5. Then,

X

(c,d)=1
*c:odd, N≥10*

*|ce** ^{iθ/2}*+

*√*

2de^{−iθ/2}*|** ^{−k}*6
X

*∞*

*N*=10

3N^{1/2}
µ*N*

3

¶_{−k/2}

6 18*√*
3
*k−*3

µ1 3

¶_{(k−3)/2}

= 162
*k−*3

µ1 3

¶_{k/2}*.*
Thus,

*|R*^{∗}_{2}*|*62 + 2
µ1

3

¶* _{k/2}*
+ 2

µ1 5

¶* _{k/2}*
+ 2

µ1 3

¶* _{Kr}*
+ 162

*k−*3
µ1

3

¶_{k/2}

*.* (1.12)

Recalling the previous section (RSD Method), we want to show that *|R*^{∗}_{2}*|<*2. However, the right-
hand side is greater than 2, so this bound is not good. The case in which (c, d) =*±(1,*1) gives a bound
equal to 2. We will consider the expansion of the method in the following sections.

### 1.3.3 Expansion of the RSD Method (1)

In the previous subsection, we could not obtain a good bound for*|R*^{∗}_{2}*|, where (c, d) =±(1,*1). Note that

“v*k*(1,1, θ) = 1 *⇔* *θ* = 3π/4”. Furthermore, “v*k*(1,1, θ) *<* 1 *⇔* *θ <* 3π/4”. Therefore, we can easily
expect that a good bound can be obtained for*θ∈*[π/2,3π/4*−x] for small* *x >*0. However, if*k*= 8n,
then we need *|R*^{∗}_{2}*|* *<* 2 for *θ* = 3π/4 in this method. In the next section, we will consider the case in
which*k*= 8n, θ= 3π/4.

Define *m*2(k) :=¥_{Kr}

8 *−*^{t}_{4}¦

, where *t* = 0,2 is chosen so that *t* *≡* *k* (mod 4), and *bnc* is the largest
integer not more than*n.*

Let*k*= 8n+*s*(n=*m*2(k), s= 4,6,0, and 10). We may assume that*k*>8.

The first step is to consider how small *x*should be. We consider each of the cases*s*= 4,6,0, and10.

When *s*= 4, for *π/2*6*θ* 63π/4, (2n+ 1)π6*kθ/2 (= (4n*+ 2)θ)6(3n+ 1)π+*π/2. So the last*
integer point (i.e. *±1) iskθ/2 = (3n*+ 1)π, then *θ*= 3π/4*−π/k. Similarly, when* *s*= 6, and 10, the
last integer points are*θ*= 3π/4*−π/2k,* 3π/4*−*3π/2k, respectively. When*s*= 0, the second to the last
integer point is*θ*= 3π/4*−π/k.*

Thus, we need*x*6*π/2k.*

Lemma 1.3.2. *Letk*>8. For all *θ∈*[π/2,3π/4*−x] (x*=*π/2k),|R*^{∗}_{2}*|<*2.

*Proof.* Let*k*>8 and*x*=*π/2k, then 0*6*x*6*π/16. If 0*6*x*6*π/16, then 1−*cos*x*>^{31}_{64}*x*^{2}.

*|e** ^{iθ/2}*+

*√*

2e^{−iθ/2}*|*^{2}>3 + 2*√*

2 cos(3π/4*−x) = 1 + 2(1−*cos*x) + 2 sinx*

>1 + 4(1*−*cos*x)*>1 + (31/16)x^{2}*.*

*|e** ^{iθ/2}*+

*√*

2e^{−iθ/2}*|** ^{k}*>¡

1 + (31/16)x^{2}¢*k/2*

>1 + (31/4)x^{2}*.* (k>8)
*v**k*(1,1, θ)6 1

1 + (31/4)x^{2} 61*−* 31*×*256
31π^{2}+ 1024*x*^{2}*.*
Thus,

2v*k*(1,1, θ)62*−*(265/9)/k^{2}*.*
Furthermore,

2 µ1

3

¶* _{k/2}*
+ 2

µ1 5

¶* _{k/2}*
+ 2

µ1 3

¶* _{Kr}*
+ 162

*k−*3
µ1

3

¶* _{k/2}*
635

µ1 3

¶_{k/2}

(k>8).

