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

九州大学学術情報リポジトリ

Kyushu University Institutional Repository

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.

Letk>4 be an even integer. Forz∈H:={z∈C; Im(z)>0}, let Ek(z) := 1

2 X

(c,d)=1

(cz+d)^−k be theEisenstein seriesassociated with SL2(Z). Moreover, let

F:={|z|>1, −1/26Re(z)60} ∪ {|z|>1, 06Re(z)<1/2}

be thestandard fundamental domainfor SL2(Z).

F. K. C. Rankin and H. P. F. Swinnerton-Dyer considered the problem of locating the zeros ofEk(z) in F[RSD]. They proved thatn zeros are on the arcA:={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 primep, we have the following:

Γ^∗₀(p) := Γ0(p)∪Γ0(p)Wp, (1)

where

Γ0(p) =©¡_{a b}

c d

¢∈SL2(Z) ; c≡0 (modp)ª

, Wp:=

³ 0 −1/√

√ p

p 0

´ .

Then, all of the zeros of the Eisenstein seriesE_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 ofp= 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/√

por 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 domainF, 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 : jΓ(z)∈R}.

However, based on the numerical calculations inChapter 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 ofE_2k^Γ in F can be proved to lie on its lower arcs (Chapter 4). However, if Γ does not satisfy the assumption (without anyacceptablefundamental domain), then the zeros ofE_2k^Γ do not always lie on{z∈H : jΓ(z)∈R}(Chapter 3).

In addition, T. Asai, M. Kaneko, and H. Ninomiya considered the problem of locating the zeros of modular functions Fm(z) for SL2(Z), which correspond to the Hecke type Faber polynomial, Pm, 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 theargu- menton 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 thereal parton the lower arc ofF.

Next, the Poincar´e seriesGk(z;t^−m) is considered. Note thatGk(z;t⁰) =Ek(z) andG0(z;t^−m) =Fm(z).

Furthermore, ifk is sufficiently large compared withm, then the distribution of zeros of Gk(z;t^−m) re- sembles that ofEk(z). On the other hand, ifmis sufficiently large compared withk, then the distribution resembles that ofFm(z). Thus, the Poincar´e seriesGk(z;t^−m) “fills the space of two modular functions discretely” (Chapter 5).

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.

LetS^d−1:={(x1, . . . , xd)∈R^d;x²₁+· · ·+x²_d= 1} be the Euclidean sphere ford>1.

Definition(Spherical design [DGS]) LetX be a non-empty finite set on the Euclidean sphereS^d−1, and lettbe a positive integer. X is called a sphericalt-design if

1

|S^d−1| Z

S^d−1

f(ξ)dξ = 1

|X|

X

ξ∈X

f(ξ) (2)

for every polynomialf(x) =f(x1, . . . , xd) of degree at mostt.

Here, the left-hand side of the above equation (2) represents the average on the sphereS^d−1, and the right-hand side represents the average on the finite subsetX. Thus, ifX is a spherical design, then X gives a certain approximation of the sphereS^d−1.

LetLbe a Euclidean lattice, which is a discrete freeZ-module. The squared norm of a vector of the lattice is called thenorm of the vector. Then, the setsm(L) of vectors of the latticeLthat take the same valuemfor their norm is called theshell of the lattice, i.e.,sm(L) :={x∈L; (x, x) =m}. Moreover, the shell ofminimummin_x∈L\{0}(x, x) of the latticeLis called theminimal shell, which is denoted byS(L).

If an integral lattice L is generated by the vectors ofsq(L), thenL is referred to as aq-lattice. For every nonempty shell sm(L) of lattice L, normalization ^√1_msm(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 bestrongly 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 A₁,A₂,D₄,E₆,E₇, orE₈. 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 areO1,O7,O16,O22, andO23. Furthermore, the minimal shell is a 7-design only for the case of lattice O23.

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.

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^k/2+ 1

³

p^k/2Ek(pz) +Ek(z)

´

be the Eisenstein series associated with Γ^∗₀(p). The region F^∗(p) :={|z|>1/√

p,−1/26Re(z)60}[

{|z|>1/√

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

is a fundamental domain for Γ^∗₀(p) whenp= 2 or 3. DefineA^∗_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)inF^∗(2) are on the arcA^∗₂. Theorem 2. Let k>4 be an even integer. All of the zeros ofE_k,3^∗ (z)inF^∗(3) are on the arcA^∗₃.

