ZEROS OF REPLICABLE FUNCTIONS BEN TOOMEY Abstract. Following the work of Asai Kaneko and Ninomiya for Faber polynomials associated

2025-05-01 0 0 1.02MB 49 页 10玖币

侵权投诉

ZEROS OF REPLICABLE FUNCTIONS

BEN TOOMEY

Abstract. Following the work of Asai, Kaneko, and Ninomiya for Faber polynomials associated

to PSL2(Z), and Bannai, Kojima, and Miezaki’s partial proof for the case of Γ∗

0(2), we show that

the zeros of certain modular functions associated to some low-level genus zero groups are all located

on the boundary of certain natural fundamental domains for Γ. The groups considered are Γ∗

0(2),

Γ∗

0(3), Γ0(2 k2), Γ∗

0(5), Γ0(6)+, Γ∗

0(7), Γ0(4 k2)+, Γ0(3 k3), and Γ0(10)+.

Contents

1. Introduction 1

2. Motivation and description of method 2

3. Groups 4

3.1. Fractional linear transformations 4

3.2. Groups 8

3.3. Additional groups 11

3.4. Fundamental domains 11

3.5. Replicable groups 15

4. Functions 23

4.1. Modular Surfaces 23

4.2. Modular Functions 25

4.3. Genus Zero Groups, Hauptmoduls 26

4.4. Replicable functions 27

4.5. Monstrous Moonshine 28

5. Faber Polynomials for Replicable Functions 30

5.1. Harmonics and Faber Polynomials 30

5.2. Beginning the approximation 31

5.3. Completing the approximation 34

6. Proofs of Theorem 1 40

6.1. The case Γ = Γ∗

0(2) 40

6.2. The case Γ = Γ0(6)+ 42

6.3. The case of Γ = Γ0(3 k3) 44

7. Appendix I: Special values of modular functions 46

References 48

1. Introduction

In 1998, Asai, Kaneko, and Ninomiya [2] located the zeros of a certain basis for the space of

weakly holomorphic modular functions for PSL2(Z), using the action of Hecke operators. A decade

later, Bannai, Kojima, and Miezaki [4] suggested that the twisted Hecke operators appearing in

Date: October 10, 2022.

2020 Mathematics Subject Classiﬁcation. Primary 11F03; Secondary 11F06.

arXiv:2210.03668v1 [math.NT] 7 Oct 2022

2 BEN TOOMEY

the Monstrous Moonshine correspondence might be used to locate zeros of weakly holomorphic

modular functions for other groups.

Let

(1.1) S={Γ∗

0(2),Γ∗

0(3),Γ∗

0(5),Γ0(6)+,Γ∗

0(7),Γ0(4 k2)+,Γ0(3 k3),Γ0(10)+}.

These groups (see §3.2 for a description) have the important property that for each Γ ∈ S, the

associated modular surface Y(Γ) is genus zero, and has only one cusp (see §4.1). As a consequence,

the C-vector space of weakly holomorphic modular functions for Γ has a basis of the form {Fn(τ) =

q−n+O(q)|n≥0}, where each Fn(τ) is uniquely determined. We deﬁne a particular fundamental

domain D(Γ) for each Γ ∈ S (see §3.4), and locate the zeros of this basis {Fn(τ)}in the given

fundamental domain D(Γ).

Theorem 1.1. Let Γ∈ S and let n∈N. Then all nzeros for the unique modular function

Fn(τ) = q−n+O(q)for Γare on the lower boundary of the fundamental domain D(Γ), except for

the cases of n= 2 and Γ=Γ∗

0(5),Γ∗

0(7), where one zero is on the lower boundary and one zero is

on the side boundary of D(Γ).

We demonstrate the particular cases of Γ∗

0(2), Γ0(6)+, and Γ0(3 k3) below in Section 6. The

other cases are similar in nature. While there are several other genus zero groups with one cusp,

the groups in Shave the property that the functions Fn(τ) are real-valued on the lower boundary

of D(Γ), allowing us to use the intermediate value theorem to locate zeros.