Then, we have

*|R*^{∗}_{2}*|*62*−*265
9

1
*k*^{2} + 35

µ1 3

¶*Kr*

*.*
Next, if we can show that

35 µ1

3

¶_{k/2}

*<* 265
9

1

*k*^{2} *or* 3* ^{k/2}*
35

*>*9

265*k*^{2}*,*
then the bound is less than 2. The proof will thus be complete.

Let*f*(x) := (1/35)3^{x/2}*−*_{265}^{9} *x*^{2}. Then,*f** ^{0}*(x) = (log 3/70)3

^{x/2}*−*

_{265}

^{18}

*x,f*

*(x) = ((log 3)*

^{00}^{2}

*/140)3*

^{x/2}*−*

_{265}

^{18}. First,

*f*

*is monotonically increasing for*

^{00}*x*>8, and

*f*

*(8) = 0.63038... >0, so*

^{00}*f*

^{00}*>*0 for

*x*>8. Second,

*f*

*is monotonically increasing for*

^{0}*x*>8, and

*f*

*(8) = 0.72785... > 0, so*

^{0}*f*

^{0}*>*0 for

*x*>8. Finally,

*f*is monotonically increasing for

*x*>8, and

*f*(8) = 0.14070... >0, so

*f >*0 for

*x*>8.

### 1.3.4 Expansion of the RSD Method (2)

For the case of “k= 8n, θ= 3π/4”, we need the following lemma:

Lemma 1.3.3. *Let* *kbe an integer such thatk*= 8n*for somen∈*N. If*nis even, thenF*_{k,2}* ^{∗}* (3π/4)

*>*0.

*On the other hand ifnis odd, then* *F*_{k,2}* ^{∗}* (3π/4)

*<*0.

*Proof.* Let*k*= 8n(n>1). By the definition of*E*_{k,2}* ^{∗}* (z), F

_{k,2}*(z) (cf. (1.4), (1.9)), we have*

^{∗}*F*

_{k,2}*(3π/4) =*

^{∗}*e*

^{i3(k/8)π}2* ^{k/2}*+ 1
µ

2^{k/2}*E**k*(−1 +*i) +E**k*

µ*−1 +i*
2

¶¶

*.*
By the *transformation rule*for SL2(Z),

*E**k*(−1 +*i) =E**k*(i), *E**k*((−1 +*i)/2) = (1 +i)*^{k}*E**k*(1 +*i) = 2*^{k/2}*E**k*(i).

Then,

*F*_{8n,2}* ^{∗}* (3π/4) = 2e

*2*

^{inπ}^{4n}

2^{4n}+ 1*F*8n(π/2), (1.13)

where _{2}4n^{2}^{4n}+1 *>*0,*F*8n(π/2) = 2 cos(2nπ) +*R*1*>*0. The question is then: “Which holds,*F**k*(π/2)*<*0 or
*F**k*(π/2)*>*0?”

F. K. C. Rankin and H. P. F. Swinnerton-Dyer showed (1.6) and (1.7) [RSD]. They then proved that

*|R*1*|<*2 for*k*>12. This was necessary only for*k*>12. Now we need *|R*1*|<*2 for*k*>8. The value of
the right-hand side of (1.7) at*k*= 8 is 1.29658... <2, which is monotonically decreasing in*k. Thus, we*
can show

*|R*1*|<*2 for all *k*>8. (1.14)

Then, the sign (±) of*F*_{k,2}* ^{∗}* (3π/4) is that of

*e*

*. Thus, the proof is complete.*

^{inπ}Next, we proved that*E*_{k,2}* ^{∗}* (z) has

*m*2(k) zeros on the arc

*A*

^{∗}_{2}. In order to determine the location of all of the zeros of

*E*

_{k,2}*(z) inF*

^{∗}*(2), we need the valence formula for Γ*

^{∗}

^{∗}_{0}(2):

Proposition 1.3.4. *Letf* *be a modular function of weightkfor*Γ^{∗}_{0}(2), which is not identically zero. We
*have*

*v** _{∞}*(f) +1

2*v*_{i/}^{√}_{2}(f) +1

4*v*_{ρ}_{2}(f) + X

*p∈Γ*^{∗}_{0}(2)\H
*p6=i/**√*

2, ρ2

*v** _{p}*(f) =

*Kr*

8 *,* (1.15)

*whereρ*2:=*e** ^{i(3π/4)}*±√

2.