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

Chapter 2.

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

Henceforth, we assume thatp= 5 or 7. The region F^∗(p) :={|z|>1/√

p, |z|>1/(2√

p), −1/26Re(z)60}

[{|z|>1/√

p,|z|>1/(2√

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

is a fundamental domain for Γ^∗₀(p). DefineA^∗_p :=F^∗(p)∩ {z∈C; |z|= 1/√

por|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>4be an even integer. Then all of the zeros ofE_k,5^∗ (z)inF^∗(5)lie on the arcA^∗₅. Conjecture 4. Letk>4be an even integer. Then all of the zeros ofE_k,7^∗ (z)inF^∗(7)lie on the arcA^∗₇. First, we prove that all but at most 2 zeros ofE_k,p^∗ (z) inF^∗(p) lie on the arcA^∗_p. Second, if (24/(p+1))| k, we prove that all of the zeros ofE_k,p^∗ (z) inF^∗(p) lie onA^∗_p. We can then prove that if (24/(p+ 1))-k, all but one of the zeros ofE_k,p^∗ (z) inF^∗(p) lie onA^∗_p. Furthermore, for most of the other cases, we prove Conjectures 3 and 4 (see Section 2.1 for details).

This result was published in the Kyushu Journal of Mathematics [SJ2].

Chapter 3.

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= 0ª .

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. LetΓbe a genus zero Fuchsian group of the first kind with∞as a cusp. If the hauptmodul JΓ takes a real value on∂FΓ, then all of the zeros of the Eisenstein series for the cusp∞forΓ inFΓ lie on the arcs

∂FΓ\ {z∈H; Re(z) =±h/2}.

Conjecture 6. LetΓbe a genus zero Fuchsian group of the first kind with∞as a cusp. If the hauptmodul JΓ takes a real value on ∂FΓ, then all but at most ch(Γ) of the zeros of the modular function from the Hecke type Faber polynomial of degree m forΓ inFΓ lie on the arcs ∂FΓ\ {z ∈H; Re(z) =±h/2} for all but a finite number ofmand for the constant number ch(Γ), which does not depend onm.

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 degreem6200 by numerical calculations.

Chapter 4.

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 ofE_2k^Γ are located.

We denote by Γ a Fuchsian group of the first kind, which has∞as a cusp with widthh. LetF be a fundamental domain of Γ contained in{z : −h/26<(z)< h/2}, and letA be the lower arcs ofF. We then definey0:= inf{y : ±h/2 +i y∈∂F}anda0:=jΓ(−h/2 +i y0). We denote byc(Γ,F) the number of equivalence classes under the action of Γ on the set of critical points ofF at which _(j ^y0(t)

Γ◦zA)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 jΓ has real Fourier coefficients. Then, all but possibly c(Γ,F) of the zeros of P(E^Γ_2k, X) lie on[a0,∞), where a0 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 polynomialof a modular formf (see [H, Section 3]).

We denote bys¹_2k(Γ) the number of cusps other than 0 and−1/2 at whichE_2k^Γ has odd multiplicity.

Then, sinceE_2k^Γ vanishes at all of the cusps other than∞, we can substitutec(Γ,F) forc(Γ,F)−s¹_2k(Γ) in the above theorem. Similarly toy0anda0, we definey1:= inf{y : i y∈∂F}anda1:=jΓ(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 jΓ has real Fourier coefficients. Then, all but at most c(Γ,F)−s¹_2k(Γ) of the zeros of P(E^Γ_2k, X)lie on (−∞, a1]. Moreover, ifm denotes the number of distinct zeros with odd multiplicity on (−∞, a1], thenm+c(Γ,F)−s¹_2k(Γ)>deg(P(E_2k^Γ, X)).

Chapter 5.

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(e2πiγ(−1/(pz))) (d√

pz−c/√ p)^k

be a Poincar´e series associated with Γ^∗₀(p), whereR(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 weightk is denoted byl. 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 m6l 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.

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.

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. Z⁷, the minimum of which is equal to one.