Bannai, Kojima, and Miezaki [4] obtained a partial proof in the case of Γ∗

0(2), locating at least

n−1 of the nzeros for each function Fn(τ) in the fundamental domain D(Γ∗

0(2)). There are some

minor errors in the write up of their proof, having to do with the bookkeeping involved with twisted

Hecke operators (their formula has the replicate function depending on n, when it instead depends

on a, where a|n). In order to avoid this bookkeeping, and the inevitable errors that would creep

in, we develop a technique to bypass this part of the proof method.

For the particular cases of Γ∗

0(2) and Γ∗

0(3), our results are previously known, due to results of

Choi and Im [6] and Hanamoto and Kuga [14], respectively. Their results cover a broader class of

modular forms, but since the methods used are suﬃciently dissimilar, we still oﬀer our own proofs

below.

2. Motivation and description of method

We brieﬂy summarize the strategies of the prior work mentioned in Section 1, which motivate

our method. Let

Hn:= a b

0da, b, d ∈Z, ad =n, 0≤b < d,

which we call the Hecke set of level n(see Def. 3.25). As noted above, Asai, Kaneko, and Ninomiya

[2] proved the result of Theorem 1.1 for the group Γ = PSL2(Z). Let

T1A(τ) = j(τ)−744 = q−1+ 196884q+ 21493760qq+···

where j(τ) is the classical j-invariant. To locate the zeros of Fn(τ), they ﬁrst observe that the

classical Hecke operators Tnfor modular forms (see, e.g. [16]) provide the identity1

Fn(τ) = q−n+O(q) = n(T1A(τ)|Tn) = X

H∈Hn

T1A(Hτ).

They also observe that when τis ‘near inﬁnity’ — that is, has suﬃciently large imaginary part

— then T1A(τ)≈q−1, and in fact, using the non-negativity of the Fourier coeﬃcients of T1A(τ),

they show T1A(τ)−e−2πiτ <1335 whenever τis in the commonly-used fundamental domain for

1The conﬂicting notations T1Aand Tnare unfortunately rather standard here.

ZEROS OF REPLICABLE FUNCTIONS 3

PSL2(Z), bounded below by the unit circle and to the left and right by |Re τ| ≤ 1

2. We denote this

fundamental domain by D(PSL2(Z)).

So, for each H∈ Hn, let GH∈PSL2(Z) be such that GHHτ ∈ D(PSL2(Z)). Then

Fn(τ)−X

H∈Hn

e−2πiGHHτ ≤X

H∈HnT1A(Hτ )−e−2πiGHHτ 

H∈HnT1A(GHHτ )−e−2πiGHHτ 

(modularity of T1A)

<1335n2,

since there are σ1(n)≤n2elements in Hn. The problem then becomes one of approximating the

left-hand sum above. By a consideration of cases, and restricting to τon the lower boundary of

D(PSL2(Z) (that is, to the unit circle), they show that for most H∈ Hn, one has e−2πiGHHτ ≤

eπnIm τ, but there are three exceptions. By bounding one of these three exceptional terms, the

remaining two serve as an approximation of Fn(τ). After some algebra, this gives a bound (see [2]

for details) Fn(τ)e−2πnIm τ−2 cos(2πnRe τ)< M n2e−2πnIm τ+n2e−πnIm τ+ 1 <2,

for n≥2 and τon the lower boundary of D(PSL2(Z)). Since Fn(τ) is real-valued on the lower

boundary, the approximation by cosine above shows Fn(τ) changes sign n+ 1 times along the unit

circle with real part in the interval [0,1

2], we deduce that Fn(τ) has nzeros on the lower boundary

of D(Γ).