The proof of this proposition is similar to that for Proposition 1.2.1 (See [Se]).

If *k* *≡* 4,6, and 0 (mod 8), then *k/8−m*2(k) *<* 1. Thus, all of the zeros of *E*^{∗}* _{k,2}*(z) in F

*(2) are on the arc*

^{∗}*A*

^{∗}_{2}. On the other hand, if

*k*

*≡*2 (mod 8), then we have

*E*

^{∗}*(i/*

_{k,2}*√*

2) = *i*^{k}*E*_{k,2}* ^{∗}* (i/

*√*

2) by the
transformation rule for Γ^{∗}_{0}(2). Then, we have*k/8−m*2(k)*−v*_{i/}^{√}_{2}(E_{k,2}* ^{∗}* )/2

*<*1.

In conclusion, for every even integer*k*>4, all of the zeros of*E*_{k,2}* ^{∗}* (z) in F

*(2) are on the arc*

^{∗}*A*

^{∗}_{2}.

8 *CHAPTER 1. ON THE ZEROS OF EISENSTEIN SERIES FOR*Γ_{0}(2)*AND*Γ_{0}(3)
Remark 1.3.1. (See Proposition 1.A.2) Let*k*>4 be an even integer. We have

*k* (mod 8) *v*_{i/}^{√}_{2}(E_{k,2}* ^{∗}* )

*v*

*ρ*2(E

^{∗}*)*

_{k,2}*k*(mod 8)

*v*

_{i/}

^{√}_{2}(E

^{∗}*)*

_{k,2}*v*

*ρ*2(E

_{k,2}*)*

^{∗}0 0 0 4 0 2

2 1 3 6 1 1

### 1.4 Γ

^{∗}_{0}

### (3) (Proof of Theorem 1.1.2)

### 1.4.1 Preliminaries

We define

*F*_{k,3}* ^{∗}* (θ) :=

*e*

^{ikθ/2}*E*

_{k,3}*³*

^{∗}*e*

^{iθ}*/√*

3´

*.* (1.16)

Similar to the case of *F*_{k,2}* ^{∗}* (θ), we can write the following:

*F*_{k,3}* ^{∗}* (θ) =1
2

X

(c,d)=1 3-c

(ce* ^{iθ/2}*+

*√*

3de* ^{−iθ/2}*)

*+1 2*

^{−k}X

(c,d)=1 3-c

(ce* ^{−iθ/2}*+

*√*

3de* ^{iθ/2}*)

^{−k}*.*(1.17)

The following lemma is then obtained:

Lemma 1.4.1. *F*_{k,3}* ^{∗}* (θ)

*is real, for all*

*θ∈*[0, π].

### 1.4.2 Application of the RSD Method

Note that*N* :=*c*^{2}+*d*^{2}.

First, we consider the case of *N*= 1. We can then write the following:

*F*_{k,3}* ^{∗}* (θ) = 2 cos(kθ/2) +

*R*

^{∗}_{3}

*,*(1.18) where

*R*

^{∗}_{3}denotes the remaining terms.

Let *v**Kr*(c, d, θ) :=*|ce** ^{iθ/2}*+

*√*

3de^{−iθ/2}*|** ^{−k}*. We will consider the following cases:

*N*= 2,5,10,13,17, and

*N*>25. Considering

*θ∈*[π/2,5π/6], we have the following:

When*N*= 2, *v**k*(1,1, θ)61, *v**k*(1,*−1, θ)*6(1/2)^{Kr}*.*
When*N*= 5, *v**k*(1,2, θ)6(1/7)^{k/2}*,* *v**k*(1,*−2, θ)*6(1/13)^{k/2}*,*

*v**k*(2,1, θ)61, *v**k*(2,*−1, θ)*6(1/7)^{k/2}*.*
When*N*= 10, *v**k*(1,3, θ)6(1/19)^{k/2}*,* *v**k*(1,*−3, θ)*6(1/28)^{k/2}*.*
When*N*= 13, *v**k*(2,3, θ)6(1/13)^{k/2}*,* *v**k*(2,*−3, θ)*6(1/31)^{k/2}*.*
When*N*= 17, *v**k*(1,4, θ)6(1/37)^{k/2}*,* *v**k*(1,*−4, θ)*6(1/7)^{Kr}*,*