2. Λ16,2,1,Λ16,2,2, andΛ16,2,3, the minima of which are equal to two.

3. O1,O7,O16,O22, andO23, 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,On3-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 forΓ^∗₀(5)andΓ^∗₀(7), Kyushu. J. Math. 61(2007), 527–549.

[SJ7] J. Shigezumi, Classification of3-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

Letk>4 be an even integer. Forz∈H:={z∈C; Im(z)>0}, let Ek(z) := 1

2 X

(c,d)=1

(cz+d)^−k (1.1)

be theEisenstein seriesassociated with SL2(Z). Moreover, let

F:={|z|>1, −1/26Re(z)60} ∪ {|z|>1, 06Re(z)<1/2}

be thestandaard fundamental domainfor SL2(Z).

F. K. C. Rankin and H. P. F. Swinnerton-Dyer considered the problem of locating the zeros ofEk(z) in F[RSD]. They proved thatn zeros are on the arcA:={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 primep, we define the following:

Γ^∗₀(p) := Γ0(p)∪Γ0(p)Wp, (1.2) where

Γ0(p) :=©¡_{a b}

c d

¢∈SL2(Z) ; c≡0 (modp)ª

, Wp:=³

0 −1/√

√ p

p 0

´

. (1.3)

Let

E_k,p^∗ (z) := 1 p^k/2+ 1

³

p^k/2E_k(pz) +E_k(z)´

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

F^∗(2) :=n

|z|>1/√

2, −1/26Re(z)60o [ n

|z|>1/√

2, 06Re(z)<1/2o , F^∗(3) :=n

|z|>1/√

3, −1/26Re(z)60o [ n

|z|>1/√

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

4 CHAPTER 1. ON THE ZEROS OF EISENSTEIN SERIES FORΓ₀(2)ANDΓ₀(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 haveA^∗₂=A^∗₂∪ {i/√

2, e^i(3π/4)/√

2}andA^∗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>4be an even integer. All of the zeros of E_k,2^∗ (z) inF^∗(2)are on the arcA^∗₂. Theorem 1.1.2. Letk>4be an even integer. All of the zeros of E_k,3^∗ (z) inF^∗(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:

Fk(θ) :=e^ikθ/2Ek

¡e^iθ¢

, (1.5)

which is real for allθ∈[0, π]. Considering the four terms withc²+d²= 1, they proved that

Fk(θ) = 2 cos(kθ/2) +R1, (1.6)

whereR1is the rest of the series (i.e. c²+d²>1). Moreover they showed

|R1|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) atk= 12 to be approximately 1.03562, which is monotonically decreasing ink. Thus, they could show that|R1|<2 for allk>12. If cos(kθ/2) is +1 or

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

In order to determine the location of all of the zeros ofEk(z) inF, we need thevalence formula:

Proposition 1.2.1 (valence formula). Let f be a modular function of weight k forSL2(Z), which is not identically zero. We have

v∞(f) +1

2vi(f) +1

3vρ(f) + X

p∈SL2(Z)\H p6=i, ρ

vp(f) = Kr

12, (1.8)

wherevp(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 thatE_k(z) hasm(k) zeros on the arcA, the valence formula, and Remark 1.2.1 below, imply that all of the zeros ofEk(z) in the standard fundamental domain for SL2(Z) are onA∪ {i, ρ}for every even integerk>4.

Remark 1.2.1. Letk>4 be an even integer. We have

k (mod 12) v_i/^√₃(Ek) vρ3(Ek) k (mod 12) v_i/^√₃(Ek) vρ3(Ek)

0 0 0 6 1 0

2 1 2 8 0 2

4 0 1 10 1 1

1.3 Γ

(2) (Proof of Theorem 1.1.1)

1.3.1 Preliminaries

We define

F_k,2^∗ (θ) :=e^ikθ/2E_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 ofEk(z), E_k,2^∗ (z) (cf. (1.1), (1.4)), we have 2(2^k/2+ 1)e^ikθ/2E_k,2^∗ ³

e^iθ/√ 2´

= 2^k/2 X

(c,d)=1

(ce^−iθ/2+√

2de^iθ/2)^−k+ 2^k/2 X

(c,d)=1

(ce^iθ/2+√

2de^−iθ/2)^−k.

Now, (c, d) = 1 is split in two cases, namely,c isoddor ciseven. We consider the case in which cis even. We have

2^k/2 X

(c,d)=1 c:even

(ce^−iθ/2+√

2de^iθ/2)^−k= X

(c,d)=1 d:odd

(√

2c⁰e^−iθ/2+de^iθ/2)^−k (c= 2c⁰)

= 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)^−k= X

(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)^−k+1 2

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)^−k and (ce^−iθ/2+√

2de^iθ/2)^−k are conjugates of each other. Thus, we have the following lemma:

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 :=c²+d².

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) whereR^∗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 vKr(c, d, θ) := |ce^iθ/2 +√

2de^−iθ/2|^−k, then vKr(c, d, θ) = 1/¡

c²+ 2d²+ 2√

2cdcosθ¢k/2

, and vKr(c, d, θ) =vKr(−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:

WhenN = 2, vk(1,1, θ)61, vk(1,−1, θ)6(1/3)^k/2. WhenN = 5, vk(1,2, θ)6(1/5)^k/2, vk(1,−2, θ)6(1/3)^Kr. WhenN >10,

6 CHAPTER 1. ON THE ZEROS OF EISENSTEIN SERIES FORΓ₀(2)ANDΓ₀(3)

|ce^iθ/2±√

2de^−iθ/2|²>c²+ 2d²−2√

2|cd||cosθ|

>(c²+d²)/N =N/3,

and the remaining problem concerns the number of terms with c²+d² =N. Because c is odd, |c| = 1,3, ...,2N⁰−16N^1/2, so the number of|c|is not more than (N^1/2+ 1)/2. Thus, the number of terms withc²+d²=N is not more than 2(N^1/2+ 1)63N^1/2, forN >5. Then,

X

(c,d)=1 c:odd, N≥10

|ce^iθ/2+√

2de^−iθ/2|^−k6 X∞ N=10

3N^1/2 µN

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

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

Define m2(k) :=¥_Kr

8 −^t₄¦

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

Letk= 8n+s(n=m2(k), s= 4,6,0, and 10). We may assume thatk>8.

The first step is to consider how small xshould be. We consider each of the casess= 4,6,0, and10.

When s= 4, for π/26θ 63π/4, (2n+ 1)π6kθ/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 areθ= 3π/4−π/2k, 3π/4−3π/2k, respectively. Whens= 0, the second to the last integer point isθ= 3π/4−π/k.

Thus, we needx6π/2k.

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

Proof. Letk>8 andx=π/2k, then 06x6π/16. If 06x6π/16, then 1−cosx>³¹₆₄x².

|e^iθ/2+√

2e^−iθ/2|²>3 + 2√

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

>1 + 4(1−cosx)>1 + (31/16)x².

|e^iθ/2+√

2e^−iθ/2|^k>¡

1 + (31/16)x²¢k/2

>1 + (31/4)x². (k>8) vk(1,1, θ)6 1

1 + (31/4)x² 61− 31×256 31π²+ 1024x². Thus,

2vk(1,1, θ)62−(265/9)/k². 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 k² + 35

µ1 3

¶Kr

. Next, if we can show that

35 µ1

3

¶_k/2

< 265 9

1

k² or 3^k/2 35 > 9

265k², then the bound is less than 2. The proof will thus be complete.

Letf(x) := (1/35)3^x/2−₂₆₅⁹ x². Then,f⁰(x) = (log 3/70)3^x/2−₂₆₅¹⁸x,f⁰⁰(x) = ((log 3)²/140)3^x/2−₂₆₅¹⁸. First,f⁰⁰ is monotonically increasing forx>8, andf⁰⁰(8) = 0.63038... >0, sof⁰⁰>0 forx>8. Second, f⁰ is monotonically increasing forx>8, andf⁰(8) = 0.72785... > 0, sof⁰ >0 forx>8. Finally, f is monotonically increasing forx>8, andf(8) = 0.14070... >0, sof >0 forx>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= 8nfor somen∈N. Ifnis even, thenF_k,2^∗ (3π/4)>0.

On the other hand ifnis odd, then F_k,2^∗ (3π/4)<0.

Proof. Letk= 8n(n>1). By the definition ofE_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/2Ek(−1 +i) +Ek

µ−1 +i 2

¶¶

. By the transformation rulefor SL2(Z),

Ek(−1 +i) =Ek(i), Ek((−1 +i)/2) = (1 +i)^kEk(1 +i) = 2^k/2Ek(i).

Then,

F_8n,2^∗ (3π/4) = 2e^inπ 2⁴ⁿ

2⁴ⁿ+ 1F8n(π/2), (1.13)

where ₂4n²⁴ⁿ+1 >0,F8n(π/2) = 2 cos(2nπ) +R1>0. The question is then: “Which holds,Fk(π/2)<0 or Fk(π/2)>0?”

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

|R1|<2 fork>12. This was necessary only fork>12. Now we need |R1|<2 fork>8. The value of the right-hand side of (1.7) atk= 8 is 1.29658... <2, which is monotonically decreasing ink. Thus, we can show

|R1|<2 for all k>8. (1.14)

Then, the sign (±) ofF_k,2^∗ (3π/4) is that ofe^inπ. Thus, the proof is complete.

Next, we proved thatE_k,2^∗ (z) hasm2(k) zeros on the arcA^∗₂. In order to determine the location of all of the zeros ofE_k,2^∗ (z) inF^∗(2), we need the valence formula for Γ^∗₀(2):

Proposition 1.3.4. Letf be a modular function of weightkforΓ^∗₀(2), which is not identically zero. We have

v_∞(f) +1

2v_i/^√₂(f) +1

4v_ρ2(f) + X

p∈Γ^∗₀(2)\H p6=i/√

2, ρ2

v_p(f) = Kr

8 , (1.15)

whereρ2:=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−m2(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, ifk ≡2 (mod 8), then we have E^∗_k,2(i/√

2) = i^kE_k,2^∗ (i/√

2) by the transformation rule for Γ^∗₀(2). Then, we havek/8−m2(k)−v_i/^√₂(E_k,2^∗ )/2<1.

In conclusion, for every even integerk>4, all of the zeros ofE_k,2^∗ (z) in F^∗(2) are on the arcA^∗₂.

8 CHAPTER 1. ON THE ZEROS OF EISENSTEIN SERIES FORΓ₀(2)ANDΓ₀(3) Remark 1.3.1. (See Proposition 1.A.2) Letk>4 be an even integer. We have

k (mod 8) v_i/^√₂(E_k,2^∗ ) vρ2(E^∗_k,2) k (mod 8) v_i/^√₂(E^∗_k,2) vρ2(E_k,2^∗ )

0 0 0 4 0 2

2 1 3 6 1 1

1.4 Γ

(3) (Proof of Theorem 1.1.2)

1.4.1 Preliminaries

We define

F_k,3^∗ (θ) :=e^ikθ/2E_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)^−k+1 2

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 thatN :=c²+d².

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) whereR^∗₃ denotes the remaining terms.

Let vKr(c, d, θ) :=|ce^iθ/2+√

3de^−iθ/2|^−k. We will consider the following cases: N = 2,5,10,13,17, andN >25. Consideringθ∈[π/2,5π/6], we have the following:

WhenN= 2, vk(1,1, θ)61, vk(1,−1, θ)6(1/2)^Kr. WhenN= 5, vk(1,2, θ)6(1/7)^k/2, vk(1,−2, θ)6(1/13)^k/2,

vk(2,1, θ)61, vk(2,−1, θ)6(1/7)^k/2. WhenN= 10, vk(1,3, θ)6(1/19)^k/2, vk(1,−3, θ)6(1/28)^k/2. WhenN= 13, vk(2,3, θ)6(1/13)^k/2, vk(2,−3, θ)6(1/31)^k/2. WhenN= 17, vk(1,4, θ)6(1/37)^k/2, vk(1,−4, θ)6(1/7)^Kr,

vk(4,1, θ)6(1/7)^k/2, vk(4,−1, θ)6(1/19)^k/2. WhenN>25, |ce^iθ/2±√

3de^−iθ/2|²>N/6,

and the number of terms withc²+d²=N is at most (11/3)N^1/2, forN >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 6N

¶_−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 considerθ∈[π/2,5π/6−x] for smallx >0. In the next subsection, we also consider the case in whichk= 12n, θ= 5π/6.

Define m3(k) :=¥_Kr

6 −^t₄¦

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

How small should xbe? Letk= 12m₃(k) +s. Considering each case, namely,s= 4,6,8,10and14, we needx6π/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/kµ

1 +256×7×13 3×127×k x²

¶ .

Proof. We have, µ3

2

¶2/k

= X∞ n=0

(2 log 3/2)ⁿ n!

1 kⁿ 61 +

µ 2 log3

2

¶ 1 Kr +1

2 µ

2 log3 2

¶2µ 3 2

¶2/k

1 k², 3 + 2√

3 cos µ5π

6 − π 3k

¶

> π

√3 1 Kr. Letx=π/3k. Then, we have

f1(k) := 4 + 2√ 3 cos

µ5π 6 − π

3k

¶

− µ3

2

¶_2/kµ

1 +256×7×13×π² 27×127

1 k³

¶ . Ifk= 8, then f1(8) = 0.00012876... >0. Next, ifk>10, then

f1(k)> 1 Kr

(

√π

3 −2 log3 2 −1

2 µ

2 log3 2

¶₂µ 3 2

¶_2/k 1

Kr −256×7×13×π² 27×127

µ3 2

¶_2/k 1 k²

)

> 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 x²

¶ .

Proof. We have

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

2(2 log 3)²3^2/k 1 k², 6 + 4√

3 cos µ5π

6 − π 3k

¶

> 2π

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

f2(k) := 7 + 4√ 3 cos

µ5π 6 − π

3k

¶

−3^2/k µ

1 + 256×7×13×π² 27×127

1 k³

¶ . Ifk= 8, then f2(8) = 0.015057... >0. Next, ifk>10, then

f2(k)> 1

Kr ×0.29437... >0.

10 CHAPTER 1. ON THE ZEROS OF EISENSTEIN SERIES FORΓ₀(2)ANDΓ₀(3) Proof of Lemma 1.4.2. Letk>8 andx=π/3k, then 06x6π/24.

By Proposition 1.4.3

|e^iθ/2+√

3e^−iθ/2|²>

µ3 2

¶_2/kµ

1 +256×7×13 3×127×kx²

¶ .

|e^iθ/2+√

3e^−iθ/2|^k >

µ3 2

¶ µ

1 +256×7×13 3×127×k x²

¶_k/2

>3

2 +64×7×13

127 x². (k>8)

vk(1,1, θ)6 2

3− (128×7×13/127)

(9/2) + (64×3×7×13/127)x²x²6 2 3−107

8 x². (x6π/24) Similarly, by Proposition 1.4.4

|2e^iθ/2+√

3e^−iθ/2|²>3^2/k µ

1 +256×7×13 3×127×kx²

¶ .

|2e^iθ/2+√

3e^−iθ/2|^k >3 + 128×7×13 127 x². vk(2,1, θ)6 1

3−107 16x². Thus,

2vk(1,1, θ) + 2vk(2,1, θ)62−107π² 24

1 k². In Eq. (1.19), replace 4 with the bound 2−^107π₂₄²_k¹2. Then,

|R^∗₃|62−107π² 24

1 k² + 176

µ1 2

¶_Kr .

Similarly to the method for Γ^∗₀(2), we can easily show that the bound is less than two fork>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= 12nfor somen∈N. Ifnis even, thenF_k,3^∗ (5π/6)>0.

On the other hand, if nis odd, thenF_k,3^∗ (5π/6)<0.

Proof. Letk= 12n(n>1). Similarly to (1.13),

F_12n,3^∗ (5π/6) = 2e^inπ 3⁶ⁿ

3⁶ⁿ+ 1F12n(2π/3), (1.20)

where ₃6n³⁶ⁿ+1 >0,F12n(2π/3) = 2 cos(4nπ) +R1>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

2v_i/^√₃(f) +1

6vρ3(f) + X

p∈Γ^∗₀(3)\H p6=i/√

3, ρ3

vp(f) = Kr

6 , (1.21)

whereρ3:=e^i(5π/6)±√

3.