Bannai, Kojima, and Miezaki [4] sought to extend this technique to families of modular func-

tions for groups other than PSL2(Z). Their key observation is that twisted Hecke operators from

Monstrous Moonshine (see §4.5 below) could replace the classical Hecke operators above. Brieﬂy,

with twisted Hecke operators, one has a family of functions {f(a)(τ), a ∈N}, with each f(a)a

normalized Hauptmodul for a genus zero group Γ(a)(analogous to the role of T1A(τ) and PSL2(Z)

above). Letting f(H)=f(a)for H=a b

0d∈ Hn, this family of functions satisﬁes the twisted Hecke

relations,

Fn(τ) = q−n+O(q) = X

H∈Hn

f(H)(Hτ),

where now Fn(τ) is the unique weakly-holomorphic modular function for Γ(1) having a pole of order

nat inﬁnity and holomorphic everywhere else. In particular, letting f(a)=T1Afor all a∈N, the

twisted Hecke operators include the case of PSL2(Z) above as a special case.

In Bannai, Kojima, and Miezaki [4], they consider the case of Γ(1) = Γ∗

0(2), where the functions

f(a)may either be T1A(τ) or T2A(τ) = q−1+ 4372q+ 96256q2+. . . [8], the unique normalized

Hauptmodul for Γ∗

0(2), depending on the parity of a. This signiﬁcantly complicates the bookkeeping,

but a consideration of cases was suﬃcient for them to locate n−1 zeros on the lower boundary.

With some extra care in our bounding procedure, we are able to locate all nzeros.

In order to directly extend [2], we will require that Γ = Γ(1) be a group having only one cusp.

This ensures there exists a fundamental domain D(Γ) which is bounded away from the real line,

similar to D(PSL2(Z)). Somewhat surprisingly, when Γ(1) has only one cusp, then Γ(a)has one cusp

for all a∈N, and moreover, each normalized Hauptmodul f(a)has non-negative Fourier coeﬃcients

(8). Taken together, this is suﬃcient to produce a bound

Fn(τ)−X

H∈Hn

e−2πiGHHτ < M n2,

4 BEN TOOMEY

for some M > 0, analogous to the case of PSL2(Z) above. Unlike for PSL2(Z), however, here

each GH∈Γ(H), which depends on H, so that a consideration of cases becomes signiﬁcantly more

error-prone.

We avoid this brute force approach by proving some group theoretic results relating to twisted

Hecke operators in Section §3. In Section §4 we review some needed facts about modular functions

and replication. We then apply these results to approximate the modular functions Fn(τ) for groups

like those in the set Sabove, in Section §5, culminating with Theorem 5.13, which is the analogue

of the approximation produced in [2] for PSL2(Z). Finally, in Section §6, we prove a few cases

to demonstrate the method, beginning with Γ∗

0(2), the simplest nontrivial case involving twisted

Hecke operators. We also present the case of Γ0(6)+, which demonstrates handling a case where

the lower boundary consists of more than one arc, as well as Γ0(3 k3), where we use additional

results (§5.1) involving ‘harmonics,’ as they were dubbed in [8]. Additional cases may be found in

our upcoming thesis [19].

3. Groups

3.1. Fractional linear transformations. The material here is substantially similar to the expo-

sition given by Duncan and Frenkel [12], with some minor diﬀerences in notation.

Deﬁnition 3.1. Let

Ω := PGL+

2(Q)

={a b

c d |a, b, c, d ∈Q, ad −bc > 0}{tI|t∈Q r {0}},

where I= ( 1 0

0 1 ) is the identity matrix.

We also deﬁne the following notation for elements of Ω. Let x, y ∈Qwith y > 0. Then

Tx:= ( 1x

0 1 ),[y] := y0

0 1 ,and S:= 0−1

1 0 .

We further deﬁne the following subgroups of Ω:

Ω∞:= {a b

c d ∈Ω|c= 0},

ΩT

∞:= {a b

0d∈Ω∞|a=d}

={Tx|x∈Q},

and

ΩD

∞:= a b

0d∈Ω∞|b= 0}

={[y]|y∈Q, y > 0}.