*v**k*(4,1, θ)6(1/7)^{k/2}*,* *v**k*(4,*−1, θ)*6(1/19)^{k/2}*.*
When*N*>25, *|ce*^{iθ/2}*±√*

3de^{−iθ/2}*|*^{2}>N/6,

and the number of terms with*c*^{2}+*d*^{2}=*N* is at most (11/3)N^{1/2}, for*N* >16. Then,
X

(c,d)=1 3-c, N≥25

*|ce** ^{iθ/2}*+

*√*

3de^{−iθ/2}*|** ^{−k}* 6
X

*∞*

*N*=25

11
3 *N*^{1/2}

µ1
6*N*

¶_{−k/2}

6352*√*

6
*k−*3

µ1 2

¶_{Kr}*.*

Thus,

*|R*^{∗}_{3}*|*64 + 2
µ1

2

¶* _{Kr}*
+ 6

µ1 7

¶_{k/2}

*· · ·*+ 2
µ1

7

¶_{Kr}

+352*√*
6
*k−*3

µ1 2

¶_{Kr}

*.* (1.19)

The cases of (c, d) =*±(1,*1), *±(2,*1) give a bound equal to 4. We will consider an expansion of the
method similar to that of Γ^{∗}_{0}(2).

### 1.4.3 Expansion of the RSD Method (1)

Similar to the method of Γ^{∗}_{0}(2), we will consider*θ∈*[π/2,5π/6−x] for small*x >*0. In the next subsection,
we also consider the case in which*k*= 12n, θ= 5π/6.

Define *m*3(k) :=¥_{Kr}

6 *−*^{t}_{4}¦

, where *t* = 0,2 is chosen so that *t* *≡k* (mod 4). We may assume that
*k*>8.

How small should *x*be? Let*k*= 12m_{3}(k) +*s. Considering each case, namely,s*= 4,6,8,10*and*14,
we need*x*6*π/3k.*

Lemma 1.4.2. *Letk*>8. For all *θ∈*[π/2,5π/6*−x] (x*=*π/3k),|R*^{∗}_{3}*|<*2.

Before proving the above lemma, we need the following preliminaries.

Proposition 1.4.3. *Let* *k*>8 *be an even integer and* *x*=*π/3k, then*
4 + 2*√*

3 cos µ5π

6 *−x*

¶

>

µ3 2

¶_{2/k}µ

1 +256*×*7*×*13
3*×*127*×k* *x*^{2}

¶
*.*

*Proof.* We have,
µ3

2

¶2/k

=
X*∞*
*n=0*

(2 log 3/2)^{n}*n!*

1
*k** ^{n}* 61 +

µ 2 log3

2

¶ 1
*Kr* +1

2 µ

2 log3 2

¶2µ 3 2

¶2/k

1
*k*^{2}*,*
3 + 2*√*

3 cos µ5π

6 *−* *π*
3k

¶

> *π*

*√*3
1
*Kr.*
Let*x*=*π/3k. Then, we have*

*f*1(k) := 4 + 2*√*
3 cos

µ5π
6 *−* *π*

3k

¶

*−*
µ3

2

¶_{2/k}µ

1 +256*×*7*×*13*×π*^{2}
27*×*127

1
*k*^{3}

¶
*.*
If*k*= 8, then *f*1(8) = 0.00012876... >0. Next, if*k*>10, then

*f*1(k)> 1
*Kr*

(

*√π*

3 *−*2 log3
2 *−*1

2 µ

2 log3 2

¶_{2}µ
3
2

¶_{2/k}
1

*Kr* *−*256*×*7*×*13*×π*^{2}
27*×*127

µ3 2

¶_{2/k}
1
*k*^{2}

)

> 1

*Kr* *×*0.24004... (k>10) *>*0.

Proposition 1.4.4. *Let* *k*>8 *be an even integer and* *x*=*π/3k, then*
7 + 4*√*

3 cos µ5π

6 *−x*

¶

>3^{2/k}
µ

1 + 256*×*7*×*13
3*×*127*×k* *x*^{2}

¶
*.*

*Proof.* We have

3^{2/k} 61 + (2 log 3) 1
*Kr* +1

2(2 log 3)^{2}3^{2/k} 1
*k*^{2}*,*
6 + 4*√*

3 cos µ5π

6 *−* *π*
3k

¶

> 2π

*√*3
1
*Kr.*
Similar to the proof of Proposition 1.4.3, let*x*=*π/3k, and write*

*f*2(k) := 7 + 4*√*
3 cos

µ5π
6 *−* *π*

3k

¶

*−*3^{2/k}
µ

1 + 256*×*7*×*13*×π*^{2}
27*×*127

1
*k*^{3}

¶
*.*
If*k*= 8, then *f*2(8) = 0.015057... >0. Next, if*k*>10, then

*f*2(k)> 1

*Kr* *×*0.29437... *>*0.

10 *CHAPTER 1. ON THE ZEROS OF EISENSTEIN SERIES FOR*Γ_{0}(2)*AND*Γ_{0}(3)
*Proof of Lemma 1.4.2.* Let*k*>8 and*x*=*π/3k, then 0*6*x*6*π/24.*

By Proposition 1.4.3

*|e** ^{iθ/2}*+

*√*

3e^{−iθ/2}*|*^{2}>

µ3 2

¶_{2/k}µ

1 +256*×*7*×*13
3*×*127*×kx*^{2}

¶
*.*

*|e** ^{iθ/2}*+

*√*

3e^{−iθ/2}*|** ^{k}* >

µ3 2

¶ µ

1 +256*×*7*×*13
3*×*127*×k* *x*^{2}

¶_{k/2}

>3

2 +64*×*7*×*13

127 *x*^{2}*.* (k>8)

*v**k*(1,1, θ)6 2

3*−* (128*×*7*×*13/127)

(9/2) + (64*×*3*×*7*×*13/127)x^{2}*x*^{2}6 2
3*−*107

8 *x*^{2}*.* (x6*π/24)*
Similarly, by Proposition 1.4.4

*|2e** ^{iθ/2}*+

*√*

3e^{−iθ/2}*|*^{2}>3^{2/k}
µ

1 +256*×*7*×*13
3*×*127*×kx*^{2}

¶
*.*

*|2e** ^{iθ/2}*+

*√*

3e^{−iθ/2}*|** ^{k}* >3 + 128

*×*7

*×*13 127

*x*

^{2}

*.*

*v*

*k*(2,1, θ)6 1

3*−*107
16*x*^{2}*.*
Thus,

2v*k*(1,1, θ) + 2v*k*(2,1, θ)62*−*107π^{2}
24

1
*k*^{2}*.*
In Eq. (1.19), replace 4 with the bound 2*−*^{107π}_{24}^{2}_{k}^{1}2. Then,

*|R*^{∗}_{3}*|*62*−*107π^{2}
24

1
*k*^{2} + 176

µ1 2

¶_{Kr}*.*

Similarly to the method for Γ^{∗}_{0}(2), we can easily show that the bound is less than two for*k*>8.

### 1.4.4 Expansion of the RSD Method (2)

For the case “k= 12n, θ= 5π/6”, we need the following lemma:

Lemma 1.4.5. *Letkbe the integer such thatk*= 12n*for somen∈*N. If*nis even, thenF*_{k,3}* ^{∗}* (5π/6)

*>*0.

*On the other hand, if* *nis odd, thenF*_{k,3}* ^{∗}* (5π/6)

*<*0.

*Proof.* Let*k*= 12n(n>1). Similarly to (1.13),

*F*_{12n,3}* ^{∗}* (5π/6) = 2e

*3*

^{inπ}^{6n}

3^{6n}+ 1*F*12n(2π/3), (1.20)

where _{3}6n^{3}^{6n}+1 *>*0,*F*12n(2π/3) = 2 cos(4nπ) +*R*1*>*0 (cf. (1.14)).

Proposition 1.4.6 (Valence formula for Γ^{∗}_{0}(3)). *Let* *f* *be a modular function of weight* *kfor* Γ^{∗}_{0}(3),
*which is not identically zero. We have*

*v**∞*(f) +1

2*v*_{i/}^{√}_{3}(f) +1

6*v**ρ*3(f) + X

*p∈Γ*^{∗}_{0}(3)\H
*p6=i/**√*

3, ρ3

*v**p*(f) = *Kr*

6 *,* (1.21)

*whereρ*3:=*e** ^{i(5π/6)}*±√

3.