That is, Ω is the group of all rational 2 ×2 matrices with positive determinant, up to scalar

equivalence. For elements of Ω, we generally choose a matrix for coset representative having

integral entries, by simultaneously scaling the entries when necessary. In particular, for p

q∈Q, we

often write

q=q p

0q,p

q=p0

0q,

with the latter expression in Ω only for p

q>0.

For any M=a b

c d , we may take as coset representative for M−1the matrix −d b

c−a, having the

same determinant as the coset representative for M. Since an involution W∈Ω satisﬁes W−1=W

(up to scalar multiplication), from the form of the inverse above we may deduce that involutions

are precisely the elements of the form W=a b

c−afor some a, b, c ∈Z. That is, involutions are the

trace zero elements of Ω.

Lemma 3.2. Let s, t, x, y ∈Qwith x, y > 0. Then

ZEROS OF REPLICABLE FUNCTIONS 5

(1) TsTt=Ts+t,

(2) (Ts)t=Tst,

(3) [x] [y]=[xy], and in particular [x]−1=x−1,

(4) S−1=S,

(5) [y]Tt=Tty [y],

(6) [y]S=S[y]−1,

(7) ST tS=(It= 0

T−1

t1

t2ST −1

tt6= 0.

Proof. In each case, one computes each side as matrices and compares. Note the identity (Ts)t=Tst

involves an appropriate choice of root when t∈Qis not an integer, that is, we are implicitly deﬁning

(Ts)1

n=Ts

nfor any nonzero n∈Z(one may check that this is the unique nth root of Tsin Ω∞). 

In preparation for Theorem 3.4 below, we introduce some further notation:

Deﬁnition 3.3. For any M=a b

c d ∈Ω, we deﬁne the following functions from Ω to P1(Q):

π(M) := −d

ρ2(M) := ad −bc

c2,

σ(M) := (a

dM∈Ω∞

1M /∈Ω∞

θ(M) := (b

dM∈Ω∞

cM /∈Ω∞

We further deﬁne2ρ: Ω →P1(R) by ρ(M) = pρ2(M).

We now deﬁne an important decomposition of elements of Ω.

Theorem 3.4 (Involutory Decomposition).Let M=a b

c d ∈Ω. Then

M=(Tθ(M)[σ(M)] M∈Ω∞

Tθ(M)ρ2(M)ST −π(M)M /∈Ω∞.

Moreover, this factorization is unique, in that if M=Tx[y]ST z(resp. M=Tx[y]) then x=θ(M),

y=ρ2(M), and z=−π(M)(resp. x=θ(M),y=σ(M)).

Proof. First, suppose M∈Ω∞, so

M=a b

0d=1b

0 1 a0

0d=Tθ(M)[σ(M)] .

Moreover, if M=Tθ(M)[σ(M)] = Tx[y], then Tx−θ(M)=σ(M)y−1, and since ΩT

∞∩ΩD

∞={I}

we must have Tx−θ(M)=I=σ(M)y−1, so x=θ(M) and y=σ(M), and the given factorization

is unique.

2We note that Duncan and Frenkel deﬁne the function %, where %(M) = ρ2(M) (not ρ(M)).

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ZEROSOFREPLICABLEFUNCTIONSBENTOOMEYAbstract.FollowingtheworkofAsai,Kaneko,andNinomiyaforFaberpolynomialsassociatedtoPSL2(Z),andBannai,Kojima,andMiezaki'spartialproofforthecaseof0(2),weshowthatthezerosofcertainmodularfunctionsassociatedtosomelow-levelgenuszerogroupsarealllocatedontheboundaryofcertai...

展开>> 收起<<

ZEROS OF REPLICABLE FUNCTIONS BEN TOOMEY Abstract. Following the work of Asai Kaneko and Ninomiya for Faber polynomials associated.pdf

共49页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

ZEROS OF REPLICABLE FUNCTIONS BEN TOOMEY Abstract. Following the work of Asai Kaneko and Ninomiya for Faber polynomials associated

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: