on vector-valued automorphic forms...on vector-valued automorphic forms by jitendra bajpai a thesis...
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On Vector-Valued Automorphic Forms
by
Jitendra Bajpai
A thesis submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Mathematics
Department of Mathematical and Statistical Sciences
University of Alberta
c©Jitendra Bajpai, 2014
To
the city we call Allahabad
For its liveliness, intellectuality, simplicity and river Ganga, in a
hope that we will always let her flow unstoppable, clean and
clear.
i
Abstract
Let G be a genus 0 Fuchsian group of the first kind , w ∈ 2Z and ρ : G −→
GLd(C) be any admissible representation of G of rank d . Then this dissertation
deduces that the space M!w(ρ) of rank d weakly holomorphic vector-valued
automorphic forms of weight w with respect to ρ is a free module of rank d
over the ring RG
of weakly holomorphic scalar-valued automorphic functions .
Note that almost every ρ is admissible .
Let H be any finite index subgroup of G and ρ be any rank d admissible
multiplier of H then this thesis establishes that the lift of any vector-valued
automorphic form of H with respect to ρ is a rank d× [G : H] vector-valued au-
tomorphic form of G with respect to the induced admissible multiplier IndG
H(ρ) .
In case G is a triangle group of type (`,m, n) we show that to classify the
rank 2 vector-valued automorphic forms is equivalent to classify the solutions
of Riemann’s differential equation of order 2 . When G is a modular triangle
group then we also classified the primes for which the denominator of Fourier
coefficients of at least one of the components of any rank 2 vector-valued mod-
ular form with respect to some rank 2 admissible multiplier ρ will be divisible
by p i.e. the Fourier coefficients will have unbounded denominators . Such
components are noncongruence scalar-valued automorphic forms of ker(ρ) .
In addition this thesis also proves the modularity of the bilateral series
associated to various mock theta functions and provide the closed formula of
the associated Ramanujan’s radial limit for all of Ramanujan’s 5th order mock
theta functions as well as few other mock theta functions of various order.
ii
Acknowledgements
This journey was quite long and there were many people who crossed my
path that have helped make it possible. However, one person is immensely
responsible for directing me to the theory of vector-valued automorphic forms ,
my supervisor Prof. Terry Gannon. Without his guidance this thesis simply
would not have been possible. I want to thank him for his exceptional support
and encouragement during my doctoral studies and not only for introducing
me to this beautiful subject but also for teaching me how to enjoy research.
I would also like to sincerely thank Prof. Arturo Pianzola who helped and
guided me in the beginning of my graduate studies in Edmonton. I have learnt
a lot from him. His professionalism, the ability to reach out to people simply
surprised me everyday.
I am fortunate to have been taught by several very good teachers and
mathematicians . They all have influenced my life in various ways : Prof. Dani
Wise and Prof. Ivo Klemes in McGill University , Prof. Dipendra Prasad and
Prof. Ravi Kulkarni in HRI , Prof. Tej Narain Trivedi in Kanpur . I thank
Prof. Ye Tian and Prof. Roman Mikhailov for their support and encourage-
ment . I would also like to thank Prof. Geoffery Mason , Prof. Ken Ono and
Prof. Amanda Folsom for support and guidance on various occasions .
Without any doubt family and close friends were a strong source of sup-
port . I am very lucky to have many good friends and it would be impossible
to name them here . I would like to thank you all for your love and encourage-
ment – you know who you are . Last but not least I would like to pay tribute to
the efforts of my whole family for their immense support and understanding.
Lastly , I would like to thank my beloved grandfather Vidyasagar Tiwari
who taught me how to work hard , keep patience and never give up . You will
always be missed .
iii
Contents
1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . 1
1.2 All about notations and symbols . . . . . . . . . . . . . . . . . 3
1.3 Purpose of this thesis . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Where is what? . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Vector-Valued Automorphic Forms 10
2.1 Fuchsian groups . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Scalar-valued automorphic forms . . . . . . . . . . . . . . . . 18
2.3 Vector-valued automorphic forms . . . . . . . . . . . . . . . . 26
2.4 An example of vector-valued modular form . . . . . . . . . . . 31
3 Construction of Vector-Valued Automorphic Forms 34
3.1 Lift of an automorphic form . . . . . . . . . . . . . . . . . . . 35
3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Vector-Valued Automorphic Forms of Triangle groups - I 49
4.1 Triangle groups . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Admissible multiplier . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Hypergeometric differential equations and functions . . . . . . 60
4.4 Monodromy group of a differential equation . . . . . . . . . . 64
4.5 Rank 2 vector-valued automorphic forms . . . . . . . . . . . . 69
5 Classification of Vector-Valued Automorphic Forms 76
5.1 Nearly holomorphic automorphic forms . . . . . . . . . . . . . 77
5.2 Weakly holomorphic automorphic forms . . . . . . . . . . . . 89
5.3 Differential operators . . . . . . . . . . . . . . . . . . . . . . . 92
5.4 Concluding remarks and conjectures . . . . . . . . . . . . . . . 96
6 Vector-Valued Automorphic Forms of Triangle groups - II 99
6.1 Fuchsian differential equations and hypergeometric functions . 100
6.2 Rank 2 vector-valued automorphic forms . . . . . . . . . . . . 105
6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7 Vector-Valued Automorphic Forms of Triangle groups -III 126
7.1 Nearly holomorphic functions at z =∞ on Riemann sphere . . 127
7.2 Nearly holomorphic automorphic forms at τ = ζ3 on H∗G
. . . . 129
7.3 Hecke triangle groups and the matrix X . . . . . . . . . . . . 131
7.4 An Explicit Example . . . . . . . . . . . . . . . . . . . . . . . 133
8 Bounded Vs. Unbounded Denominators 137
8.1 Introduction and historical background . . . . . . . . . . . . . 137
8.2 Sufficiently integral Fuchsian groups . . . . . . . . . . . . . . . 141
8.3 Integrality of the hauptmodul . . . . . . . . . . . . . . . . . . 152
8.4 Modular vs. hypergeometric differential equations . . . . . . . 157
8.5 End on a high note with p-curvature . . . . . . . . . . . . . . 169
9 Bilateral series and Ramanujan’s Radial Limits 178
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
9.2 Modularity of the bilateral series . . . . . . . . . . . . . . . . 183
9.3 Proof of theorems 9.1.1 and 9.1.2 . . . . . . . . . . . . . . . . 187
9.4 Results for χ0 and χ1 . . . . . . . . . . . . . . . . . . . . . . . 189
9.5 Mock theta functions of other orders . . . . . . . . . . . . . . 190
v
Chapter 1
Introduction
1.1 Background and Motivation
Scalar-valued modular forms and their generalizations are one of the most
central concepts in number theory and perhaps in mathematics. It took almost
300 years to cultivate the mathematics lying behind the classical (i.e. scalar-
valued) modular forms. Why is the notion of vector-valued modular forms one
of the most natural generalizations of scalar-valued modular forms? History of
modern mathematics answers this question naturally. All of the most famous
modular forms have a multiplier, for example:
η
(aτ + b
cτ + d
)=√cτ + d · ρ
(a
c
b
d
)· η(τ) for
(a
c
b
d
)∈ SL2(Z).
In this case the multiplier ρ is a 1-dimensional representation of the double
cover of SL2(Z). These examples suggest having multipliers ρ of dimension d ≥
1 and the corresponding modular forms are called vector-valued modular
forms (vvmf). These are much richer mathematically and more general
than scalar-valued modular forms.
In spite of its naturality the theory of vvmf has been slow in coming.
In the 1960’s, Selberg [45] called for a theory of vvmf, as a way to study
the noncongruence scalar-valued modular forms as these are intractable by
classical methods. In the 1980’s, Eichler-Zagier [10] explained how Jacobi
1
forms and Siegel modular forms for Sp(4) can be reduced to vvmf. Since
then the theory has been in demand to be developed, to a large extent thanks
to the work of Borcherds and the rise of the string theory in physics. This
theory of vvmf has applications in various fields of mathematics and physics
such as vertex operator algebra, conformal field theory, Borcherds-Kac-Moody
algebras, etc. There are plenty of vvmf in “nature”. For instance the characters
of a rational conformal field theory (RCFT) form a vvmf of weight 0. The
Borcherds lift associates vvmf for a Weil representation to automorphic forms
on orthogonal groups with infinite product expansions, which can arise as
denominator identities in Borcherds-Kac-Moody algebras.
It is evident from conformal field theory and vertex operator algebras that
there are plenty of examples available of vvmf . For example, the charac-
ter vectors of rational vertex operator algebras will be vector-valued modular
functions for the modular group Γ(1) .
In terms of developing the theory of vvmf, some efforts have been made
to lift to vvmf, classical results like dimension formulas and growth estimates
of Fourier coefficients of vvmf of the modular group. For example we refer
[27, 26, 39] to mention a few of these efforts. In much of the development of
the theory of vvmf, differential equations have played a dominant role and this
thesis is also a continuation of this trend although we use Fuchsian differential
equations on the sphere instead on the upper half plane . In some sense,
this thesis strengthens the relation between differential equations and vector-
valued automorphic forms (vvaf) by showing that for any triangle group
G, defining a Riemann’s differential equation is equivalent to rank 2 vvaf of G.
Much of this analysis also suggests that this relation is extendable in order to
study higher rank vvaf of G through defining a Fuchsian differential equation
of higher rank on the sphere, the curve associated to G.
This thesis will show that a vvaf X(τ) of a finite index subgroup H of any
2
Fuchsian group of the first kind G can be lifted to one of the vvaf X(τ) of
G by inducing the multiplier. Similarly a vvaf X(τ) of G can be restricted
to one of the vvaf X(τ)′ of any of the finite index subgroup H by reducing
the multiplier. However, lifting of a vvaf increases the rank of vvaf by the
factor equal to the index of H in G whereas the restriction does not affect
the rank of vvaf . These arguments give an easy construction of vvaf of any
finite index subgroup H of G. The lifting argument can also be used to verify
the existence of scalar-valued noncongruence modular forms . Usually, for any
multiplier ρ : Γ(1) → GLd(C), kerρ will be a noncongruence subgroup of
Γ(1). Since all the components of vvmf of G are scalar-valued modular forms
of kerρ, this gives a different approach and direction to develop the theory
of scalar-valued noncongruence modular forms of Γ(1) and hence some hope
to contribute substantially in the development of the long standing Atkin-
Swinnerton-Dyer conjecture about the unbounded denominator property of
modular forms of Γ(1).
1.2 All about notations and symbols
Throughout this thesis G will denote a genus-0 Fuchsian group of first kind
as well as a triangle group, a special class of genus-0 Fuchsian group. Let
ζn = exp(2πin
) denote the n-th root of unity where exp(x) = ex for any x ∈ C.
For any z ∈ C, the complex powers zw are defined by zw = |z|w exp(wiArg(z))
for −π < Arg(z) ≤ π. Let us fix
t = ±(
1
0
1
1
), s = ±
(0
−1
1
0
)and u = st−1 = ±
(0
1
−1
−1
). (1.2.1)
These are elements of the group PSL2(R). The matrices t, s and u are of order
∞, 2 and 3 respectively. We record all the notations in the following table for
the reader’s convenience and reference .
3
Z the set of integersZm the finite cyclic group of order mQ the field of rational numbersR the field of real numbersC the field of complex numbersC[x] the ring of polynomials in variable x and coefficients in
CG genus-g Fuchsian group of the first kindH the complex upper half planeFn free group in n generatorsH∗
Gthe extended upper half plane of G, i.e. a subset of H ∪R ∪ ∞
i a square root of −1 in CF
Gfundamental domain of G
FG
closure of the fundamental domain FG
q exp(2πiτ), for any τ ∈ Hζm the standard m-th primitive root of unity in C, i.e., ζm =
exp(2πim
)Md(R) the set of d× d-matrices with coefficients in the ring RAt transpose of matrix Aξ, η indices of matrices, where 1 ≤ ξ, η ≤ dc a cusp of Gω` elliptic fixed point of G of order `hc cusp width of cusp ch cusp width of cusp ∞Λc the exponent matrix with respect to the cusp cΛ,Ω the exponent matrix with respect to the cusp ∞Gc the stabilizer group of cusp c in Gρ the admissible multiplier, a representation ρ : G →
GLd(C)
N (c)
w (ρ) the space of nearly holomorphic vvaf of G of weightw ∈ 2Z with respect to cusp c ∈ C
Gand admissible
multiplier ρM!
w(ρ) the space of all weakly holomorphic vvaf of G of weightw ∈ 2Z with respect to the admissible multiplier ρ
Hw(ρ) ,H(ρ) the space of weight w holomorphic and respectively allholomorphic vvaf of G with respect to the admissiblemultiplier ρ
J(c)
G(τ), z
(c)
G(τ),
(c)
G(τ) the hauptmoduls of G with respect to the cusp c
JG
(τ), zG
(τ), G
(τ) the hauptmoduls of G with respect to the cusp ∞X(τ),Y(τ) vector-valued automorphic formsX
[n]the nth vector-valued Fourier coefficients of vvaf X(τ)
4
We end this section by fixing the notation for the modular group PSL2(Z)
by Γ(1) as well as for any number N ∈ Z≥1, the notation Γ0(N),Γ(N) is
customary to denote the congruence and principal congruence subgroups of
Γ(1) respectively defined as follows
Γ0(N) =
±(a
c
b
d
)∈ Γ(1)
∣∣ c ≡ 0 (mod N)
, (1.2.2)
Γ(N) =
±(a
c
b
d
)∈ Γ(1)
∣∣∣∣ b ≡ c ≡ 0 (mod N) ,
a ≡ d ≡ ±1 (mod N)
. (1.2.3)
1.3 Purpose of this thesis
Broadly speaking, this thesis is mainly concerned with developing the theory
of vector-valued automorphic forms (vvaf) of Fuchsian groups. In order to be
explicit about the use of the terms vvmf and vvaf, we will make the following
distinction between them : vvaf for a group commensurable with Γ(1)
will usually be referred as vvmf .
The motivation for this thesis (about the theory of vvaf) was largely the
question asked by Prof. Gannon : Let G be any Fuchsian group of the first
kind and let ρ : G→ GLd(C) be any representation, then classify all the vvaf
of G with respect to the multiplier ρ . In an attempt to answer the above , this
is answered with certain restrictions and qualifications . More precisely,
• first restricted only to the world of genus-0 Fuchsian groups of the first
kind , then
• focus is put on the special class of these called triangle groups , and
• only with respect to certain type of their representations which is called
admissible multiplier (for the definition see 2.3.1) .
5
1.4 Where is what?
This thesis is divided into three different parts. The first part is about the
theory of vvaf of genus-0 Fuchsian groups of the first kind and the classifi-
cation of rank 2 vvaf of triangle groups. The second part is an attempt to
discuss the unbounded denominator conjecture and the related aspects about
the growth of Fourier coefficients of rank 2 vvaf of modular triangle groups .
The last part discusses the mock theta functions . This chapter is influenced
by a recent exposure to the theory of mock modular forms during the partici-
pation in the Arizona Winter School in March 2013. This project resulted in
a publication [6] .
Chapter 2 begins by introducing the basics facts about Fuchsian groups ,
in particular genus 0 Fuchsian groups of the first kind . We review quickly the
theory of scalar-valued auttomorphic forms of Fuchsian groups of the first kind
and prepare the reader for advancing this scalar-valued theory of automorphic
forms to vector-valued automorphic forms . This chapter is ended with an
example of vector-valued modular forms .
Chapter 3 discusses the lift of an automorphic form of any Fuchsian group
G of the first kind . This chapter also shows how the induction of any represen-
tation (multiplier) of rank d of any finite index subgroup H of G will become
a multiplier of G of rank d× [G : H] . Theorem 3.1.3 is the main result of this
chapter , which together with Lemma 3.1.5 is one of the beautiful results in
this chapter .
Chapters 4 , 5 ,and 6 are one of the crucial chapters of this thesis . Chapter 4
gives all the details about triangle groups . More precisely, their fundamental
domains, hauptmoduls, scalar-valued automorphic forms and their admissible
6
multipliers are discussed in detail. The highlight of the chapter is section 4.2.2
where the existence of infinitely many rank 2 even integer weight, nearly holo-
morphic vvaf has been given . This is done by building a connection between
the theory of Riemann’s differential equation and the theory of vvaf . As a
basic result and the starting point in the world of classification of vvaf we
proved Theorem 4.5.2 . Chapter 7 also follow the same methodology .
Chapter 5 discusses the module structure of vvaf for any genus-0 Fuchsian
group G with respect to any admissible representation. We show that the
space of even integer weight nearly respectively weakly holomorphic vvaf of G
with respect to any rank d admissible ρ is a free module of rank d over the
ring of nearly respectively weakly scalar-valued automorphic functions of G .
We end this chapter with a brief introduction along with some speculations of
the theory of holomorphic vvaf .
Chapter 6 discusses the module structure of vvaf of any triangle group with
respect to any rank 2 admissible multiplier . The rank 2 vvaf of any triangle
groups are explicitly classified by using the existence theory of free module
structure of the space of vvaf of any G . The classification is being achieved
by building an abstract connection with hypergeometric and Riemann’s dif-
ferential equations by exploiting the information from the theory of classical
hypergeometric and Riemann’s differential equations of order 2 and the exis-
tence of free basis with respect to any rank d admissible multiplier of G form
chapter 5 .
Chapter 7, once again looks at the triangle groups and classification of
their rank 2 vvaf with respect to admissible multiplier . This time through
a more direct approach starting with an admissible multiplier , following the
7
strategy developed in chapter 4 , this multiplier is associated with two differ-
ent Riemann’s differential equations . The basis of the solution space of these
two equations are the possible candidate to generate the whole module of even
integer weight rank 2 nearly and weakly holomorphic automorphic forms of
any triangle group .
Chapter 8 deals with the behaviour of Fourier coefficients of vvaf. In par-
ticular, let X(τ) =(
X1
X2
)be a nontrivial rank 2 vvaf of a modular triangle
group G with respect to multiplier ρ : G → GL2(C). It is demonstrated that
the components X1(τ) and X2(τ) have integral Fourier coefficients, only when
kernel of ρ is a congruence group (see the introduction of this chapter for
details and qualifications). To accomplish this the theory of hypergeometric
differential equations is used again . A detailed exposition on modular triangle
groups is given. This chapter ends with a brief introducing of the p-curvature
with some speculations and conjectures towards the study of the behaviour of
the Fourier coefficients of modular triangle groups . These two approaches are
complimentary : the approach in this chapter applies to primes dividing the
order ρ(t∞) , where t∞ is the generator of the subgroup G∞ while p-curvature
restricts to those primes which are coprime to order ρ(t∞) . It is expected
that these two approaches cover the possible behaviour of most of the primes
in the denominators of the components of vvaf for modular triangle groups .
The work of this chapter was initiated by a joint project started with Chris
Marks during his postdoctoral studies in this department and will appear as
a joint publication in the near future . I would like to take this opportunity to
thank him for introducing me to the theory of noncongruence modular forms
and their natural connection with the theory of vvaf to study the behaviour
of Fourier coefficients of noncongruence modular forms .
8
As mentioned briefly in the very beginning of this section , in chapter 9
the advances in the Ramanujan radial limits in case of various mock theta
functions are discussed . This work has already been published [6] titled “Bi-
lateral Series and Ramanujan’s Radial Limits” . This is shown by exploiting
the connection between the mock theta functions and their associated modular
bilateral series . I would like to take this opportunity to thank my coauthors
Susie Kimport , Jie Liang , Ding Ma and James Ricci for making this collabo-
rative work a rewarding experience . At the same time I would like to extend
my appreciation to Amanda Folsom , Ken Ono and Robert Rhoades for in-
troducing me to the theory of mock theta functions and mock modular forms
as well as to the organizers of Arizona winter school 2013 for giving me the
opportunity to participate . This will influence my future research directions .
9
Chapter 2
Vector-Valued AutomorphicForms
2.1 Fuchsian groups
The study of Fuchsian groups begins by looking at the discrete group of mo-
tions of the upper half plane H = z = x + iy ∈ C | y > 0 in the complex
plane C equipped with the Poincare metric ds2 = dx2+dy2
y2 . The semicircles
orthogonal to the real axis, as well as the vertical half lines, are the geodesics
for this metric . The group of all orientation-preserving isometries of H for this
metric coincides with the group PSL2(R) = SL2(R)/±I , where
SL2(R) =
(a
c
b
d
) ∣∣∣∣ a, b, c, d ∈ R , ad− bc = 1
.
Roughly speaking , a Fuchsian group is a discrete subgroup G of PSL2(R) for
which G\H is topologically a Riemann surface with finitely many punctures .
For detailed exposition on the theory of Fuchsian groups , see [25, 46, 50] . A
group G in PSL2(R) is called discrete , if G is a discrete subgroup of PSL2(R)
with respect to the induced topology of PSL2(R) . More explicitly , to define
the discreteness of a subgroup G of PSL2(R) , we mean :
given any matrix A ∈ G , there is an εA > 0 such that all the matrices B(6= A)
10
in G have dist(A,B) > εA , where
dist(A,B) = min
∑i,j
| Aij −Bij |,∑i,j
| Aij +Bij |.
The action of any subgroup of SL2(R) on H is the Mobius action , defined by(a
c
b
d
)· τ =
aτ + b
cτ + d. (2.1.1)
Define H∗ = H∪R∪∞ to be the extended upper half plane of PSL2(R) and
this action can easily be extended to H∗ . For any γ = ±(acbd
)∈ PSL2(R) ,
the action of γ on ∞ is defined by the following equation :
γ · ∞ = limτ 7→∞aτ + b
cτ + d=a
c∈ R ∪ ∞ , (2.1.2)
and for any x ∈ R, the action is defined similarly by taking the limit τ 7→ x
in equation (2.1.2) .
The two elements ±(acbd
)of SL2(R) can be grouped together , since their
actions on H coincide . This allows for the study of the action of PSL2(R)
and its subgroups G , on H . Unlike SL2(R) , PSL2(R) has a faithful action
on H . Moreover , the discrete subgroups of PSL2(R) tend to have a simpler
structure than the subgroups of SL2(R) . For example : PSL2(Z) ∼= Z2 ∗Z3 the
free product of Z2 and Z3 whereas SL2(Z) ∼= Z4 ∗Z2Z6 , the amalgamated free
product of Z4 and Z6 over Z2 .
The free product G1∗G2 of the groups G = 〈g1, g2 , · · · , gn;R1 , R2 , · · · , Rp〉
and G2 = 〈h1 , h2 , · · · , hm;S1 , S2 , · · · , Sq〉 is the group
G1 ∗G2 = 〈g1, g2 , · · · , gn , h1 , h2 , · · · , hm;R1 , R2 , · · · , Rp , S1 , S2 , · · · , Sq〉
where ∀1 ≤ i ≤ p and ∀1 ≤ j ≤ q Ri and Sj denote the various relations
among the generators of groups G and H respectively . G1 and G2 are called
the free factors of the group G1 ∗ G2 . The free product G1 ∗ G2 is uniquely
determined by the groups G1 and G2 . Moreover , G1 ∗G2 is generated by two
subgroups A and B such that A ∼= G1 and B ∼= G2 and A ∩B = 1 .
11
Let G1,G2,H be any three groups and for i = 1, 2, let φi : H → Gi be
any homomorphism. Let N be the normal subgroup of G1 ∗ G2 generated
by elements of the form φ1(h)φ2(h)−1 for h ∈ H; then the amalgamated free
product G1 ∗HG2 of G1,G2 over H is G1 ∗G2/N . Note that the free product
G1 ∗ G2 is a special case of the amalgamated free product of two groups over
H = 1. The amalgamated free product of Gi’s for i ∈ I over a group H can
be defined inductively. For example:
SL2(Z) =⟨a, b | a4 = b6 = 1, a2 = b3
⟩ ∼= Z4 ∗Z2Z6
where Z2∼= 〈a2〉 ∼= 〈b3〉 and Z4
∼= 〈a〉 and Z6∼= 〈b〉 . For more details on free
product and amalgamated free product of groups see chapter 4 of [33] .
The elements of PSL2(R) can be divided into three classes: elliptic, parabolic
and hyperbolic elements. An element γ ∈ PSL2(R) is elliptic, parabolic or hy-
perbolic, if the absolute value of the trace of γ is respectively less than, equal
to or greater than 2. A point τ ∈ H∗ is said to be a fixed point of γ ∈ PSL2(R)
if γ · τ = τ . Let γ = ±(acbd
)∈ PSL2(R) and τ ∈ H∗ be one of its fixed points .
Then γ · τ = τ implies that τ =a−d ±
√(a+d)2−4
2c. If γ is an elliptic element
then
τ =
a−d −i
√4−(a+d)2
2c, if c < 0
a−d +i√
4−(a+d)2
2c, if c > 0
. (2.1.3)
If γ is a parabolic element then τ = a∓1c
when a+d = ±2 and c 6= 0 , in addition
τ =∞ when c = 0 . If γ is an hyperbolic element then τ =a−d ±
√(a+d)2−4
2care
two different points on the boundary R ∪ ∞ .
Therefore with respect to the action of PSL2(R) on H∗ , the hyperbolic
elements have two fixed points (both inside R∪∞) , the parabolic elements
have one fixed point on R ∪ ∞ whereas the elliptic elements have one fixed
point in H . Note that PSL2(R) fixes R ∪ ∞ . Note that in H∗ there is only
one notion of ∞ usually denoted by i∞ but for notational convenience it will
12
be written ∞ .
Note 2.1.1. Following the equation (2.1.2) , for any x ∈ R it is observed that
there exists an element γ = ±(x1−10
)such that γ ·∞ = x which means PSL2(R)
acts transitively on R ∪ ∞ . For any x ∈ R such γ is denoted by Ax .
Definition 2.1.2. Let G be a subgroup of PSL2(R) . A point τ ∈ H is called
an elliptic fixed point of G if it is fixed by some nontrivial elliptic element of
G, and c ∈ R∪∞ is called a cusp (respectively hyperbolic fixed point) of G
if it is fixed by some nontrivial parabolic (respectively hyperbolic) element of
G . Moreover , EG
and CG
denote the set of all elliptic fixed points and cusps
of G and define H∗G
= H ∪ CG
to be the extended upper half plane of G .
Example 2.1.3.
• If G = PSL2(R) then CG
= R ∪ ∞ and EG
= H .
• If G = PSL2(Z) then CG
= Q ∪ ∞ and EG
= G · i ∪ G · ω , i.e. CG
consists of the G-orbit of cusp∞ and EG
consists of the G-orbits of i and
ω = 1+i√
32
.
For any τ ∈ H∗G
let Gτ = γ ∈ G|γ ·τ = τ be the stabilizer subgroup of τ in
G . For each τ = x+iy ∈ H Gτ is a cyclic subgroup of G of finite order generated
by γτ = AτKmA−1τ where m = m(τ) is the unique positive integer called the
order of τ , Aτ = 1√y
(y0x1
)such that Aτ (i) = τ and Km = ±
(cos( π
m)
−sin( πm
)
sin( πm
)
cos( πm
)
).
For any c ∈ CG
Gc is an infinite order cyclic subgroup of G . If c =∞ then G∞ is
generated by γ∞ = ±(
10h∞1
)= th∞ for a unique real number h∞ > 0 called the
cusp width of the cusp∞ . In case of c 6=∞ , Gc is generated by γc = ActhcA−1
c
for some smallest real number hc > 0 , called the cusp width of the cusp c such
that γc ∈ G where Ac = ±(c1−1
0
)∈ PSL2(R) so that Ac(∞) = c , as defined in
the Note 2.1.1 . From now on for convenience h∞ will be denoted by h .
13
Observation 2.1.4. For every c ∈ CG\∞, the elements of Gc depend on
c. Since c ∈ R ∪ ∞, there are two possibilities: c ∈ R or c = ∞. Con-
sider c ∈ R. Let γ be any element in Gc then γ = (γc)r for some inte-
ger r, i.e. γ = Ac(thc)rA−1
c . Observe that (thc)r = ±(
10rhc1
)and therefore
γ = ±(
1−crmhc−rmhc
c2rmhc1+crmhc
). When c ∈ Q, this is simplified as follows: without
loss of generality assume that c = pq
with gcd(p, q) = 1. Let γ be any element
in Gc and Ac = ±(acbd
)gives Ac(∞) = a
cso that we may choose a = p and
c = q. Since gcd(p, q) = 1 there exist integers x, y such that px+ qy = 1. This
implies Ac = ±(pq−xy
), and hence γ = Act
rhcA−1c = ±
(1−pqrhc−q2hc
p2hc1+pqrhc
). When
c = ∞, taking p = 1 and q = 0, γ ∈ G∞ is of the form ±(
10rh∞
1
). In either
case ∀c ∈ CG
, Gc is a nontrivial infinite cyclic subgroup of G and the trace of
every γ ∈ Gc is ±2.
2.1.1 Fuchsian groups of the first kind
The class of all Fuchsian groups is divided into two categories, namely Fuchsian
groups of the first and of the second kind. To distinguish between them, a
fundamental domain of Fuchsian groups is defined. The fundamental domain,
denoted by FG
, exists for any discrete group G acting on H and is defined as
follows:
Definition 2.1.5. Let G be any discrete subgroup of PSL2(R). Then a domain
(connected open set) FG
in H is called the fundamental domain of G, if
1. no two elements of FG
are equivalent with respect to G .
2. any point in H is equivalent to a point in the closure of FG
with respect
to G i.e. any G-orbit in H intersects with the closure of FG
.
The hyperbolic area of FG
may be finite or infinite. When FG
has finite
area then such G is a Fuchsian group of the first kind otherwise of the second
kind. For example G = 〈t〉 is the simplest example of a Fuchsian group of the
14
second kind. This thesis mainly concerned with Fuchsian groups of the first
kind. More precisely,
Definition 2.1.6. A discrete subgroup G of PSL2(R) is a Fuchsian group of
the first kind if there exists a fundamental domain FG
of finite hyperbolic area.
A Fuchsian group G will have several different fundamental domains but
this can be observed that their area will always be the same . From FG
a
(topological) surface ΣG
is obtained by identifying the closure FG
of FG
using
the action of G on FG
, i.e. ΣG
= FG/∼ (equivalently Σ
G= G\H∗
G) . In fact
ΣG
can be given a complex structure, for details see chapter 1 of [46] . ΣG
has
genus-g where as surface G\H is of genus-g with finitely many punctures. Due
to Fricke, any Fuchsian group of the first kind is finitely generated . In fact, if
G is a Fuchsian group of the first kind its group presentation is written as
G =
⟨ai, bi, rj, γk
∣∣∣∣ Πgi=1[ai, bi] · Πl
j=1rj · Πnk=1γk = 1 , r
mjj = 1
⟩(2.1.4)
where 1 ≤ i ≤ g, 1 ≤ j ≤ l, 1 ≤ k ≤ n and [a, b] = aba−1b−1 . Here ai, bi are
generators of the stabilizer group of the 2g orbits of hyperbolic fixed points .
Each rj is the generator of the stabilizer group of l orbits of elliptic fixed points,
each γk is the generator of the stabilizer group of n orbits of cusps of G and
∀j mj ∈ Z≥2 denotes the order of elliptic element rj.
The set of numbers (g;m1, . . . ,ml; n) is called the signature of G. For
example, the signature of Γ(1), Γ0(2) and Γ(2) are (0; 2, 3; 1), (0; 2; 2) and
(0; ; 3) respectively, where ‘ ’ represents the nonexistence of any nontrivial
elliptic element in Γ(2). By using the Gauss-Bonnet formula, the area of any
FG
can be computed in terms of its signature. Namely,
Area(FG
) = 2π
[2g− 2 +
l∑j=1
(1− 1
mj
)+ n
].
With respect to a set of generators of G, FG
can be chosen to be the interior
of a convex polygon bounded by (4g+2l+2n−2) geodesics, the sides of which
15
−ωω
−12
0 12
i
Figure 2.1: Fundamental domain of Γ(1). All 4 geodesics can be describedas follows: straight lines ω to ∞ and −ω to ∞ contribute to two geodesicsand the arcs ω to i and i to −ω contribute to the other two geodesics, hereω = −1+i
√3
2.
are pairwise identified under the action of the generators of G. For G = Γ(1)
the polygon is bounded by 4 sides, which can be seen in figure 2.1, although
the sides (ω, i) and (i,−ω) lie on the same geodesic. G is called a co-compact
group if n = 0. In addition if l = 0 then G is called a strictly hyperbolic group.
In general, FG
has exactly n-vertices on R ∪ ∞. These vertices correspond
to the inequivalent cusps of G.
2.1.2 Genus-0 Fuchsian groups of the first kind
One of the basic properties of Fuchsian groups of the first kind is that their
action on H gives rise to a surface ΣG
of genus-g of finite area. The genus-0
Fuchsian groups of the first kind are those which give the Riemann sphere with
finitely many special points . These special points correspond to the G-orbits
of elliptic fixed points and cusps of G. Let G be a genus-0 Fuchsian group of
the first kind. Let EG
:= ej ∈ H | 1 ≤ j ≤ l be a set of all inequivalent
elliptic fixed points of G where ∀j, ej’s are representatives of distinct orbits
of elliptic fixed points of G with respect to its action on H and CG
:= ck ∈
16
R∪∞ | 1 ≤ k ≤ n be a set of all inequivalent cusps of G where ∀k, ck’s are
representatives of the distinct orbits of cusps of G with respect to its action
on R ∪ ∞ . For example
• If G = Γ(1) , then CG
= ∞ and EG
= i, 1+i√
32 ,
• If G = Γ0(2) , then CG
= 0,∞ and EG
= 1+i2 ,
• If G = Γ(2) , then CG
= −1, 0,∞ and EG
= φ .
Then G\H∗G− (E
G∪ C
G) is a Riemann sphere with l + n punctures . As an
abstract group, the presentation of G can be obtained by taking g = 0 in the
presentation (2.1.4) . If n ≥ 1 then
G ∼= Zm1 ∗ . . . ∗ Zml∗ F(n−1) ,
where for every integer k ≥ 0 , Fk ∼= Z ∗ Z ∗ . . . ∗ Z︸ ︷︷ ︸k−copies
, is known as a free group
in k generators .
Example 2.1.7. For some values of integer N ≥ 1 , congruence groups Γ0(N)
and Γ(N) are a few examples of genus-0 Fuchsian groups of the first kind .
• For N = 1 , Γ0(1) = Γ(1) ∼= Z2 ∗ Z3 .
• For N = 2 , Γ0(2) ∼= Z2 ∗ Z and Γ(2) ∼= F2 .
• For N = 3 , Γ0(3) ∼= Z3 ∗ Z and Γ(3) ∼= F3 .
• For N = 4 , Γ0(4) ∼= F2 and Γ(4) ∼= F5 .
• For N = 5 , Γ0(5) ∼= Z2 ∗ Z2 ∗ Z and Γ(5) ∼= F11 .
The class of triangle groups (hyperbolic) is defined as genus-0 Fuchsian
groups of the first kind with sum n + l = 3 . In the above example 2.1.7 , all
the congruence groups except Γ(3) ,Γ(4) ,Γ(5) and Γ0(5) are triangle groups .
17
The theory of vvaf of triangle groups is discussed in detail later in this thesis .
There are uncountably many inequivalent genus-0 Fuchsian groups of the first
kind which have the sum n+ l > 3 . One such group is Γ(3), which has 4 cusps
namely ∞ , 0 , 1 and −1 , and has no elliptic fixed points . The modular group
Γ(1) is probably the most widely used Fuchsian group , and it appears in most
branches of mathematics , if not in all .
2.2 Scalar-valued automorphic forms
The basics of the theory of classical (i.e. scalar-valued) automorphic forms
is being reviewed before discussing the details of the theory of vvaf . Every
automorphic form possesses an infinite series expansion . To understand this
series expansion of any scalar-valued meromorphic function f : H → C , a
notion of growth condition is required . Therefore we begin with the following
Definition 2.2.1. Let f(τ) be scalar-valued meromorphic function in H . De-
fine
• f(τ) to have moderate growth at ∞ if there is some ν ∈ C and some real
number Y such that |f(x+ iy)| < exp(Im(ντ)) for all y > Y .
• f(τ) to have moderate growth at c ∈ R if f(A−1
cτ) has moderate growth
at ∞ .
Definition 2.2.2. Let G be any genus-0 Fuchsian group of the first kind and
w ∈ 2Z . Let σ : G → C× be a 1-dimensional representation . Then , a
scalar-valued meromorphic function f : H → C is a weight w meromorphic
scalar-valued automorphic form of G with respect to σ , if
(i) f(γτ) = σ(γ)(cτ + d)wf(τ) , ∀τ ∈ H and ∀γ ∈ G .
(ii) f(τ) has finitely many poles in FG∩H .
18
(iii) f(τ) has moderate growth at all cusps c of G .
Remark 2.2.3. Usually the representation σ is referred as multiplier of the
automorphic forms of G . This thesis mainly explore the theory of automorphic
forms with respect to the higher rank multipliers σ and call them vector-valued
automorphic forms .
Meromorphicity and holomorphicity at other cusps can be defined by mov-
ing the cusp to ∞ , i.e. conjugating G by Ac and this gives
CAcGA
−1c
= A−1
c (x) | x ∈ CG .
Moreover , f(A−1c τ) will define a scalar-valued meromorphic automorphic form
of AcGA−1
c , for every f(τ) of G . To be more explicit , this observation is
recorded in Lemma 2.2.4 but before stating the lemma two important notions
namely the co-cycle function j and cusp-factor function ςc for any cusp c ∈ CG
are introduced for notational convenience .
1. Let j : G×H→ C× be the function such that for every γ = ±(acbd
)∈ G
and τ ∈ H
j(γ, τ) = cτ + d .
One of the important properties of the j-function which we will make
use of is
j(γ1γ2 , τ) = j(γ1 , γ2τ) · j(γ2 , τ) (2.2.1)
where γ1, γ2 ∈ G and τ ∈ H .
2. Let ςc : H→ C be the function such that for every τ ∈ H
ςc(τ) = A−1
c τ =1
c− τ
when c 6=∞ and in case of c =∞ ςc(τ) = 1 .
19
Lemma 2.2.4. Let w ∈ 2Z . f(τ) is a meromorphic scalar-valued automorphic
form for G of weight w with respect to a multiplier σ if and only if for every c ∈
CG\∞ , ςc(τ)wf(A
−1
c τ) is a meromorphic scalar-valued automorphic form of
weight w for AcGA−1
c with respect to the multiplier ρ where ρ(AcγA−1
c ) = σ(γ) ,
∀γ ∈ G .
Proof. Since w ∈ 2Z , ςc(τ)w = (τ − c)−w . Let g(τ) = (τ − c)−wf(A−1
c τ)
for c of G . Suppose f(τ) is a meromorphic scalar-valued automorphic form
for G of weight w with respect to a multiplier σ . It is required to show
that g(τ) satisfies all three conditions in the Definition 2.2.2 . We begin with
the first condition where to show that g(γ′τ) = ρ(γ′)j(γ′, τ)wg(τ) for every
γ′ ∈ AcGA−1
c . Clearly γ′ = AcγA−1
c for some γ ∈ G . This follow from the
definition of g(τ) and property (2.2.1) of the j-function .
g(γ′τ) = (γ′τ − c)−wf(γA−1
c τ)
= σ(γ)(γ′τ − c)−wj(γ,A−1
c τ)wf(A−1
c τ)
= ρ(γ′)j(γ′, τ)wj(Ac, A−1
c τ)wf(A−1
c τ)
= ρ(γ′)j(γ′, τ)w(τ − c)−wf(A−1
c τ) .
For the second part note that A−1
c (FG
) is the fundamental domain of AcGA−1
c .
Since f(τ) has finitely many poles in FG∩H therefore f(A
−1
c τ) has finitely many
poles in A−1
c (FG
)∩H . Hence g(τ) has finitely many poles in FAcGA
−1c∩H . For
the third part , we wish to show that g(τ) is of moderate growth at all cusps of
AcGA−1
c . From the first part of the definition 2.2.1 this is equivalent to show
that g(τ) has moderate growth at cusp c and from part 2 of the definition 2.2.1
this is equivalent to show that g(Acτ) = τwf(τ) has moderate growth at cusp
∞ . For the converse interchange the roles of f(τ) and ςc(τ)wf(A−1
c τ) .
As a consequence of growth condition and functional behaviour , f(τ) has
an infinite series expansion at the cusp c ∈ CG
. These expansions which are
20
essentially Laurent series expansions , will be referred to as “Fourier series
expansions” since they have an exponential form . Often these expansions are
referred to as qz -expansion with respect to z ∈ H∗G
where in case of z ∈ EG∪C
G
qz =
exp
(2πiA−1
zτ
hz
)if z ∈ C
G(τ−zτ−z
)`if z ∈ E
Gof order `
. (2.2.2)
Here and throughout this thesis we write qΛ
= exp(
2πiΛτh
)similarly q
r
cfor
any r ∈ C .
Theorem 2.2.5. Let f(τ) be a scalar-valued meromorphic function on H
which has no poles when Im(τ) ≥ Y for some Y > 0 and which obeys
f(τ +h) = exp(2πiΛ)f(τ) for every τ ∈ H for some Λ ∈ C . Suppose f(τ) has
moderate growth at ∞ . Then
q−Λ
f(τ) =∞∑
n=−M
f[n] qn
, (2.2.3)
for some f[n]∈ C , and this sum converges absolutely in Im(τ) > Y .
Proof. Since f(τ) has moderate growth at∞ , there is an integer M such that
F (τ) = qM−Λ
f(τ) tends to 0 as Im(τ) → ∞ for 0 ≤ x ≤ h . Note that
F (τ + h) = F (τ) therefore g(q) = F (τ) is a well defined and holomorphic
function in the punctured disc 0 < |q| < exp(−2πiYh
) , about q = 0 and is
bounded there (because it tends to 0 as q goes to 0) . This means that q = 0
is a removable singularity thus defining g(0) = 0 gives g(q) is holomorphic in
the disc |q| < exp(−2πiYh
) . This means that g(q) has a Taylor expansion in q
which converges absolutely in that disc .
Corollary 2.2.6. Let G be a Fuchsian group of the first kind and w ∈ 2Z. Let
f(τ) be a weight w scalar-valued meromorphic automorphic form with respect
to the multiplier σ . Let c be any cusp of G and hc be its cusp width such that
σ(tc) = exp(2πiΛc) . Then the following is true.
21
1. If c =∞ then there exists an integer M such that
q−Λ
f(τ) =∞∑
n=−M
f[n] qn
, (2.2.4)
converges absolutely for Im(τ) > Y for some nonzero real number Y and
for all τ ∈ H where q = exp(
2πiτh
).
2. If c 6=∞ then there exists an integer Mc such that
(τ − c)wq−Λc
cf(τ) =
∞∑n=−Mc
fc
[n] qn
c, (2.2.5)
converges absolutely for Im(A−1c τ) > Yc for some nonzero real number
Y and for all τ ∈ H where qc = exp(
2πiA−1c τ
hc
).
Proof. The case c = ∞ is a direct corollary of Theorem 2.2.5 and the case
c 6=∞ follows from Lemma 2.2.4 .
Thanks to Corollary 2.2.6 , we write the following
Definition 2.2.7. f(τ) is meromorphic at c if it has moderate growth there .
Likewise , we say it is Λc-holomorphic at c if the expansion in (2.2.5) holds
for integer Mc = 0 . Let Λhol
cbe the unique complex number which satisfies
σ(tc) = exp(2πiΛ
hol
c
)and 0 ≤ Re(Λ
hol
c) < 1 . Then f(τ) is holomorphic at τ = c
if and only if it is Λhol
c-holomorphic i.e. f
c
[n] = 0 for every integer n < Re(Λc) .
Definition 2.2.8. The expansion (2.2.4) and (2.2.5) above are called the
“Fourier expansion of f(τ) at q and qc” respectively. If f(τ) is holomorphic in
H then the expansions (2.2.4) and (2.2.5) are valid in all of H, i.e. in this case
Y = 0 = Ycmay be chosen .
Remark 2.2.9. We are particularly interested in the case where f(τ) is a
component of a vvaf X(τ) of some G, and exp(2πiΛc) is the corresponding
eigenvalue of ρ(tc) for some representation ρ : G → GLd(C), which will be
explained in the next section.
22
Definition 2.2.10. Let G be any Fuchsian group of the first kind and σ :
G → C× be any multiplier of rank 1. Let f(τ) be a meromorphic scalar-
valued automorphic form of G of weight w ∈ 2Z, then :
(a) f(τ) is a weakly holomorphic scalar-valued automorphic form if f(τ) is
holomorphic in H and meromorphic at the cusps of G, i.e. the only poles
of f(τ) lie inside the set CG
. Let M!w(σ) denote the space of all weight w
weakly holomorphic scalar-valued automorphic forms of G with respect to the
multiplier σ.
(b) f(τ) is a holomorphic scalar-valued automorphic form if f(τ) is holomorphic
at every point in H∗G
. Let Hw(σ) denote the space of all such f(τ) .
(c) f(τ) is a scalar-valued cusp form if f(τ) is holomorphic in H∗G
such that in
its Fourier expansion of Theorem (2.2.5), fc
[n] = 0 for all integer n ≤ Re(Λc) .
Let Sw(σ) denote the space of all such f(τ) .
(d) in addition, f(τ) is a nearly holomorphic scalar-valued automorphic form
with respect to the cusp c ∈ CG
if f(τ) is holomorphic in H∗G\G · c and
meromorphic at c . The space of all such f(τ) is denoted by N (c)
w (σ) .
Note 2.2.11. When w = 0, an automorphic form will be referred as an au-
tomorphic function. For any G with σ = 1, M!0(1) forms a ring , N (c)
0 (1) is
a subring and H(1) =∐∞
k=0H2k(1) forms a ring graded by weight whereas∐∞k=0 S2k(1) forms a subring of H(1) . It has been shown that H0(1) = C,
see [3, 28, 46] .
In this thesis , we are interested in G being genus-0 . The key feature of G
being genus-0 , is the existence of a hauptmodul which is denoted by JG
and
is defined in the following
Definition 2.2.12. Let G be a Fuchsian group of the first kind . A haupt-
modul of G is any scalar-valued meromorphic automorphic function JG
with
23
respect to the trivial multiplier such that the space of all scalar-valued mero-
morphic automorphic functions (i.e. weight 0 automorphic forms) of G with
trivial multiplier is C(JG
) , the ring of rational functions in JG
.
Geometrically , the hauptmodul identifies G\H∗G
with the Riemann sphere
P1(C) which is implicit in chapter 2 in [46] and this fact is recorded without
proof in the form of the following
Lemma 2.2.13. G possesses a hauptmodul if and only if G is genus-0 Fuch-
sian group of the first kind .
Given any c ∈ CG
, we can uniquely fix such a J(c)
G(τ) to be a nearly
holomorphic modular function on H with respect to the cusp c (i.e. holo-
morphic in H∗G\c) such that its Fourier expansion is of the form
J(c)
G(τ) = q
−1
c+∞∑n=1
cnqn
c. (2.2.6)
We call such J(c)
G(τ) the normalized hauptmodul for G with respect to the
cusp c . In such a case , all nearly holomorphic modular functions for G with
respect to the cusp c form the polynomial ring N (c)
0(1) = C[J
(c)
G] . Similarly, the
analogue of the normalized hauptmodul can be defined for any G with respect
to any τ ∈ H∗G
i.e. holomorphic everywhere in H∗G\G · τ with qτ -expansion
of the form (2.2.6) .
One of the many important features of G being genus-0 is that all types
of scalar-valued automorphic and quasi-automorphic (defined below) forms of
G can be computed if a hauptmodul of G is known . Moreover , if J(τ) is a
hauptmodul so is any aJ+bcJ+d
for every(acbd
)∈ GL2(C) . In case of G = Γ(1) this
is discussed in the following
Example 2.2.14. Consider G = Γ(1) . Then
1. Hauptmoduls: It has normalized hauptmodul
J(∞)
G(τ) := J(τ) = q−1 + 196884q + . . . .
24
Alternate choices of hauptmodul are j(τ) = 744+J(τ)1728
and z(τ) = 984−J(τ)1728
. Their
values at the special points are given in the table 2.1.
τ J(τ) j(τ) z(τ)
∞ ∞ ∞ ∞
i 984 1 0
1+i√
32
−744 0 1
Table 2.1: Hauptmoduls of Γ(1)
2. Spaces of automorphic forms: M!0(1) = C[J], H(1) = C[E4, E6], and
S(1) = ∆C[E4, E6], where
E4(τ) = 1 + 240∞∑n=1
σ3(n) qn
= 1 + 240(q + 9q2 + 10q3 + 73q4 + · · · ), (2.2.7)
E6(τ) = 1− 504∞∑n=1
σ5(n) qn
= 1− 504(q + 33q2 + 244q3 + 1057q5 + · · · ) (2.2.8)
∆(τ) = qΠ∞n=1(1− qn)24 (2.2.9)
= q − 24q2 + 252q3 − 1472q4 + 4830q5 − 6048q6 + . . . ,
E4, E6 are called the Eisenstein series respectively of weight 4 and 6 of Γ(1)
and, for any positive integer k, σk(n) =∑
d|n dk. These are the holomorphic
modular forms of Γ(1) whereas ∆(τ) is the cusp form of weight 12 of Γ(1),
well known as Ramanujan’s Delta function.
Knowing the hauptmodul for G is enough to recover all scalar-valued auto-
morphic and quasi-automorphic (defined below) forms of any G , see [9] where
this is done explicitly for triangle groups . The scalar-valued (quasi-) auto-
morphic forms contained in the following Lemma (taken from [5]) are such
examples which will be needed later in this thesis .
25
Lemma 2.2.15. Let G be a genus-0 Fuchsian group of the first kind, and
z ∈ EG∪ C
G. Then,
1. Ramanujan G-Delta forms: there exists some ∆(c)G
(τ) ∈ M(1) nonzero
everywhere except at the G-orbit of c with weight kc = 2 · lcmm1, · · · ,ml
where l = ||EG|| and lcm is taken over the orders of all the elliptic fixed points .
In case of an elliptic fixed point z = ωr, the weight of ∆(z)G
(τ) is the lcm of the
orders of all the elliptic fixed points except the order of the elliptic fixed point
ωr.
2. scalar-valued quasi-automorphic forms: define
EG
(2,z)(τ) =
1
2πi
1
∆(z)
G
d∆(z)
G
dτ=
qz∆(z)
G
d∆(z)
G
dqz.
Then EG
(2,z)(τ) is holomorphic in H∗
G, and E
G
(2,z)(τ) vanishes at any cusp c 6= z.
When z ∈ CG
,
EG
(2,z)(z) =
kz2·(n + l− 2−
∑j=1
1
mj
).
Moreover, EG
(2,z)(τ) is a quasi-automorphic form of weight 2 and depth 1 for
G: i.e. for all γ = ±(acbd
)∈ G and τ ∈ H,
EG
(2,z)(γ · τ) =
kzc
2πi(cτ + d)E
G
(2,z)(τ) + (cτ + d)2E
G
(2,z)(τ) . (2.2.10)
Example 2.2.16. Consider G = Γ(1) and z = ∞ then EG
(2,z)(τ) := E2(τ) =
1∆(τ)· q d
dq∆(τ). Using the series expansion of ∆(τ) from example 2, E2(τ) =
1 − 24q − 72q2 − · · · and E2(z) = 1. For other groups G, their ∆ and E2 at
cusp ∞ are described in the table 2.2 below.
2.3 Vector-valued automorphic forms
Throughout this chapter, G will denote a Fuchsian group of the first kind with
a cusp at ∞, unless otherwise mentioned. More precisely, as long as G has at
26
G c qc ∆c
G(τ) E
G
(2,c)(τ) E
G
(2,c)(c)
Γ(1) ∞ exp(2πiτ) q − 24q2 + 252q3 − 1472q4 + · · · 1− 24q − 72q2 − · · · 1
Γ0(2) ∞ exp(2πiτ) q + 8q2 + 28q3 + 64q4 + · · · 1 + 8q − 8q2 + 32q3 + · · · 1
Γ0(3) ∞ exp(2πiτ) q2 + 6q3 + 27q4 + 80q5 + · · · 2 + 6q + 18q2 − 30q3 + · · · 2
Γ(2) ∞ exp(πiτ) q + 4q 3 + 6q 5 + 8q 7 + · · · 1 + 8q + 24q 2 + 32q 3 + · · · 1
Table 2.2: Scalar-valued automorphic and quasi-automorphic forms of G
least one cusp then that cusp can (and will) be moved to∞ without changing
anything, simply by conjugating the group by the matrix Ac = ±(c0−1
1
)∈
PSL2(R) if c ∈ R is a cusp of G . Lemma 2.2.4 explains the minor way
this changes the automorphic forms . Roughly speaking a vvaf for G of any
weight w ∈ 2Z with respect to a multiplier ρ is a meromorphic vector-valued
function X : H→ Cd which satisfies a functional equation of the form X(γτ) =
ρ(γ)(cτ +d)wX(τ) for every γ = ±(acbd
)∈ G and is also meromorphic at every
cusp of G . The multiplier ρ is a representation of G of arbitrary rank d and is
an important ingredient in the theory of vvaf . This thesis classify the vvaf of G
of any even integer weight w with respect to a generic kind of multiplier which
we call an admissible multiplier . This amounts to little loss of generality .
This is defined in the following
Definition 2.3.1 (Admissible Multiplier). Let G be any Fuchsian group of
the first kind with a cusp at∞ and ρ : G→ GLd(C) be a rank d representation
of G. We say that ρ is an admissible multiplier of G if it satisfies the following
properties :
1. ρ(t∞) is a diagonal matrix, i.e. there exists a diagonal matrix Λ∞ ∈ Md(C)
such that ρ(t∞) = exp(2πiΛ∞) and Λ∞ will be called an exponent matrix of
cusp ∞. From now on the exponent matrix Λ∞ will be denoted by Λ.
2. ρ(tc) is a diagonalizable matrix for every c ∈ CG\∞, i.e. there exists an
invertible matrix Pc ∈ GLd(C) and a diagonal matrix Λc ∈ Md(C) such that
P−1c ρ(tc)Pc = exp(2πiΛc). Λc will be called an exponent matrix of cusp c.
27
Note 2.3.2. Note that each exponent Λc for every c ∈ CG
, is defined only up
to changing any diagonal entry by an integer and therefore Λhol
cis defined to
be the unique exponent satisfying 0 ≤(Λ
hol
c
)ξξ< 1 for all 1 ≤ ξ ≤ d .
Remark 2.3.3.
1. Dropping the diagonalizability does not introduce serious complications . The
main difference is the coefficient f(z)
[n] in qz -expansions become polynomials in
τ . A revealing example of such a vvaf is X(τ) =(τ1
)of weight w = −1 for any
G with respect to the multiplier ρ which is the defining representation of G .
2. Obviously, if ρ(t∞) was also merely diagonalizable , ρ could be replaced with an
equivalent representation satisfying the assumption 1 of the multiplier system .
Thus in this sense , assumption 1 is assumed without the loss of generality for
future convenience. Since almost every matrix is diagonalizable, the generic
representations are admissible. For example: the rank 2 admissible irreducible
representations of Γ(1) fall into 3 families parameterized by 1 complex param-
eter, and only 6 irreps are not admissible.
3. The reason for assumptions 1 and 2 in the definition 2.3.1 of the multiplier
system is that any vvaf X(τ) for ρ will have qc-expansions , as explained by
Corollary 2.3.5.
The meromorphic vvaf is defined in the following
Definition 2.3.4. Let G be any Fuchsian group of the first kind, w ∈ 2Z
and ρ : G → GLd(C) be any rank d admissible multiplier of G. Then a
meromorphic vector-valued function X : H → Cd is a meromorphic vvaf of
weight w of G with respect to multiplier ρ, if X(τ) has finitely many poles in
FG∩H and has the following functional and growth behaviour :
1. Functional behaviour
X(γτ) = ρ(γ)j(γ, τ)wX(τ), ∀γ ∈ G & ∀τ ∈ H,
28
2. Moderate growth behaviour
Every component of X(τ) is of moderate growth at every cusp c of G .
We say X(τ) is “meromorphic at c” if Xξ(τ) has moderate growth at c for
every 1 ≤ ξ ≤ d .
Corollary 2.3.5. Let G be any Fuchsian group of the first kind and ρ : G→
GLd(C) be an admissible multiplier. Then for any vector-valued meromorphic
automorphic form X(τ) of weight w ∈ 2Z with respect to ρ, the following is
true
1. at cusp ∞ the Fourier series
q−ΛX(τ) =
∞∑n=−M
X[n] qn, where q = exp
(2πiτ
h
),
converges absolutely in the region Im(τ) > Y for some nonnegative real number
Y and some integer M ,.
2. at any cusp c ∈ CG\∞ the expansion
(τ − c)wP−1
c q−Λc
cPc X(τ) =
∞∑n=−Mc
Xc
[n] qn
c, where qc = exp
(2πiA−1
c τ
hc
)
converges absolutely in the region Im(A−1
c τ) > Yc for some nonnegative real
number Yc and some integer Mc .
Proof. Since G∞ = 〈t∞ := ±(
10h1
)= th〉 and ρ(th) = exp(2πiΛ), by the
functional equation satisfied by X(τ) we get X(t∞τ) = ρ(t∞)X(τ) which is
X(τ + h) = exp(2πiΛ)X(τ). Therefore, q −ΛX(τ) is periodic with period h.
Also, this implies that Xξ(τ + h) = exp(2πiΛξ) Xξ(τ), ∀1 ≤ ξ ≤ d. Also,
∀ξ, Xξ(τ) is a scalar-valued meromorphic function which satisfies the growth
condition at ∞ . Then applying Theorem 2.2.5 at each component of X(τ) we
get Xξ(τ) = qΛξ ∑
n=−M Xξ [n] qn
. This gives the desired Fourier expansion
29
of X(τ) with respect to the cusp ∞ . This proves part 1 . For part 2 , using
Lemma 2.2.4 along with Theorem 2.2.5 applied to each component of X(τ)
gives the desired expansion .
Note 2.3.6. In Corollary 2.3.5 , Yc can be taken maxξ(Yξ
c) for every c ∈ C
G,
and if X(τ) is holomorphic in H then we may choose Yc = 0 . Also , if X(τ) is
holomorphic at c then the integer Mc can be taken so that Re(Λc) ≥Mc .
Definition 2.3.7 (Cuspidal behaviour). Note that the growth behaviour
in Definition 2.3.4 can be replaced by the following cuspidal behaviour of the
weight w meromorphic vvaf X(τ) of G :
1. at the cusp ∞:
X(τ) = q Λ
∞∑n=−M
X[n] qn , X[n] ∈ Cd
2. at the cusp c(6=∞):
X(τ) = (τ − c)−wPcqΛcP−1
c
∞∑n=−Mc
Xc
[n] qn
c, Xc
[n] ∈ Cd .
The reason why for qc-expansions imply moderate growth follows because
X(qc) = q Λcc
∞∑n=−M
X[n] qn
c
will have a removable singularity at qc = 0 and therefore will be meromorphic
at qc = 0 .
Following this weakly holomorphic and holomorphic vvaf of even integer
weight with respect to an admissible multiplier is now defined in the following
Definition 2.3.8. Let G be any Fuchsian group of the first kind , ρ be an
admissible multiplier of G of rank d and w ∈ 2Z . Then
1. A meromorphic vvaf X(τ) is said to be weakly holomorphic vvaf for G of
weight w and multiplier ρ if X(τ) is holomorphic throughout H . Let M!w(ρ)
30
denote the set of all such weakly holomorphic vvaf for G of weight w and
multiplier ρ .
2. X(τ) ∈M!w(ρ) is called a holomorphic vvaf if X(τ) is holomorphic through-
out H∗G
. Let Hw(ρ) denote the set of all such holomorphic vvaf for G of weight
w and multiplier ρ .
Remark 2.3.9. Let RG
denote the ring of scalar-valued weakly holomorphic
automorphic functions of G. Then RG
:=M!0(1) = C[J
c1
G, . . . , J
cn
G], where n =
||CG|| and J
c
Gis the normalized hauptmodul of G with respect to c ∈ C
G. There
is an obvious RG
-module structure on M!w(ρ) . Without loss of generality we
may assume that c1 =∞.
2.4 An example of vector-valued modular form
Two examples of a genus-0 Fuchsian group of the first kind are the principal
congruence subgroup Γ(1) and Γ(2) . Γ(2) is a subgroup of index 6 of Γ(1).
Following equation (1.2.3)
Γ(1) =⟨t, s, u := t−1s
∣∣ s2 = 1 = u3⟩≈⟨t, s⟩≈ Z2 ∗ Z3 ,
and
Γ(2) =⟨t∞, t0, t−1
∣∣ t−1t0t∞ = 1⟩≈⟨t∞, t−1
⟩≈ Z ∗ Z ,
where s, t are same as defined earlier and t∞ = ±(
10
21
)= t2, t0 = ±
(1−2
01
)=
st2s−1 and t−1 = (t−1s)t2(t−1s)−1 = ±(
3−2
2−1
). Γ(2) has three inequiva-
lent cusps, all of cusp width 2 therefore the Fourier expansion of any weakly
holomorphic vvaf X(τ) will have qc-expansion where q = exp(πiτ) , q0 =
exp(πis−1τ) = exp(−πiτ
) and q−1 = exp(πi(t−1s)−1τ) = exp(− πiτ+1
) whereas
Γ(1) has one inequivalent cusp ∞ of cusp width 1 therefore the Fourier ex-
pansion of any weakly holomorphic vvaf X(τ) will have q-expansion where
q = q .
31
Consider the following well known θ-functions
θ2(τ) =∞∑
n=−∞
q (n+ 12
)2
= 2q 1/4(1 + q 2 + q 6 + · · ·
)θ3(τ) =
∞∑n=−∞
q n2
= 1 + 2q + 2q 4 + 2q 9 + · · ·
θ4(τ) =∞∑
n=−∞
(−1)nq n2
= 1− 2q + 2q 4 − 2q 9 + · · · (2.4.1)
where q = exp(πiτ). These are the scalar-valued modular forms of Γ(2) of
weight 1/2 , for complete definition about half integer weight modular forms
see chapter 4 of [28] . Consider the Dedekind eta function
η(τ) = q1/24Π∞n=1(1− qn) = q 1/12Π∞n=1(1− q 2n)
= q1/24 − q25/24 − q49/24 + q121/24 + · · ·
= q 1/12 − q 25/12 − q 49/12 + q 121/12 + · · · (2.4.2)
η(τ) is a weight 1/2 scalar-valued modular form of Γ(1) . Thus
θ2(τ)
η(τ),θ3(τ)
η(τ),θ4(τ)
η(τ)
are scalar-valued modular functions of Γ(2). Now consider
X(τ) =1
η(τ)
θ2(τ)θ3(τ)θ4(τ)
.
Then X(τ) is a vector-valued modular function of Γ(1) and multiplier ρ where
ρ : PSL2(Z)→ GL3(C) is a rank 3 representation of Γ(1). Recall the transfor-
mation properties of θ2(τ), θ3(τ), θ4(τ), η(τ) under the matrices s, t
θ2(τ + 1) = exp
(πi
4
)θ2(τ) , θ2
(− 1
τ
)=
√τ
iθ4(τ) ,
θ3(τ + 1) = θ4(τ) , θ3
(− 1
τ
)=
√τ
iθ3(τ) ,
32
θ4(τ + 1) = θ3(τ) , θ4
(− 1
τ
)=
√τ
iθ2(τ) ,
η(τ + 1) = exp
(πi
12
)η(τ) , η
(− 1
τ
)=
√τ
iη(τ) ,
Using the above transformation properties in X(τ) the multiplier ρ is de-
fined by
ρ(s) = S :=
0 0 10 1 01 0 0
, and
ρ(t) = T :=
exp(πi6
) 0 00 0 exp(− πi
12)
0 exp(− πi12
) 0
.
To read the Fourier series expansion of X(τ) we have to find the equivalent
admissible representation ρ′ of ρ and the corresponding exponent matrix Λ.
Consider ρ′ = P−1ρP where
P =
1 0 00 1 −10 1 1
.
Then ρ′ is an admissible multiplier of Γ(1) which yields a vector-valued mod-
ular function PX(τ) . In this case the exponent Λ can be obtained from the
expression
ρ′(t) = P−1TP =
exp(πi6
) 0 00 exp(− πi
12) 0
0 0 − exp(− πi12
)
= exp(2πiΛ)
where
Λ =
112
0 00 − 1
240
0 0 1124
.
33
Chapter 3
Construction of Vector-ValuedAutomorphic Forms
The existence and construction of a vvaf for any arbitrary G , and any ρ of
finite image can be established by lifting a scalar-valued automorphic form of
the finite index subgroup ker(ρ) of G . In other words, the following question
has been partially answered : For which representations ρ of a given G, is
there a nontrivial vvaf exist ?
In this chapter vvaf for any ρ of finite image are explicitly constructed .
This is done by lifting (inducing) a scalar-valued automorphic form from the
kernel of ρ up to G . Because this is a fundamental question , more examples
are included than in later chapters . Section 3.2 is devoted to the examples.
One of the advantages of vvaf is that (unlike scalar-valued modular forms) it
is closed under inducing. For example θ2(τ) and η(τ) are scalar-valued modular
forms of weight 1/2 of Γ(2). However, θ2(τ) is not a scalar-valued modular
form of Γ(1), i.e. θ2(τ), η(τ) ∈ M!1/2(Γ(2), 1) but θ2(τ) /∈ M!
1/2(Γ(1), 1) . But
their lifts θ2(τ), η(τ) are vvmf of Γ(1) with respect to the rank 6 multiplier
1 = IndΓ(1)
Γ(2)(1).
This chapter begins with developing the theory of induction of vvmf. As
throughout this thesis, we work with even integer weights - the same construc-
tion works for fractional weights but extra technicalities obscure the underlying
34
ideas . It is also shown below that the spaces M!w(Γ(1), 1) and M!
w(Γ(1), 1)
are naturally isomorphic modules over the ring RΓ(1)
.
3.1 Lift of an automorphic form
Let G be any genus-0 Fuchsian group of the first kind with a cusp at ∞ and
H be any finite index subgroup of G . In this section the relation between
weakly holomorphic vvaf of H and G is established . Let n be the number of
inequivalent cusps and l be the number of inequivalent elliptic fixed points of
G , and let mj, 1 ≤ j ≤ l denote the orders of the elliptic fixed points . Recall
the cusp form ∆G
of weight 2 · L where L = lcm[mj, 1 ≤ j ≤ l] from the
Lemma 2.2.15 . Think of it as the analogue for G of the cusp form ∆(τ) =
q∏∞
n=1(1 − qn)24 of Γ(1) of weight 12 = 2 · lcm[2, 3]. Like ∆(τ) , ∆G
(τ) is
holomorphic throughout H∗G
and nonzero everywhere except at the cusp ∞ .
Because ∆G
(τ) is holomorphic and nonzero throughout the simply con-
nected domain H it possesses a holomorphic logarithm log∆G
(τ) in H . For
any w ∈ C define ∆G
(τ)w = exp(w log∆G
(τ)) . Then ∆G
(τ)w is also holo-
morphic throughout H . A little work (see [7]) shows that it is a holomorphic
automorphic form . For any G , let ν : G → C× denote the multiplier of the
scalar-valued automorphic form ∆1
2L
G. It is calculated in [7] for any G . For
example , in case of G = Γ(1) the multiplier ν for any γ = ±(acbd
)∈ Γ(1) is
explicitly defined as follows:
ν(γ) =
exp[πi(a+d
12c)− 1
2−∑c−1
i=1ic(dic− bdi
cc − 1
2)] if c 6= 0
exp[πi(a(b−3)+312
)] if c = 0
Using the above technical information, the following is obtained
Lemma 3.1.1. For any w ∈ 2Z , M!w(ρ) and M!
0(ρ⊗ ν−w) are naturally iso-
morphic as RG
-modules, where the isomorphism is defined by X(τ) 7→ ∆G
(τ)− w
2LX(τ).
We now state the two theorems which are the main results and focus of
this section. The proofs will commense in a couple of pages.
35
Theorem 3.1.2. Let G be any genus-g Fuchsian group of the first kind and
H be any finite index subgroup of G, i.e. [G : H] = m. Then, if ρ is a rank d
admissible representation of H then the induced representation ρ = IndG
H(ρ) of
G of rank dm is also an admissible representation.
Theorem 3.1.3. Let G be a genus-0 Fuchsian group of the first kind and w
be an even integer. Let H be any finite index subgroup of G, i.e. [G : H] =
m < ∞, and ρ be a rank d admissible representation of H. Then there is a
natural RG
-module isomorphism betweenM!w(ρ) andM!
w(ρ) where the induced
representation ρ = IndG
H(ρ) is an admissible representation of G of rank dm.
Theorem 3.1.2 is an important tool to prove Theorem 3.1.3 which estab-
lishes the relation between weakly holomorphic vvaf of H and G. More im-
portantly, the isomorphism between M!w(ρ) and M!
w(ρ) is given by X(τ) 7→(X1(γ−1
1 τ),X1(γ−12 τ), · · · ,X1(γ−1
m τ)
)t
, where γ1, · · · , γm are distinct coset
representatives of H in G.
Before giving the proofs of Theorems 3.1.2 and 3.1.3 let us recall why
ρ = IndG
Hρ defines a representation . Write G = γ1H ∪ γ2H ∪ · · · ∪ γmH .
Without loss of generality we may assume that γ1 = 1 . The representation
ρ : H→ GLd(C) can be extended to a function on all of G , i.e. ρ : G→ GLd(C)
by setting ρ(x) = 0 ,∀x /∈ H. The induced representation ρ = IndG
Hρ is defined
by
ρ(x) =
ρ(γ−1
1 xγ1) ρ(γ−11 xγ2) . . . ρ(γ−1
1 xγm)ρ(γ−1
2 xγ1) ρ(γ−12 xγ2) . . . ρ(γ−1
2 xγm)...
.... . .
...ρ(γ−1
m xγ1) ρ(γ−1m xγ2) . . . ρ(γ−1
m xγm)
, ∀x ∈ G. (3.1.1)
Due to the extension of ρ for any x ∈ G and ∀1 ≤ i ≤ m there exists a unique
1 ≤ j ≤ m such that ρ(γ−1i xγj) 6= 0 . Therefore, exactly one nonzero d × d
block appear in every row and every column of (3.1.1) .
Before going into the details of the proofs of the Theorems 3.1.2 and 3.1.3,
let us confirm that ρ does not depend on the choice of the coset representatives.
36
Lemma 3.1.4. Let G,H be as in Theorem 3.1.2. Let R = g1, · · · , gm,
R = γ1, · · · , γm be two different coset representatives of H in G. Let ρ : H→
GLd(C) be an admissible representation . Then the induced representation ρ =
IndG
H(ρ) and ρ = Ind
G
H(ρ) with respect to the coset representatives R and R
respectively are equivalent representations of G .
Proof. For each 1 ≤ i ≤ m there exists xi ∈ H such that γi = gixi up to
reordering gi’s and γi’s . Then ρ(g) = D−1ρ(g)D for every g ∈ G where
block diagonal matrix D = Diag(ρ(x1), · · · , ρ(xm)
)is the conjugating matrix
between ρ and ρ .
Lemma 3.1.5. Let G and H be as in Theorem 3.1.2. Fix any cusp c ∈ CG
and let c1, · · · , cnc be the representatives of the H-inequivalent cusps which are
G-equivalent to the cusp c, so
G · c = ∪nci=1H · ci. (3.1.2)
Let kc be the cusp width of c in G and hci be the cusp width of ci in H. Write
hi =hcikc∈ Z, Gc = 〈tc〉 and Ai(c) = ci where Ai ∈ G. Then m =
∑i hi and
coset representatives of H in G can be taken to be gij = tjcA−1
i for all i and
0 ≤ j < hi.
Proof. Let g be any element of G. Because of the decomposition (3.1.2) there
is a unique i such that g−1c = γ · ci for some γ ∈ H . Then Aigγ fixes ci and
so it equals AitjcA−1i for some j ∈ Z . Recall that tc = Act
kcA−1c where Ac =(
c1−10
)∈ PSL2(R) such that Ac(∞) = c . Note that Ait
jcA−1i and Ait
j+hic A−1
i =
AitjcA−1i Ait
hic A
−1i lie in the same coset of H because hi is the least positive
integer such that Aithic A
−1i ∈ H . Moreover Hci = 〈ti = Ait
hic A
−1i 〉 and ti =
(AiAc)tkchi(AiAc)
−1 . Thus we can restrict 0 ≤ j < hi . This means that every
coset gH of H in G contains an element of the form tjcA−1i := gij for some
0 ≤ j < hi and some 1 ≤ i ≤ nc . This implies that m ≤∑nc
i=1 hi .
37
In addition, for all the ranges of i, j as above the cosets gijH are distinct .
Suppose this is not true , then for some i, j, k, l in the range as above let
gijH = gklH, i.e. tjcA−1i H = tlcA
−1k H, i.e. Akt
j−lc A−1
i ∈ H . Then Aktj−lc A−1
i · ci =
ck . Hence ci and ck are H-equivalent cusps which implies that i = k. This
implies that Aitj−lc A−1
i ∈ Hci . hi is the smallest positive integer for which
Aithic A
−1i ∈ H and 0 ≤ j, l < hi therefore 0 ≤ |j − l| < hi and Ait
j−lc A−1
i ∈ H
is possible only when j − l = 0. This implies that j = l. Hence for i, j, k, l
ranged as above gijH = gklH requires i = k and j = l . Thus∑nc
i=1 hi ≤ m and
we are done.
Example 3.1.6. As an illustration of Lemma 3.1.5, Table 3.1 shows data for
certain finite index subgroups H of G = Γ(1). In this case CG
= ∞, kc = 1
and ci ∈ CH, hi = hci .
H m CH
hi
Γ0(2) 3 0,∞ 2, 1
Γ(2) 6 −1, 0,∞ 2, 2, 2
Γ0(3) 4 0,∞ 3, 1
Γ(3) 12 −1, 0, 1,∞ 3, 3, 3, 3
Γ0(4) 6 − 12 , 0,∞ 1, 4, 1
Γ(4) 24 −1,− 12 , 0, 1, 2,∞ 4, 4, 4, 4, 4, 4
Γ0(5) 6 0,∞ 5, 1
Γ0(8) 12 − 14 ,−
12 , 0,∞ 1, 2, 8, 1
Table 3.1: Relation between index and cusp widths of H in G.
Proof of Theorem 3.1.2. We need to show that for each cusp c of G, ρ(tc) is
diagonalizable. Let c be any cusp of G. Due to Lemma 3.1.4, it is sufficient to
choose the coset representatives as in Lemma 3.1.5. Then ρ(tc) can be written
38
in block form as
ρ(g−1ij tcgkl) =
I , if i = k and j 6= hi − 1
ρ(ti) , if i = k, j = 0 and l = hi − 1
0 , otherwise
, (3.1.3)
where i, j, k, l range as in Lemma 3.1.5 I is the identity matrix of order d× d
and ti = Aithic A
−1i is the generator of the stabilizer Hci in H . Thus ρ(tc) is in
block form , one for each i of order dhi × dhi as shown in the matrix given in
the Table 3.2 . Also , for every 1 ≤ i ≤ nc ρ(ti) := Ti is diagonalizable by the
admissibility hypothesis . So , for every i let v(i,k), 1 ≤ k ≤ d , be a basis of
eigenvectors respectively with eigenvalues λ(i,k) of ρ(ti) . Let ζ be any hth
i root
of unity and let V(i,k,ζ) be the column vector of order dm×1, defined as follows.
Its nonzero entries appear only in the ith block of order dhi× 1. That block is
given by(λ
1/hi
(i,k)v(i,k), ζλ
1/hi
(i,k)v(i,k), ζ
2λ1/hi
(i,k)v(i,k), · · · , ζhi−1λ
1/hi
(i,k)v(i,k)
)t. From (3.1.3)
it is clear that V(i,k,ζ) is an eigenvector of ρ(tc) with eigenvalue ζλ1/hi
(i,k). Hence,
for every i there are exactly dhi eigenvectors of order dm × 1 formed with
respect to the dhi eigenvalues ζλ1/hi
(i,k)for ζ = exp
(2πijhi
)with 0 ≤ j < hi . Since
V(i,k,ζ)
are linearly independent , ρ(tc) is indeed diagonalizable .
Recall from the definition of admissible multiplier system that the exponent
Λc for any cusp c of G is a diagonal matrix such that P−1c ρ(tc)Pc = exp(2πiΛc)
for some diagonalizing matrix Pc.
Corollary 3.1.7. For any cusp c of G an exponent Ωc of the induced represen-
tation IndG
H(ρ) , of a rank d admissible representation ρ of H , has components
(Λi )kk+j
hi, where 1 ≤ i ≤ nc, 0 ≤ j < hi , 1 ≤ k ≤ d and Λi is an exponent of ρ
at cusp ci .
Proof. From Theorem 3.1.2 , for every c ∈ CGρ(tc) is a diagonalizable matrix
with the eigenvaluesξλ
1/hi
(i,k)
∣∣ 1 ≤ i ≤ nc, 1 ≤ k ≤ d, and ξ = exp
(2πij
hi
), 1 ≤ j < hi
.
39
ρ(t
c):=
Tc
=
00
0···
0T
10
00···
00
······
00
0···
00······
00
0···
00
0I
0···
00
00
0···
00
······
00
0···
00······
00
0···
00
00
I···
00
00
0···
00
······
00
0···
00······
00
0···
00
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
00
0···
I0
00
0···
00
······
00
0···
00······
00
0···
00
00
0···
00
00
0···
0T
2······
00
0···
00······
00
0···
00
00
0···
00
I0
0···
00
······
00
0···
00······
00
0···
00
00
0···
00
0I
0···
00
······
00
0···
00······
00
0···
00
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
00
0···
00
00
0···
I0
······
00
0···
00······
00
0···
00
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
00
0···
00
00
0···
00
······
00
0···
0T
i······
00
0···
00
00
0···
00
00
0···
00
······
I0
0···
00······
00
0···
00
00
0···
00
00
0···
00
······
0I
0···
00······
00
0···
00
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
00
0···
00
00
0···
00
······
00
0···
I0······
00
0···
00
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
00
0···
00
00
0···
00
······
00
0···
00······
00
0···
0T
nc
00
0···
00
00
0···
00
······
00
0···
00······
I0
0···
00
00
0···
00
00
0···
00
······
00
0···
00······
0I
0···
00
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
. . .. . .
00
0···
00
00
0···
00
······
00
0···
00······
00
0···
I0
Tab
le3.
2:T
he
shap
eof
the
mat
rixρ(t
c)
40
For all i there exists a diagonalizing matrix Pi such that P−1i ρ(ti)Pi = exp(2πiΛi),
where the exponent matrix Λi = Diag(Λi1, . . . ,Λid). Therefore, λ(i,k)
= exp(2πiΛik
).
This implies that ξλ1/hi
(i,k)= exp
(2πi
Λik
+j
hi
). Hence the exponent Ωc of ρ(tc)
has dm diagonal entries of the form(Λi )kk+j
hi.
A formal proof of Theorem 3.1.2 in complete generality made the argu-
ment seem more complicated than it really is. Thus, this simple idea will be
illustrated with examples in Subsection 3.2.1.
Proof of Theorem 3.1.3. ρ : G → GLdm(C) is an induced representation of G
of an admissible representation ρ : H→ GLd(C) of H . For any representation
ρ : H→ GLd(C) we wish to find an isomorphism betweenM!w(ρ) andM!
w(ρ) .
Lemma 3.1.1 gives M!w(ρ) ≈ M!
0(ρ ⊗ ν−w
H) using the isomorphism X(τ) 7→
∆− w
2L
GX(τ) , where ν
−w
His the restriction of ν
−w
Gto H . Similarly , M!
w(ρ) ≈
M!0(ρ⊗ν−w
G). Therefore to show a one-to-one correspondence betweenM!
w(ρ)
and M!w(ρ) it is enough to establish a one-to-one correspondence between
M!0(ρ⊗ ν−w
H) andM!
0(ρ⊗ ν−wG
) . Note that Ind(ρ⊗ ν−wH
) = IndG
H(ρ)⊗ ν−w
G. In
other words , it is enough to show that the following diagram commutes .
M!w(ρ)
≈//M!0(ρ⊗ ν
− w2L )
M!w(ρ)
≈//M!0(ρ⊗ ν
− w2L )
Let X(τ) ∈M!0(ρ) then define X(τ) =
(X(γ−1
1 τ),X(γ−12 τ), . . . ,X(γ−1
m τ)
)t
.
We claim that X(τ) ∈M!0(ρ
′) . Since every component is weakly holomorphic
therefore X(τ) is also weakly holomorphic . Hence it suffices to check the
functional behaviour of X(τ) under G , i.e. ∀γ = ±(acbd
)∈ G, X(γτ) =
ρ(γ)X(τ) . Consider X(γτ) for γ ∈ G , then by definition
X(γτ) =
X(γ−1
1 γτ)X(γ−1
2 γτ)...
X(γ−1m γτ)
=
X(γ−1
1 γγj1γ−1j1τ)
X(γ−12 γγj2γ
−1j2τ)
...X(γ−1
m γγjmγ−1jmτ)
=
ρ(γ−1
1 γγj1)X(γ−1j1τ)
ρ(γ−12 γγj2)X(γ−1
j2τ)
...ρ(γ−1
m γγjm)X(γ−1jmτ)
41
This implies that X(τ) ∈ M!0(ρ) . Conversely, for any X(τ) ∈ M!
0(ρ) define
X(τ) by taking the first d components of X(τ), i.e. X(τ) =
(X1(τ), . . . , Xd(τ)
)t
.
Since γ1 = 1 therefore ∀ γ ∈ H, ρ(γ) will appear as the first d× d block in the
dm×dm matrix ρ(γ) such that all the other entries in the first row and column
are zeros and first d components on both sides of X(γτ) = ρ(γ)X(τ), ∀γ ∈ H
give the required identity X(γτ) = ρ(γ)X(τ), ∀γ ∈ H. To see whether thus
defined X(τ) will have Fourier expansion at every cusp of H, first notice that
CH
= γ−1j c | c ∈ C
G& 1 ≤ j ≤ m and C
G⊂ C
H. Since ∀j, γj /∈ H we obtain
X(γjτ) =
(X1(γjτ), X2(γjτ), . . . , Xd(γjτ)
)t
.
Since for any c ∈ CG
, c and γ−1j c are G-equivalent cusps, every component of
X(γjτ) inherits the Fourier expansion at cusp c from the Fourier expansion of
X(γjτ). Hence, X(τ) is a weakly holomorphic vvaf and has Fourier expansion
at every cusp of G.
3.2 Examples
3.2.1 Exponent matrix of a lift
1. Let G = Γ(1),H = Γ0(2) and K = Γ(2). Recall the definition of Γ(2) and the
notation used in this example from example 2.4. H and K both are congruence
subgroups of G of index 3 and 6 respectively and
CG
= ∞, CH
= ∞, 0, CK
= ∞, 0,−1.
• Since [G : K] = 6 write G = K ∪ tK ∪ sK ∪ tsK ∪ stK ∪ tstK where the coset
representatives are I = ±(
10
01
), s = ±
(01−1
0
), t = ±
(10
11
), st = ±
(01−1
1
), ts =
±(
11−1
0
), tst = ±
(11
01
). Let ρ : H→ GLd(C) be any admissible representation
and write ρ(tc) = Tc for the cusp c = −1, 0,∞. Since ρ is an admissible
representation of K, there exist diagonal matrices Λ,Λ0,Λ−1 and diagonalizing
matrices P0,P−1 ∈ GLd(C) such that T∞ = exp(2πiΛ), P0T0P−10 = exp(2πiΛ0)
42
and P−1T−1P−1−1 = exp(2πiΛ−1) are diagonal matrices, where Λ,Λ0, and Λ−1
are called respectively the exponent matrices of cusps ∞, 0 and −1. Then,
from the definition of induced representation of ρ = IndG
H(ρ) : G → GL6d(C)
defined by the equation (3.1.1)
ρ(t) := T∞ =
0 T∞ 0 0 0 0I 0 0 0 0 00 0 0 T0 0 00 0 I 0 0 00 0 0 0 0 T−1
0 0 0 0 I 0
To assure the admissibility of ρ we need to show that ρ(t) is diagonalizable.
From Theorem 3.1.2 it follows that ρ(t) is diagonalizable and from Corol-
lary 3.1.7, the exponent matrix of cusp ∞ of G with respect to the admissible
representation ρ is
Ω = Diag
(Λ
2,1 + Λ
2,Λ0
2,1 + Λ0
2,Λ−1
2,1 + Λ−1
2
).
• Since [G : H] = 3 write G = H ∪ sH ∪ tsH where the coset representatives are
I = ±(
10
01
), s = ±
(01−1
0
), ts = ±
(11−1
0
). Following the definition of Γ0(N)
from equation (1.2.2) for N = 2
Γ0(2) ∼=⟨t∞, t0, tω
∣∣ t2ω = 1 = tωt0t∞⟩ ∼= ⟨tω, t∞ ∣∣ t2ω = 1
⟩ ∼= Z2 ∗ Z,
where t∞ = t = ±(
10
11
), t0 = st2s−1 = ±
(1−2
01
)and tw = ±
(1−2
1−1
). H has
two cusps 0 and ∞ and an elliptic fixed point ω = −1+i2
of order 2 . Let ρ :
H→ GLd(C) be any admissible representation. From the definition of induced
representation ρ = IndG
H(ρ) : G→ GL3d(C) defined by the equation (3.1.1)
ρ(t) := T∞ =
T∞ 0 00 0 T0
0 I 0
.
From Theorem 3.1.2, it follows that ρ(t) is diagonalizable and from Corol-
lary 3.1.7, the exponent matrix of cusp ∞ of G with respect to the admissible
43
representation ρ is
Ω = Diag
(Λ,
Λ0
2,1 + Λ0
2
)2. Consider G = Γ(1) and H = Γ0(3) then [G : H] = 4 and G = H ∪ sH ∪
tsH ∪ t2sH where the coset representatives are I, s, ts are same as before and
t2s = ±(
21−1
0
). Following the definition of Γ0(N) from equation (1.2.2) for
N = 3
Γ0(3) ∼=⟨t∞, t0, tω
∣∣ t3ω = 1 = tωt0t∞⟩ ∼= ⟨tω, t∞ ∣∣ t3ω = 1
⟩ ∼= Z3 ∗ Z,
where t∞ = t = ±(
10
11
), t0 = st3s−1 = ±
(1−3
01
)and tw = ±
(2−3
1−1
). H has
two cusps 0 and ∞ and an elliptic fixed point ω = −3+i√
36
of order 3 . Let ρ :
H→ GLd(C) be any admissible representation . From the definition of induced
representation ρ = IndG
Hρ : G→ GL4d(C) defined by the equation (3.1.1)
ρ(t) := T∞ =
T∞ 0 0 00 0 0 T0
0 I 0 00 0 I 0
.
Similarly ρ(t) is diagonalizable and the exponent matrix of cusp∞ with respect
to the admissible representation ρ is
Ω = Diag
(Λ,
Λ0
3,1 + Λ0
3,2 + Λ0
3
).
3.2.2 An easy construction of vvmf
This section provides an explaination of the above ideas by constructing a rank
2 and 3 vvaf of Γ0(2) and Γ(1) by lifting an appropriate scalar-valued modular
form of index 2 and 3 subgroups of Γ0(2) and Γ(1) respectively.
44
Consider
G = Γ(1) ≈ Z2 ∗ Z3
index−3
H = Γ0(2) ≈ Z ∗ Z2
index−2
K = Γ(2) ≈ Z ∗ Z
For the definition of G,H,K and their set of inequivalent cusps, see Sub-
section 2.1.2 and Example 2.4 . Write H = K ∪ tK. Hauptmoduls and the
normalized hauptmodul of H with their values at the elliptic fixed point ω and
cusps ∞ and 0 are recorded in the table 3.3, where J(τ) := J∞
H(τ) = q−1 +
τ J(τ) j(τ) z(τ)
∞ ∞ ∞ ∞
0 24 1 0
ω -40 0 1
Table 3.3: Hauptmoduls of Γ0(2)
276q−2048q2 + · · · is the normalized hauptmodul and z(τ) := z∞
H(τ) = 24−J(τ)
64,
j(τ) := j∞
H(τ) = 1 − z(τ) = 40+J(τ)
64are hauptmoduls of H with respect to the
cusp ∞. Let σ : K −→ C× be a trivial character, i.e. trivial one dimensional
representation of K. Consider ρ = IndH
K(σ) : G → GL2(C) to be the rep-
resentation induced by the character σ. Therefore, by definition of induced
representation defined by equation (3.1.1)
ρ(γ) =
(σ(I−1γI)
σ(t−1γI)
σ(I−1γt)
σ(t−1γt)
); ∀ γ ∈ Γ0(2) . (3.2.1)
This implies that
T∞ = ρ(t) =
(σ(t)
σ(I)
σ(t2)
σ(t)
)=
(0
σ(I)
σ(t2)
0
)=
(0
1
1
0
)T0 = ρ(t0) =
(σ(t0)
σ(t−1t0)
σ(t0t)
σ(t−1t0t)
)=
(σ(t0)
0
0
σ(t−1)
)=
(1
0
0
1
)45
Now, we record hauptmoduls and normalized hauptmodul of K with their
values at the cusps -1, 0 and ∞ in the table 3.4, where JK
(τ) = q−1/2
+
τ JK
(τ) jK
(τ) zK
(τ)
∞ ∞ ∞ ∞
0 8 1 0
−1 -8 0 1
Table 3.4: Hauptmoduls of Γ(2)
20q1/2 − 62q3/2
+ 216q5/2
+ . . . = q−1
+20 q − 62q3
+ · · · , with q = exp(2πiτ)
and q = q1/2 is the normalized hauptmodul of K and jK
(τ) =8+J
K(τ)
16=
θ43(τ)
θ42(τ)
=
116q−1
+ 12
+ 54q− 31
8q
3+ . . ., z
K(τ) = 1− j
K(τ) are the equivalent hauptmoduls
of K with respect to the cusp ∞.
Now consider X(τ) = zK
(τ), a weight 0 scalar-valued modular form of K ,
then X(τ) =
(X(τ),X(t−1τ)
)t
is a weight 0 rank 2 vvmf of G with respect
to an equivalent admissible multiplier ρ ′ = P−1ρ P of ρ given by the equa-
tion (3.2.1), where P =(
11−1
1
)and the exponent matrix Ω of ρ ′ is
(1/20
01
).
Similarly, we know that G = H ∪ sH ∪ (ts)H. Therefore if we consider a
trivial multiplier 1 : H → C× and the induced representation ρ = IndG
H(1) :
G → GL3(C) then for X(τ) = z(τ) X(τ) =
(X(τ),X(s−1τ),X((ts)−1τ)
)t
is a weight 0 rank 3 vvmf of Γ(1) with respect to an equivalent admissible
multiplier ρ ′ of ρ where, by using the definition of induced representation
defined by the equation (3.1.1) ,
T∞ = ρ(t) =
1(t) 0 00 0 1(t0)0 1(1) 0
=
1 0 00 0 10 1 0
and therefore ρ ′ = P−1ρ P where
P =
1 0 00 1 −10 1 1
.
In this case, an exponent matrix Ω of ρ ′ is the diagonal matrix Diag(1, 1, 1/2) .
46
3.3 Existence
Theorem 3.3.1. Let G be a Fuchsian group of the first kind and ρ : G →
GLd(C) be any admissible representation of finite image . Then there exists a
weakly holomorphic vector-valued automorphic function for G with multiplier
ρ , whose components are linearly independent over C .
Proof. First, note that if f(z) is any nonconstant function holomorphic in
some disc then the powers f(z)1, f(z)2, ... are linearly independent over C.
To see this let z0 be in the disc; it suffices to prove this for the powers of
g(z) = f(z) − f(z0), but this is clear from Taylor series expansion of g(z) =∑∞n=k(z − z0)nan where ak 6= 0 (k is the order of the zero at z = z0). In
particular, if f(z) is any nonconstant modular function for any Fuchsian group
of the first kind then its powers are linearly independent over C .
Moreover, suppose G,H are distinct Fuchsian groups of the first kind with
H normal in G with index m . Fix any τ0 ∈ H\EG such that all m points
γiτ0 are distinct where γi are m inequivalent coset representatives . Then there
is a modular function f(τ) for H such that the m points f(γiτ0) are distinct.
This is because distinct Fuchsian groups must have distinct sets of modular
functions. Define g(τ) =∏
i(f(γiτ) − f(γiτ0))i. Then g(τ) is also a modular
function for H, and manifestly the m functions g(γiτ) are linearly independent
over C (since they have different orders of vanishing at τ0).
Let H = ker(ρ). Then ρ defines a representation of the finite group
K = G/H, so ρ decomposes into a direct sum ⊕imiρi of K-irreps (mi is the
multiplicity in ρ of the irrep ρi of K) .
Suppose that the Theorem is true for all irreps ρi of K . Let Xi(τ) =
(Xi1(τ), · · · ,Xidi(τ))t be a vvmf for the ith-irrep of K with linearly independent
components . Changing basis , ρ can be written in the block-diagonal form (mi
47
blocks for each ρi) . Choose any nonconstant modular function f(τ) of G . Then
X(τ) = (f(τ)X1(τ), · · · , f(τ)m1X1(τ), f(τ)X2(τ), · · · , f(τ)m2X2(τ), ...)t
will be a vvmf for G with multiplier ρ (or rather ρ written in block-diagonal
form), and the components of X(τ) will be linearly independent over C .
So it suffices to prove the theorem for irreps of K . Let m = [G : H] = |K|
and write G = γ1H∪ γ2H∪ · · · ∪ γmH . Let g(τ) be the modular function for H
defined above which is such that the m functions g(γiτ) are linearly indepen-
dent over C . Induce g(τ) (which transforms by the trivial H-representation)
from H to a vvmf Xg(τ) of G ; by definition its m components Xg,i
(τ) = g(γi.τ)
are linearly independent over C . Inducing the trivial representation of H gives
the regular representation of K and the regular representation of a finite group
(such as K) contains each irrep (in fact with a multiplicity equal to the dimen-
sion of the irrep) . To find a vvmf for the K-irrep ρi find a subrepresentation of
regular representation equivalent to ρi and project to that component ; the re-
sulting vvmf (resulting from the projection applied to Xg(τ)) will have linearly
independent components .
Final remarks
In the next chapter another more general approach is provided to show the
existence of infinitely many vector-valued automorphic functions . This new
approach does not require the image of ρ to be finite . We will restrict to a
special class of genus-0 Fuchsian groups of the first kind called triangle groups
and to rank 2 , where we can be more explicit but the arguments are more
general . Eventually in chapter 7 the set of two weakly holomorphic vvaf for
these G and ρ will be constructed from this approach which generate the space
M!w(ρ) .
48
Chapter 4
Vector-Valued AutomorphicForms of Triangle groups - I
In this chapter we introduce triangle groups rigourously and give the details
about their normalized and regularized (defined in 4.1.1) hauptmoduls with
their first few coefficients , scalar-valued automorphic forms along with the
classification of their admissible multipliers . It is shown that there is a natural
connection between the rank 2 vvaf and hypergeometric as well as Riemann’s
differential equations . By exploring this connection it is explicitly shown that
there exist infinitely many rank 2 nearly holomorphic vvaf of any G with re-
spect to any admissible multiplier ρ . It is also shown that the admissible
multiplier for any G is equivalent to any monodromy representation associ-
ated to Riemann’s (equivalently to Papperitz’s) differential equation defined
on the Riemann sphere with three punctures—these punctures on the Rie-
mann sphere correspond to the three regular singular points of the associated
differential equation . The monodromy representation is defined on the first
fundamental group of G\(H − EG
) i.e. it is a representation of the free group
on two generators with respect to a fixed basis of the solution space of the
associated differential equation with regular singular points at 0 , 1 , and ∞ .
We begin our exploration first by defining the triangle groups .
49
4.1 Triangle groups
In 1856, Hamilton introduced the triangle groups in his paper on Icosian Calcu-
lus, where he presented the icosahedral group as the (spherical) (2,3,5) triangle
group. We are interested in the hyperbolic case: the symmetries of a tiling of
the hyperbolic plane H by congruent triangles .
By a triangle group G we mean a genus-0 Fuchsian group of the first
kind with exactly 3 distinct orbits of elliptic and parabolic fixed points, i.e.
|| EG∪ CG ||= 3 . Any such triangle group G can be realized (in many ways) as
a subgroup of PSL2(R), the isometries of H, in such a way that G\H∗G
is the
Riemann sphere . All of these realizations are conjugate in PSL2(R) . As an
abstract group, the hyperbolic triangle group denoted by G of type (`,m, n)
has a presentation
G = 〈t1, t2, t3 | t`1 = tm2 = tn3 = 1 = t1t2t3〉 , (4.1.1)
where 2 ≤ ` ≤ m ≤ n ≤ ∞ .
There are four types of hyperbolic triangle groups, namely of type (`,m, n),
(`,m,∞), (`,∞,∞) and (∞,∞,∞), i.e with no cusp, 1 cusp, 2 cusps and 3
cusps respectively and these can be easily distinguished from their fundamental
domain, sketched in figure 4.1.
Because of our interest in Fourier coefficients focus is put on the case where
there is at least one cusp, i.e. at least one of the `,m, n is ∞ and in the
presentation of that type of G any relation of the form t∞i = 1 is dropped . For
example without loss of generality we may fix n =∞ and therefore a triangle
group G of type (`,m,∞) is isomorphic to Z` ∗Zm, where we define Z∞ := Z.
The triples (2, 2,∞) are excluded as they are not hyperbolic. The cardinality
of the set CG is at most 3. A fundamental domain of G, denoted by FG
is
naturally a quadrilateral and exactly double a hyperbolic triangle .
All realizations of a triangle group G of type (`,m,∞) are equivalent (con-
50
Figure 4.1: Fundamental domain of triangle groups of all types
jugate) in PSL2(R) . Fix the one which is generated by
t1 = ±(
2cos(π` )
−1
1
0
), t2 = ±
(0
−1
1
2cos( πm )
), t3 = ±
(1
0
2cos(π` ) + 2cos( πm )
1
).
(4.1.2)
The corresponding hyperbolic triangle has vertices
ζ1 = − exp
(− πi
`
), ζ2 = exp
(πi
m
), ζ3 =∞ . (4.1.3)
These vertices form a set of representatives for the orbits of the elliptic and
parabolic points of extended upper half plane H∗G
(defined in chapter 2) under
the action of G .
The most famous triangle group is the modular group PSL2(Z) which is
of type (2, 3,∞) with generators t1 = ±(
0−1
10
), t2 = ±
(01−1−1
), t3 = ±
(10
11
).
Other famous triangle groups are Γ(2) and Γ0(4) which are triangle groups of
type (∞,∞,∞), Γ0(2) which is a triangle group of type (2,∞,∞) and the
Hecke triangle groups Hm which are of type (2,m,∞) . The details of the
following two sections is taken from [9].
4.1.1 Hauptmodul
For each such G, there exists a hauptmodul
z = zz
G(τ) := z
z
[−1]q−1
z+ z
z
[0]+ z
z
[1]qz + z
z
[2]q
2
z+ · · · (4.1.4)
51
with respect to any z ∈ ζ1, ζ2, ζ3 where zz
[n]∈ C for all n ≥ −1 and
qz =
exp
(2πiA−1
zτ
hz
)if z ∈ C
G(τ−zτ−z
)`if z ∈ E
Gof order `
. (4.1.5)
which gives an equivalence z : G\H∗G−→ P1(C) between two compact Riemann
surfaces and determined by 3 independent complex parameters by demanding
for i = 1, 2, 3
z(ζi) ∈ 0, 1, ∞ such that z(z) =∞ . (4.1.6)
There are two distinct choices of hauptmodul z(τ) . We call hauptmodul z(τ)
satisfying (4.1.6) the regularized hauptmodul of G. The following theorem
is stated mutatis-mutandis from [9] which explain the explicit computation of
the regularized hauptmodul z(τ) of choice
z(ζ1) = 1, z(ζ2) = 0 and z(ζ3) =∞ . (4.1.7)
Theorem 4.1.1. Let G be a fixed realization of triangle group of hyperbolic
type (`,m,∞) with 2 ≤ ` ≤ m ≤ ∞. For i = 1, 2, 3 let qi
be the local coordi-
nates about the points ζi ∈ H∗G
and the regularized hauptmodul z(τ) described
by equation (4.1.7) has local expansions
z(τ) = 1 + α1 q1 +∞∑k=2
ak αk1 q
k1
= α2 q2 +∞∑k=2
bk αk2 q
k2
=1
α3
q−1
3+∞∑k=0
ck αk3 q
k3, (4.1.8)
where αi defined as follows:
• If ζi =∞ then
αi = b′d′Πb′−1k=1
(2− 2cos(
2πk
b′)
)− 12cos( 2πka′
b′ )
Πd′−1l=1
(2− 2cos(
2πl
d′)
)− 12cos( 2πlc′
d′ )
(4.1.9)
52
where the integers a′, b′, c′, d′ are defined by a′
b′= 1
2
(1 + 1
`− 1
m
)and c′
d′=
12
(1 + 1
`+ 1
m
)• If ζi = − exp(−πi
`) then
αi =cos(π
2(1`
+ 1m
))
cos(π2(1`− 1
m))·
Γ(1 + 1`)
Γ(1− 1`)·
Γ(14(1− 1
`+ 1
m)2)
Γ(14(1 + 1
`+ 1
m)2). (4.1.10)
• If ζi = − exp(−πim
) then
αi =cos(π
2(1`
+ 1m
))
cos(π2(1`− 1
m))·
Γ(1 + 1m
)
Γ(1− 1m
)·
Γ(14(1− 1
m+ 1
`)2)
Γ(14(1 + 1
`+ 1
m)2). (4.1.11)
and the coefficients ak , bk , ck are uniquely determined by
− 2(D3z)(Dz) + 3(D2z)2− (Dz)2
| Gz |2= (Dz)4
(1− 1
m2
z2+
1− 1`2
(z− 1)2+
1`2
+ 1m2 − 1
z(z− 1)
)(4.1.12)
for the choice z = ζ1, ζ2, ζ3 respectively, where D = qiddqi
and | Gz | is the order
of the stabilizer subgroup of z in G.
Coefficients of regularized hauptmodul z(τ)
The coefficients ak, bk, ck are universal (i.e. type independent) polynomials in
Q[1`, 1m
], and are also unchanged if we replace G by any of its conjugate. The
first few coefficients are given below. For notational convenience we write
α± = v22 ± v2
3 , β± = v21 ± v2
3 , γ± = v21 ± v2
2
where v1 = 1`, v2 = 1
m, v3 = 1
nif G is a triangle group of type (`,m, n).
a2 =v2
1 + α− − 1
2(v21 − 1)
,
a3 =(−11 + 32α− − 2α+ − 19α2
−) + (25− 40α− + 7α2− + 4α+)v2
1
16(v21 − 4)(v2
1 − 1)2
+(−17− 2α+ + 8α−)v4
1 + 3v61
16(v21 − 4)(v2
1 − 1)2,
a4 =−162− 108α+ + 898α− − 1458α2
− + 172α−α+ + 658α3−
96(v21 − 4)(v2
1 − 9)(v21 − 1)3
53
+558 + 336α+ − 2148α− − 372α−α+ + 2106α2
− − 416α3−
96(v21 − 4)(v2
1 − 9)(v21 − 1)3
v21
+−708− 360α+ + 1632α− + 228α−α+ − 702α2
− + 46α3−
96(v21 − 4)(v2
1 − 9)(v21 − 1)3
v41
+(396 + 144α+ − 412α− + 54α2
− − 28α−α+)v61 + (−90 + 30α− − 12α+)v8
1 + 6v101
96(v21 − 4)(v2
1 − 9)(v21 − 1)3
,
b2 =1− β− − v2
2
2(v22 − 1)
,
b3 =(−11 + 32β− − 2β+ − 19β2
−) + (25 + 4β+ − 40β− + 7β2−)v2
2
16(v22 − 4)(v2
2 − 1)2
+(−17− 2β+ + 8β−)v4
2 + 3v62
16(v22 − 4)(v2
2 − 1)2,
b4 =81− 449β− + 54β+ + 729β2
− − 86β−β+ − 329β3−
48(v22 − 4)(v2
2 − 9)(v22 − 1)3
+−279 + 1074β− − 168β+ + 186β−β+ − 1053β2
− + 208β3−
48(v22 − 4)(v2
2 − 9)(v22 − 1)3
v22
+354− 816β− + 180β+ + 351β2
− − 114β−β+ − 23β3−
48(v22 − 4)(v2
2 − 9)(v22 − 1)3
v42
+(−198 + 206β− − 72β+ + 14β−β+ − 27β2
−)v62 + (45 + 6β+ − 15β−)v8
2 − 3v102
48(v22 − 4)(v2
2 − 9)(v22 − 1)3
,
c0 =−1 + γ− + v2
3
2(v23 − 1)
,
c1 =(5− 2γ+ − 3γ2
−) + (−6 + 2γ+)v23 + v4
3
16(v23 − 1)(v2
3 − 4),
c2 =(−2γ− + γ+γ− + γ3
−) + (2γ− − γ+γ−)v23
6(v23 − 9)(v2
3 − 1)2,
c3 =−31 + 76γ+ + 690γ2
− − 28γ2+ − 404γ2
−γ+ − 303γ4−
128(v23 − 16)(v2
3 − 4)2(v23 − 1)3
+100− 244γ+ + 88γ2
+ − 1052γ2− + 660γ2
−γ+ + 192γ4−
128(v23 − 16)(v2
3 − 4)2(v23 − 1)3
v23
+−114 + 276γ+ − 96γ2
+ + 390γ2− − 288γ2
−γ+ − 24γ4−
128(v23 − 16)(v2
3 − 4)2(v23 − 1)3
v43
+(52− 124γ+ + 40γ2
+ − 24γ2− + 32γ2
−γ+)v63 + (−7 + 16γ+ − 4γ2
+ − 4γ2−)v8
3
128(v23 − 16)(v2
3 − 4)2(v23 − 1)3
,
When there are cusps these above coefficients simplify considerably. For
example,
54
• when n =∞ then
c0 =1
2
(1− γ−
),
c1 =1
64
(5− 2γ+ − 3γ2
−
),
c2 =1
54
(− γ3
− − γ+γ− + 2γ−
),
c3 =1
32768
(− 31 + 76γ+ − 28γ2
+ + 690γ2− − 404γ+γ
2− − 303γ4
−
),
c4 =1
216000
(− 274γ− + 765γ+γ− − 314γ2
+γ− + 2807γ3− − 1865γ+γ
3− − 1119γ5
−
);
• when m = n =∞, then
b2 =1
2
(v2
1 − 1),
b3 =1
64
(11− 30v2
1 + 19v41
),
b4 =1
1728
(− 81 + 395v2
1 − 643v41 + 329v6
1
),
b5 =1
884736
(9693− 70372v2
1 + 196926v41 − 249156v6
1 + 112909v81
); and
• when ` = m = n =∞, then
a2 =1
2, a3 =
11
64, a4 =
3
64, a5 =
359
32768, a6 =
75
32768.
Note 4.1.2. The regularized hauptmodul of triangle group G of type (` ,m , n)
for 2 ≤ ` ≤ m ≤ n <∞ such that 1`
+ 1m
+ 1n< 1 can also be computed in the
similar manner with some efforts . For suggestions see the Appendix B in [9] .
4.1.2 Scalar-valued automorphic forms
Details are given about all other automorphic forms and functions of triangle
group G . It is a well known fact that one can construct all the automorphic
functions by knowing a hauptmodul of G—they are simply the rational func-
tions in the hauptmodul . On the contrary, in the following theorem, which is
taken in its exact form from [9] the generators of the ring of all holomorphic
55
automorphic forms, the analogue of the cusp form ∆G
(τ) of the Delta func-
tion ∆(τ) = η(τ)24 for Γ(1) are constructed and the analogue EG
2(τ) of the
quasi-automorphic form E2 of Γ(1) is given .
Theorem 4.1.3.
1. For each k ∈ Z, write d2k = k − dk`e − d k
me and let
f2k = (−1)k(Dz)kzdkme−k(z− 1)d
k`e−k = q d2k +O(q d2k+1), (4.1.13)
where D = q ddq
. Then a basis for the C-vector space H2k(G) of holomorphic
automorphic forms of weight 2k for G is f2k(τ)z(τ)l for each 0 ≤ l ≤ d2k. In
particular,
dim(H2k(G)) =
d2k + 1 if k ≥ 0
0 if k < 0, (4.1.14)
The algebra H(G) of holomorphic automorphic forms has the following mini-
mal set of generators:
• f2, zf2 when G is of type (∞,∞,∞).
• f2, f4, · · · , f2` when G is of type (`,∞,∞) for ` <∞.
• f2`|2 ≤ l ≤ m ∪ zd2`f2`|3 ≤ l ≤ ` when G is of type (`,m,∞) for ` ≤ m <
∞.
2. Define
L =
` if m =∞ and ` <∞1 if ` =∞ = m
lcm(`,m) if ` ≤ m <∞.
Then ∆G
(τ) := f2L(τ) is a holomorphic automorphic form of weight 2L,
nonzero everywhere in H∗G
except in the G-orbit of the cusp ∞, where ∆G
(τ)
has a zero of order n∆G
= L(1 − 1
`− 1
m
). Define E
G
2= 1
2πi1
∆G
d∆G
dτ. Then E
G
2
is holomorphic in H∗G
, EG
2vanishes at any cusp ζ1, ζ2 not in the G-orbit of the
56
cusp ∞ and EG
2(∞) = n
∆G
. Moreover, EG
2is a quasi-automorphic form of
weight 2 and depth 1 for G: i.e. for all(acbd
)∈ G,
EG
2
(aτ + b
cτ + d
)=n
∆Gc
2πi(cτ + d)E
G
2(τ) + (cτ + d)2E
G
2(τ) . (4.1.15)
The derivation Dk = 12πi
ddτ− k
LE
G
2sends weight k automorphic forms to weight
k + 2 ones . The space of all holomorphic quasi-automorphic forms of G is
H(G)[EG
2].
4.2 Admissible multiplier
Let G be any triangle group of type (`,m, n). Then the rank 1 and 2 admissible
irreps of G are classified . Let ρ : G −→ GLd(C) be a rank d admissible irrep
such that ρ(ti) = Ti with
T `1 = 1 = Tm2 = T n3 and T1T2T3 = 1 (4.2.1)
where from Definition 2.3.1 T3 is a diagonal matrix and T1, T2 are diagonaliz-
able matrices in GLd(C) . This means that there exist invertible matrices P1,
P2 and diagonal matrices A,B,Λ satisfying the following condition
P−11 T1P1 = exp(2πiA), P−1
2 T2P2 = exp(2πiB) and T3 = exp(2πiΛ)
(4.2.2)
Note that in case one or all of the ` ,m , n is ∞ any relation of the form T∞i
will be dropped in equation (4.2.1) .
4.2.1 Rank 1
In case of d = 1, Ti ∈ C× . Then
• If G is of type (∞,∞,∞) then there are infinitely many inequivalent rank
1 admissible multipliers given by the choices of 2 nonzero complex numbers
T1 , T2 ∈ C× as T3 = (T2T1)−1 . Hence , the moduli space of rank 1 admissible
multiplier consists 1 connected component of complex dimension 2 .
57
• If G is of type (`,∞,∞) then there are infinitely many rank 1 admissible
multipliers given by the choice of one nonzero complex number T2 ∈ C× with
T1 = exp(2πij`
) for j ∈ Z` and T3 = (T2T1)−1 . Hence , the moduli space of rank
1 admissible multipliers consists of ` connected components each of complex
dimension 1.
• If G is of type (`,m,∞) then there are `m distinct rank 1 admissible multipliers
given by the choices T1 = exp(2πij`
) , T2 = exp(2πikm
) for j ∈ Z` , k ∈ Zm as
T3 = (T2T1)−1 . Hence , the moduli space of rank 1 admissible multipliers
consists of `m points .
4.2.2 Rank 2
In case of d = 2, let λ3, λ4 be the eigenvalues of T1; λ5, λ6 be the eigen-
values of T2 and λ1, λ2 be the eigenvalues of T3 . Let us write
T3 =
(λ1
0
0
λ2
), T1 =
(a1
c1
b1
d1
)and T2 =
(a2
c2
b2
d2
). (4.2.3)
The relation in equation (4.2.1) and the fact det(T1) = λ3λ4 give
a2 =d1
λ1λ3λ4
, b2 = − b1
λ2λ3λ4
, c2 = − c1
λ1λ3λ4
, d2 =a1
λ2λ3λ4
.
Since a1 + d1 = λ3 + λ4 and a1d1 − b1c1 = λ3λ4 then
T1 =
(a1yb1
b1
λ3 + λ4 − a1
). (4.2.4)
where y = b1c1 = −a21 +(λ3 +λ4)a1−λ3λ4 subject to the condition that b1 6= 0
(the case b1 = 0 considered later in the proof of Theorem 4.2.1).
Now, from equation (4.2.1) λ5λ6 = 1λ1λ2λ3λ4
and
T2 =1
λ3λ4
(λ3+λ4−a1
λ1
− yb1λ1
− b1λ2a1
λ2
). (4.2.5)
This implies that
λ5 + λ6 =1
λ1λ2λ3λ4
[λ2(λ3 + λ4) + a1(λ1 − λ2)
]. (4.2.6)
58
and therefore
a1 =1
λ1 − λ2
[1
λ5
+1
λ6
− λ2(λ3 + λ4)
]. (4.2.7)
Say r1 = λ1λ2λ3λ4 and r2 = λ2(λ3+λ4)+a1(λ1−λ2) then λ6 = r2r1−λ5 and using
this in equation (4.2.6) with the value of λ5λ6, we get λ25− r2
r1λ5− 1
r1= 0 , solving
this quadratic equation gives λ5 =r2±√r22−4r1
2r1. From this it is concluded that
λ5 is one choice of the sign and λ6 is the other. Hence, with the help of above
discussion the classification of the rank 2 admissible multiplier of G can be
summarized in the following
Theorem 4.2.1. For d = 2, ρ defined by the equation (4.2.1) is equivalent to
one of the following types of admissible multiplier
(a) If λi ∈ C× for i 6= 2 satisfy
λ`3 = λ`4 = 1 = λm5 = λm6 (4.2.8)
and λ2 := 1λ1λ3λ4λ5λ6
6= λ1, λ3 6= λ4, λ5 6= λ6 then T1 , T2 and T3 can be
defined respectively by the equation (4.2.4), (4.2.5) and (4.2.3) such that a1 ∈
λ3, λ4 . Such ρ is an irreducible admissible multiplier which is denote by
ρirred
:= ρirred
(λ1,λ3,λ4,λ5,λ6).
(b) If λi ∈ C× for i 6= 1, 2 satisfy (4.2.8) and λ3λ5 6= λ4λ6 such that
T1 =
(λ3
0
0
λ4
), T2 =
(λ5
0
0
λ6
)(4.2.9)
then T3 =(λ1
00λ2
)where λ1 = 1
λ3λ5, λ2 = 1
λ4λ6and such ρ is a reducible admis-
sible multiplier which is denoted by ρred
:= ρred
(λ3,λ4,λ5,λ6).
(c) If λi ∈ C× for i 6= 1, 2 satisfy (4.2.8) and λ3 6= λ4, λ5 6= λ6 such that
T1 =
(λ3
0
1
λ4
), T2 =
(λ5
0
−λ6
λ3
λ6
)(4.2.10)
then T3 =(λ1
00λ2
)where λ1 = 1
λ3λ5, λ2 = 1
λ4λ6. Such ρ is an indecomposable
but reducible admissible multiplier which is denoted by ρind
:= ρind
(λ3,λ4,λ5,λ6).
59
Proof. If λ1 = λ2 then T1 can be diagonalized in which case T2 = T−11 T−1
3
is also diagonal and therefore ρred
is obtained . Likewise if b1 = 0 = c1 then
this gives ρred
. So without loss of generality λ1 6= λ2 and b1 6= 0 can be
considered and in this case we can force b1 = 1 because different values of b1
corresponds to equivalent representation. . If c1 = 0 this gives the ρind . In all
other cases , following from the earlier discussion a ρ will be recovered which
will be equivalent to ρirred
.
Remark 4.2.2. The multiplier ρred
is the direct sum of 1-dimensional repre-
sentations . In the relation (4.2.8), any relation of the form λ∞ = 1 is dropped .
If both `,m < ∞ then there are(`2
)(m2
)components to the moduli space of
irreps, each of which is 1-dimensional . If ` < m = ∞ then there are(`2
)components, each of which is 3-dimensional . If ` = m =∞ then there is one
5-dimensional connected component .
4.3 Hypergeometric differential equations and
functions
In this part , we are interested in hypergeometric differential equations on the
sphere with three regular singular points . The term “hypergeometric” first
was used by the Oxford professor John Wallis in 1655 in his work Arithmetica
Infinitorum to denote any series which was beyond the ordinary geometric
series 1 + x+ x2 + x3 + . . .. The series
2F1(a, b; c; z) := 1 + abcz1!
+a(a+1)b(b+1)c(c+1)
z2
2!
+a(a+1)(a+2)b(b+1)(b+2)c(c+1)(c+2)
z3
3!+ . . .
is called the ordinary hypergeometric series. It satisfies the following second
order differential equation with 3-regular singular points 0, 1 and ∞,
z(1− z)d2W
dz2+[c− (a+ b+ 1)z
]dWdz− abW = 0 . (4.3.1)
60
This equation is a standard form of hypergeometric differential equation which
is also known as Euler’s hypergeometric differential equation. It has
regular singularities at 0, 1 and∞ with the exponents 0, 1−c at z = 0, 0, c−
a−b at z = 1 and a, b at z =∞. For detailed exposition on hypergeometric
differential equations and functions see [1, 48], chapter 16 of [42] and chapter
14 of [52].
The point made below is that the theory of hypergeometric differential
equations is equivalent to the theory of rank 2 vvaf for triangle groups. We
begin with the following
Definition 4.3.1 (Papperitz’s and Riemann’s differential equation). Let Y =
f(z) be a complex valued function. A second order differential equation of Y
with three singular points α, β, γ and the exponents u1, u2 at z = γ, u3, u4
at z = α and u5, u6 at z = β of the form
d2Ydz2 +dY
dz
(1−u3−u4
z−α + 1−u5−u6
z−β + 1−u1−u2
z−γ
)+ Y
(z−α)(z−β)(z−γ)
((α−β)(α−γ)u3u4
z−α + (β−α)(β−γ)u5u6
z−α + (γ−α)(γ−β)u1u2
z−γ
)= 0 ,
(4.3.2)
satisfying the condition
u1 + u2 + u3 + u4 + u5 + u6 = 1 (4.3.3)
is called Papperitz’s differential equation of second order. Let us consider
the regular singular points z = 0, 1 and ∞ then taking α = 0, β = 1 and
γ →∞ in the differential equation (4.3.2) can be written in the form
z2(z − 1)2 d2
dz2Y + ddzY (1− u3 − u4)z(z − 1)2 + (1− u5 − u6)z2(z − 1)
+Y u3u4(1− z) + u5u6z + u1u2z(z − 1) = 0 .(4.3.4)
which we call Riemann’s differential equation of second order. This equa-
tion has exponents u3, u4 at z = 0, u5, u6 at z = 1 and u1, u2 at
61
z =∞ and satisfy the condition (4.3.3). The hypergeometric differential equa-
tion (4.3.1) is a special type of Riemann’s differential equation which possesses
an exponent 0 at singular points z = 0 and 1.
In 1857, Riemann introduced a way to denote the set of all solutions of the
equation (4.3.2) by
P
α β γu3 u5 u1 ; zu4 u6 u2
(4.3.5)
which is known as Riemann scheme . Using the Riemann scheme the so-
lutions of Riemann’s and hypergeometric differential equation, namely (4.3.4)
and (4.3.1) can be described respectively by the Riemann schemes
P
0 1 ∞u3 u5 u1 ; zu4 u6 u2
and P
0 1 ∞0 0 a ; z
1− c c− a− b b
(4.3.6)
4.3.1 Solutions of equation (4.3.1)
The exponents of this hypergeometric differential equation are 0, 1 − c,
0, c − a − b and a, b at regular singular points 0, 1 and ∞ respectively .
This hypergeometric equation has two linearly independent solutions. Around
each of its regular singular points, these two solutions can be represented by
hypergeometric functions. Now using the Riemann scheme (4.3.6) its solutions
is described as follows
• At z = 0 with respect to the exponent 0 and 1− c the solutions are
W01(z) = 2F1(a, b, c; z) ,
W02(z) = z1−c2F1(a+ 1− c, b+ 1− c, 2− c; z) . (4.3.7)
• At z =∞ with respect to the exponent a and b the solutions are
W∞1(z) = z−a2F1
(a, a+ 1− c; a+ b− 1;
1
z
),
W∞2(z) = z−b2F1
(b, b+ 1− c; b+ 1− a;
1
z
). (4.3.8)
62
• At z = 1 with respect to the exponent 0 and c− a− b the solutions are
W11(z) = 2F1(a, b, a+ b− c+ 1; 1− z) ,
W12(z) = (1− z)c−a−b2F1(c− a, c− b, c− a− b+ 1; 1− z) . (4.3.9)
According to Kummer , solutions of (4.3.1) can be represented by 24 different
hypergeometric series . These 24 hypergeometric series are known as Kum-
mer’s 24 solutions which can be described by using the properties of Riemann
schemes. For more details on Kummer’s 24 solutions of hypergeometric differ-
ential equation (4.3.1) see section 2.9 of chapter 2 in [11] or subsection 16.1.6
of chapter 16 in [42] .
4.3.2 Solutions of equation (4.3.4)
The Riemann scheme bears various interesting properties. One of the most
famous, amongst other properties, which we will make use of in this thesis is
the following
P
0 1 ∞u3 u5 u1 ; zu4 u6 u2
= zu3(1− z)u5P
0 1 ∞0 0 u1 + u3 + u5 ; z
u4 − u3 u6 − u5 u2 + u3 + u5
(4.3.10)
Now note that from property (4.3.10) it is clear that any solution of Riemann’s
differential equation (4.3.4) can be obtained from a solution of the hyperge-
ometric differential equation of the form (4.3.1) with c = 1 + u3 − u4 , a =
u1 + u3 + u5 , b = u2 + u3 + u5 . The solutions of (4.3.4) can be explicitly
written around the points z = 0 , 1 and ∞ as follows
• at z = 0 with respect to the exponent u3 and u4 the solutions are
Y01(z) = zu3(1− z)u52F1(a, b, c; z)
= zu3(1− z)u52F1(u1 + u3 + u5 , u2 + u3 + u5 , 1 + u3 − u4 ; z)
Y02(z) = zu3(1− z)u5z1−c2F1(a+ 1− c, b+ 1− c, 2− c; z)
= zu4(1− z)u52F1(u1 + u4 + u5 , u2 + u4 + u5 , 1 + u4 − u3; z) .
63
(4.3.11)
• at z =∞ with respect to the exponent u1 and u2 the solutions are
Y∞1(z) = zu3(1− z)u5z−a2F1
(a, a+ 1− c; a+ 1− b; 1
z
),
= z−u1
(1
z− 1
)u5
2F1
(u1 + u3 + u5 , u1 + u4 + u5; 1 + u1 − u2;
1
z
),
Y∞2(z) = zu3(1− z)u5z−b2F1
(b, b+ 1− c; b+ 1− a;
1
z
),
= z−u2
(1
z− 1
)u5
2F1
(u2 + u3 + u5 , u2 + u4 + u5 , 1− u1 + u2 ;
1
z
)(4.3.12)
• at point z = 1 with respect to the exponent u5 and u6 the solutions are
Y11(z) = zu3(1− z)u52F1(a, b, a+ b− c+ 1; 1− z)
= zu3(1− z)u52F1(u1 + u3 + u5 , u2 + u3 + u5 , 1 + u5 − u6; 1− z) ,
Y12(z) = zu3(1− z)u5(1− z)c−a−b2F1(c− a, c− b, c− a− b+ 1; 1− z)= zu3(1− z)u6
2F1(1− u4 − u5 − u1 , 1− u4 − u5 − u2 , 1 + u6 − u5 ; 1− z) .(4.3.13)
4.4 Monodromy group of a differential equa-
tion
Let G be any triangle group of type (`,m, n) defined by (4.1.1) . Note that
G\(H − EG
) is a Riemann sphere with three punctures . Let ρ : G −→
GL2(C) be any admissible multiplier with the exponent matrices with respect
to t1 , t2 , t3 are being A ,B ,Λ respectively . We can choose the eigenvalues
u1 , u2 ,u3 , u4 and u5 , u6 respectively of matrices Λ ,A and B such that∑6i=1 ui = 1 i.e. tr(A + B + Λ) = 1 . Since ρ is an admissible multiplier of G
then without loss of generality we may assume that ρ(t3) is a diagonal ma-
trix and therefore Λ = diag(u1 , u2) . Now consider the following differential
64
equations on G\H∗G
with respect to z with regular singular points z = 0, 1,∞
d2Ydz2 +dY
dz
(1−u3−u4
z+ 1−u5−u6
z−1
)+Y
(− u3u4
z2(z−1)+ u5u6
z(z−1)2 + u1u2
z(z−1)
)= 0
(4.4.1)
The point is that the monodromy representation of (4.4.1) is equivalent to
the admissible multiplier ρ with the eigenvalues of the exponent matrices as
defined above which will be discussed in the next few pages . Note that the
solutions of (4.4.1) are described by the Riemann scheme (4.3.5) which are
written explicitly in subsection 4.3.2 . In the next subsection the monodromy
group of equation (4.4.1) is computed . We begin by stating the following
Definition 4.4.1. Let Vz = 〈Yz1 , Yz2〉 be the basis of solution space of (4.4.1)
with respect to z = 0, 1,∞ . Then the monodromy group of (4.4.1) is the image
of M(V ) : F2 −→ GL(V) where F2 represents the free group in two generators
and M(V ) is a representation of the fundamental group π1(G\(H − EG
)) =
〈l0, l1, l∞ | l0l1l∞ = 1〉 ≈ F2 with respect to the solution space V . l0, l1 and
l∞ represents the loop in G\(H − EG
) respectively around 0, 1 and ∞ . The
monodromy group of (4.4.1) with respect to the solution basis V is denoted
by MG(V ) .
It is clear from the definition of monodromy group that
MG(V ) = 〈M0 ,M1 ,M∞ | M0M1M∞ = 1〉 ∼= 〈M0,M∞〉
where Mz := M(V )(lz) for z = 0, 1,∞ and therefore MG(V ) is a homomorphic
image of free group F2 generated by the two matrices M0,M∞ in GL2(C) .
4.4.1 Monodromy group of (4.4.1)
Let V := V∞ = 〈Y∞1, Y∞2〉 , U := V0 = 〈Y01, Y02〉 and W := V1 = 〈Y11, Y12〉
over C . To compute MG(V ) it is enough to compute any two of its three
65
generators however for the pedagogical reasons all the three generators will be
computed here explicitly .
• M∞: To compute M∞ we are looking for the action of M(V )(l∞) on the basis
vectors Y∞1, Y∞2 and
M(V )(l∞) :
(Y∞1
Y∞2
)7→(
exp(2πiu1)
0
0
exp(2πiu2)
)(Y∞1
Y∞2
).
Therefore
M∞ =
(exp(2πiu1)
0
0
exp(2πiu2)
)= exp(2πiΛ) .
• M0: To compute M0 we are looking for the action of M(V )(l0) on the basis
vectors Y∞1, Y∞2 and
M(U)(l0) :
(Y01
Y02
)7→(
exp(2πiu3)
0
0
exp(2πiu4)
)(Y01
Y02
).
Now , by analytic continuation of Y∞1, Y∞2 inside the loop around z = 0 ,
Y∞1 = B1Y01 +B2Y02
Y∞2 = B3Y01 +B4Y02
for some yet to be determined Bξ ∈ C for 1 ≤ ξ ≤ 4 . Also, this is equivalent
to (Y∞1
Y∞2
)= B
(Y01
Y02
)with B =
(B1
B3
B2
B4
)where the values of B1, B2, B3, B4 taken from section 2.9 of chapter 2 in [11]
are
B1 =Γ(1− c)Γ(a+ 1− b)Γ(1− b)Γ(a+ 1− c)
exp(−πia) ,
B2 =Γ(c− 1)Γ(a+ 1− b)
Γ(c− b)Γ(a)exp(iπ(c− a− 1)) ,
B3 =Γ(1− c)Γ(b+ 1− a)
Γ(1− a)Γ(b+ 1− c)exp(−πib) ,
B4 =Γ(c− 1)Γ(b+ 1− a)
Γ(c− a)Γ(b)exp(iπ(c− b− 1)) . (4.4.2)
66
where c = 1 + u3 − u4 , a = u1 + u3 + u5 , b = u2 + u3 + u5 . Now, M0 can be
computed explicitly as follows: since(Y01
Y02
)= B−1
(Y∞1
Y∞2
)therefore under M(V )(l0),
B−1
(Y∞1
Y∞2
)7→(
exp(2πiu3)
0
0
exp(2πiu4)
)B−1
(Y∞1
Y∞2
).
This implies that
M(V )(l0) :
(Y∞1
Y∞2
)7→ B
(exp(2πiu3)
0
0
exp(2πiΛ4)
)B−1
(Y∞1
Y∞2
).
Therefore
M0 = B
(exp(2πiu3)
0
0
exp(2πiu4)
)B−1 = B exp(2πiA)B−1 .
• M1: To compute M1 we are looking for the action of M(V )(l1) on the basis
vectors Y∞1, Y∞2 and
M(W )(l1) :
(Y11
Y12
)7→(
exp(2πiu5)
0
0
exp(2πiu6)
)(Y11
Y12
).
Now , by analytic continuation of Y∞1, Y∞2 inside the loop around z = 1 ,
Y∞1 = C1Y11 + C2Y12
Y∞2 = C3Y11 + C4Y12
for some yet to be determined Cξ ∈ C for 1 ≤ ξ ≤ 4 . Same as before this is
equivalent to write in the matrix form as follows(Y∞1
Y∞2
)= C
(Y11
Y12
)with C =
(C1
C3
C2
C4
)and the values of C1, C2, C3, C4 can be obtained from section 2.9 of chapter 2
in [11] as follows
C1 =Γ(c− a− b)Γ(a+ 1− b)
Γ(1− b)Γ(c− b),
67
C2 =Γ(a+ b− c)Γ(a+ 1− b)
Γ(a+ 1− c)Γ(a)exp(iπ(c− a− b)) ,
C3 =Γ(c− a− b)Γ(b+ 1− a)
Γ(1− a)Γ(c− a),
C4 =Γ(a+ b− c)Γ(b+ 1− a)
Γ(b+ 1− c)Γ(b)exp(iπ(c− a− b)) , (4.4.3)
where a, b, c are same as before and therefore
M1 = C
(exp(2πiu5)
0
0
exp(2πiu6)
)C−1 = C exp(2πiB)C−1 .
4.4.2 Monodromy vs. admissible multiplier
We want to see now how the original representation ρ : G −→ GL2(C) and the
monodromy M(V ) : π1(G\H) −→ GL(V ) are related to each other i.e. whether
they are equivalent representations or not . Therefore, we want to find a map
L : GL(V ) −→ GL2(C) such that the following diagram commutes:
π1(G\H)
M // GL(V )
L
Gρ // GL2(C)
An invertible matrix L is being looked such that LM1L−1 = T2, LM∞L
−1 = T3
and LM0L−1 = T1. Since, M∞ = T3 and they both are diagonal therefore
the only possible candidate for L is some diagonal matrix which can be cho-
sen of the following form : L =(
10
0χ
)for some unknown χ ∈ C . To find
this χ consider LM0L−1 = T1 . This implies that LB exp(2πiA)B−1L−1 =
P−11 exp(2πiA)P1 . Note that P1, exp(2πiA) and B are all known matrices
therefore to find an α is equivalent to solve P1LB exp(2πiA) = exp(2πiA)P1BL .
Say P1 =(P1
P3
P2
P4
)and B =
(B1
B3
B2
B4
)then solving this gives
1
χ= −P2B4
P1B2
= −P4B3
P3B1
. (4.4.4)
68
4.5 Rank 2 vector-valued automorphic forms
Let G be any triangle group of type (`,m, n) as defined by the equation (4.1.1)
such that ζ1 , ζ2 , ζ3 are three of the corners of the FG
. Let ρ be its rank 2
admissible multiplier such that ρ(ti) = Ti for i = 1, 2, 3 defined by the equa-
tion (4.2.3) with λj = exp(2πiΛj) for j = 1 , 2 i.e. T3 = exp(2πiΛ) where
Λ =(
Λ1
00
Λ2
)is a 2 × 2 diagonal matrix and T1, T2 are 2 × 2 diagonaliz-
able matrices. More precisely, by the definition of our admissible multiplier,
there exist diagonal matrices A,B and diagonalizing matrices P1,P2 satisfying
equation (4.2.2) . The matrices A ,B ,Λ are exponent matrices respectively of
ζ1, ζ2, ζ3 . We require them to satisfy
tr(A+ B + Λ) = 1 . (4.5.1)
Let λ1, λ2 be the eigenvalues of T3, λ3, λ4 be the eigenvalues of T1 and
λ5, λ6 be the eigenvalues of T2, then from Theorem 4.2.1, T1, T2, T3 can be
expressed in terms of some subset of the eigenvalues λ1, λ3, λ4, λ5, λ6 . Let
Λ3,Λ4 be the eigenvalues of A and Λ5 ,Λ6 are the eigenvalues of B .
Note 4.5.1. There are infinitely many different choices for Λi for every 1 ≤ i ≤
6 which satisfy the above conditions . For example for every k ∈ Z replacing
Λ1 ,Λ2 by Λ1 + k ,Λ2 − k respectively and leave Λi , for all 3 ≤ i ≤ 6 fixed .
This means that the construction in this chapter will result in infinitely many
independent vvaf of G with respect to the multiplier ρ .
4.5.1 Nearly holomorphic vector-valued automorphic formsat ∞
Choose the unique Λi ∈ C for 3 ≤ i ≤ 6 such that λi = exp(2πiΛi) and
0 ≤ Re(Λi) < 1 . Choose any Λ1 ,Λ2 ∈ C such that
λ1 = exp(2πiΛ1) , λ2 = exp(2πiΛ2) and6∑i=1
Λi = 1 .
69
For example if ` < ∞ then Λ3,Λ4 ∈
0`, 1`, 2`, · · · , `−1
`
whereas if n < ∞
then Λ1 ,Λ2 ∈ 1nZ . Now, fix the regularized hauptmodul z(τ) := z
(3)(τ) of
G by demanding
z(ζ1) = 0, z(ζ2) = 1 and z(ζ3) =∞ . (4.5.2)
With respect to the regularized hauptmodul z(τ) write the normalized haupt-
modul J(τ) and its values as follows
J(ζ1) = α(3)
1 , J(ζ2) = α(3)
2 and z(ζ3) = α(3)
3 =∞ . (4.5.3)
Now, consider the following Riemann’s differential equation on G\H∗G
with
respect to z(τ) with regular singular points at z = 0, 1,∞
d2Ydz2
+dYdz
(1−Λ3−Λ4
z+ 1−Λ5−Λ6
z−1
)+Y
(− Λ3Λ4
z2(z−1)+ Λ5Λ6
z(z−1)2 + (Λ1+k)(Λ2−k)z(z−1)
)= 0
(4.5.4)
whose solutions can be described through the following Riemann scheme
P
0 1 ∞
Λ3 Λ5 Λ1 + k ; zΛ4 Λ6 Λ2 − k
. (4.5.5)
Therefore , the solutions of equation (4.5.4) can be easily read off by replacing
ui = Λi for 3 ≤ i ≤ 6 and u1 = Λ1 + k , u2 = Λ2 − k in the equation (4.3.12)
from subsection 4.3.2 . Hence , at point z = ∞ with respect to the exponent
Λ1 + k and Λ2 − k the solutions are spanned by
Y∞1(z) = z−(Λ1+k)
(1− 1
z
)Λ5
×
2F1
(Λ1 + Λ3 + Λ5 + k,Λ1 + Λ4 + Λ5 + k; 1 + Λ1 − Λ2 + 2k; 1
z
),
Y∞2(z) = z−(Λ2−k)
(1− 1
z
)Λ5
×
2F1
(Λ2 + Λ3 + Λ5 − k,Λ2 + Λ4 + Λ5 − k; 1− 2k + Λ2 − Λ1; 1
z
),
(4.5.6)
70
and at z = 0 with respect to the exponent Λ3 and Λ4 the solutions are
Y01(z) = zΛ3(1− z)Λ5×2F1(Λ1 + Λ3 + Λ5 + k ,Λ2 + Λ3 + Λ5 − k , 1 + Λ3 − Λ4 ; z)
Y02(z) = zΛ4(1− z)Λ5×2F1(Λ1 + Λ4 + Λ5 + k ,Λ2 + Λ4 + Λ5 − k , 1 + Λ4 − Λ3; z) .
(4.5.7)
Now, consider
X(τ) =
X(3)
1(z(τ))
X(3)
2(z(τ))
(4.5.8)
whose components are defined by
X(3)
1(z) = ((α1 − α2)z)−(Λ1+k)
(1− 1
z
)Λ5
×
2F1
(Λ1 + Λ3 + Λ5 + k,Λ1 + Λ4 + Λ5 + k; 1 + Λ1 − Λ2 + 2k; 1
z
),
X(3)
2(z) = χ((α1 − α2)z)−(1+Λ2−k)
(1− 1
z
)Λ5
×
2F1
(Λ2 + Λ3 + Λ5 − k,Λ2 + Λ4 + Λ5 − k; 1− 2k + Λ2 − Λ1; 1
z
)(4.5.9)
for some soon to be determined constant χ := χ(3)
and similarly write
Y(τ) =
Y(3)
1(z(τ))
Y(3)
2(z(τ))
(4.5.10)
whose components are defined as
Y(3)
1(z(τ)) = Y01(z(τ)) ,Y(3)
2(z(τ)) = Y02(z(τ)) .
Let B be the connection matrix between Y(3)
1(τ) ,Y(3)
1(τ) and X1(τ) :=
X(3)
1(τ) ,X2(τ) := χ−1X(3)
2(τ) i.e.X1(z(τ))
X2(z(τ))
= B ·
Y(3)
1(z(τ))
Y(3)
2(z(τ))
,
and by using subsection 4.4.1
B =
((α1 − α2)−(Λ1+k)
0
0
(α1 − α2)−(1+Λ2−k)
)·B
71
where B is defined by (4.4.2) .
Now claim that X(τ) is a rank 2 nearly holomorphic vector-valued auto-
morphic function at ζ3 (equivalently at∞) of G with respect to the admissible
multiplier ρ . The reason for rescaling the solutions W∞1(τ),W∞2(τ) by the
factor (α1 − α2) is so that q−Λ
3X(τ) have the desired Fourier series expansion
of the form
q−Λ
3X(τ) =
qk
3+ X[1,k+1]q
k+1
3+ · · ·
χq−k
3+ X[1,−k+1]q
−k+1
3+ · · ·
. (4.5.11)
The reason for the constants χ := χ(3)
is to make sure that these vectors
transform correctly with respect to t1 . Also, the hauptmodul z(τ) is a weight
0 nearly holomorphic scalar-valued automorphic form of G therefore it is obvi-
ous from the expressions of the components of X(τ) that X(τ) is also a weight
0 rank 2 vector-valued automorphic form of G if it satisfies the following prop-
erties
1. X(γ · τ) = ρ(γ)X(τ) for every τ ∈ H and γ ∈ G .
2. X(τ) has the moderate growth at ζ3 .
3. X(τ) is holomorphic in H∗G\G · ζ3 .
Note that any solution to equations (4.5.4) is automatically holomorphic at
z 6= 0, 1,∞ and has branch points at 0, 1,∞ . Hence the lift Y (τ) of any
solution Y (z(τ)) to H will be holomorphic away from the elliptic fixed points .
The growth of any solution Y (z) to equation (4.5.4) as z tends to 0, 1 or∞ is :
• |Y (z)| < C∞|z|max[−Re(Λ1)−k ,k−Re(Λ2)] for |z| > 2 ,
• |Y (z)| < C0|z|max[Re(Λ3) ,Re(Λ4)] for |z| < 1/2 , and
• |Y (z)| < C1|1− z|max[Re(Λ5) ,Re(Λ6)] for |1− z| < 1/2 .
This means that the lift Y (z(τ)) will be of moderate growth at EG∪ C
G.
If say ` < ∞ then in that case ζ1 is an elliptic fixed point but Λ3 ,Λ4 ∈
72
0`, 1`, 2`, · · · , `−1
`
and z(τ) has a Taylor expansion in q1 =
(τ−ζ1τ−ζ1
)`, so Y (z(τ))
have Taylor expansion in (τ−ζ1) and hence it is holomorphic at τ = ζ1 . If say
m =∞ then z(τ) has a Taylor expansion in q2 and from this we see that Y (z(τ))
is holomorphic at ζ2 . Thus any lift of Y (z(τ)) to H∗G
is nearly holomorphic
at ζ3 . Hence , X(τ) is a rank 2 nearly holomorphic vector-valued function
which has the moderate growth . Finally , it is shown that X(τ) satisfies the
functional property under G . Note that G = 〈t1 , t3〉 therefore it is enough to
show that
X(t1 · τ) = T1X(τ) and X(t3 · τ) = T3X(τ) ,
for every τ ∈ H . This is equivalent to show thatX(3)
1(t1τ)
X(3)
2(t1τ)
= T1
X(3)
1(τ)
X(3)
2(τ)
and
X(3)
1(τ + h)
X(3)
2(τ + h)
= T3
X(3)
1(τ)
X(3)
2(τ)
Let V = 〈X1(z(τ)) ,X2(z(τ))〉 and U = 〈Y1(z(τ)) ,Y2(z(τ))〉 be the two
bases of the solution space of equation (4.5.4) . Let M(V ) : F2 −→ GL(V ) and
M(U) : F2 −→ GL(U) are two monodromy representations of F2 = π1(G\H−
EG) . Since these monodromy representations are equivalent to admissible
multiplier ρ therefore Ti for i = 1, 2, 3 is respectively equivalent to M(V )(lj)
and M(U)(lj) for i = 0, 1,∞ . Observe that Y(t1τ) = exp(2πiA)Y(τ) i.e.Y(3)
1(t1 τ)
Y(3)
2(t1 τ)
= exp(2πiA)
Y(3)
1(τ)
Y(3)
2(τ)
B−1
X1(t1 τ)
X2(t1 τ)
= exp(2πiA)B−1
X1(τ)
X2(τ)
,
B−1L−1X(t1τ) = exp(2πiA)B−1L−1X(τ) , where L =
(1
0
0
χ
)X(t1τ) = LB exp(2πiA) B−1L−1X(τ) ,
X(t1τ) = LM(V )(l0)L−1X(τ) , where M(V )(l0) = B exp(2πiA) B−1 ,
X(t1τ) = T1 X(τ) , since LM(V )(l0)L−1 = T1 .
73
Whereas to show T3X(τ) = X(τ + h) = X(t3 τ) is a straightforward compu-
tation performed by replacing τ to τ + h in the series expansion (4.5.11) of
X(τ) . However , in the spirit of the proof of X(t1 · τ) = T1X(τ) considerX(3)
1(t3 τ)
X(3)
2(t3 τ)
= exp(2πiΛ)
X(3)
1(τ)
X(3)
2(τ)
L−1X(t3 τ) = M(V )(l∞)L−1X(τ) ,
X(t3 τ) = LM(V )(l∞)L−1 X(τ) ,
X(t3 τ) = T3 X(τ) .
4.5.2 Coefficient χ in X(τ)
Now, the only thing left to determine is the constant χ explicitly which is
found by analytic continuation of X1(τ) and X2(τ) from z(τ) =∞ to z(τ) = 0 ,
following section 4.4 . Since t1 · ζ1 = ζ1 and z(ζ1) = 0 therefore applying γ = t1
and τ = ζ1 in X(τ) give X(ζ1) = T1X(ζ1) and
X(ζ1) =
(X1(ζ1)
X2(ζ2)
)where each component X
i(ζi) can be computed by analytic continuation of the
solutions of the Riemann’s differential equation from z(τ) = ∞ to z(τ) = 0
and therefore(Y∞1(z)
Y∞2(z)
)= B ·
(Y01(z)
Y02(z)
)where B =
(B1
B3
B2
B4
). (4.5.12)
The values of B1, B2, B3, B4 are defined from (4.4.2) . Note that
X1(z(τ)) = (α1 − α2)−(Λ1+k)Y∞1(z(τ))
X2(z(τ)) = χ(α1 − α2)−(Λ2−k)Y∞2(z(τ)) . (4.5.13)
From (4.5.12) in the neighbourhood of z(τ) = 0
Y∞1(z(τ)) = B1Y01(z(τ)) +B2Y02(z(τ))
74
therefore
limτ→ζ1
Y∞1(z(τ))
B1Y01(z(τ)) +B2Y02(z(τ))= 1 .
This implies that Y∞1(ζ1) = B1 . Similarly we show that Y∞2(ζ1) = B3 .
Hence ,
X(ζ1) =
((α1 − α2)−(Λ1+k)B1
χ(α1 − α2)−(Λ2−k)B3
). (4.5.14)
Then solving X(ζ1) = T1 X(ζ1) gives
(α1 − α2)−Λ1−kB1 = a1 (α1 − α2)−Λ1−kB1 + b1 χ(α1 − α2)−Λ2+kB3
⇒ χ = (α1 − α2)Λ2−Λ1−2k · 1− a1
b1
· B1
B3
. (4.5.15)
The discussion of this section is summarized in the following
Theorem 4.5.2. X(τ) ∈ N (3)
0 (ρ) for the admissible multiplier ρ as defined
above , where the components are X(3)
ξ(τ) , ξ = 1, 2 are defined by equation (4.5.9)
which are obtained from the solutions of equation (4.5.4) .
And the story begins
So far we have shown that for any triangle group G there exist infinitely many
rank 2 nearly holomorphic vvaf with respect to any admissible multiplier . In
chapters 6 and 7, it will be shown how with respect to integer k = 0 and when
choosing the eigenvalues Λi , for 1 ≤ i ≤ 6 , in such a way that
6∑i=1
Λi = 0 and u1 + u2 = Λ1 + Λ2 + 1 (4.5.16)
the corresponding nearly holomorphic vvaf generate the module N (3)
0 (ρ) of
nearly holomorphic vvaf with respect to ζ3 over the polynomial ring C[z(3)
(τ)] .
More precisely, for the two distinct choices of u1 , u2 satisfying the condi-
tion (4.5.16) it is shown that with respect to integer k = 0 the nearly holo-
morphic vvaf constructed as discussed in Theorem 4.5.2 will form a free basis
of N (3)
0 (ρ) . Moreover it is shown that this is true in general irrespective to ζ3 .
75
Chapter 5
Classification of Vector-ValuedAutomorphic Forms
Throughout this chapter , let G denote a genus-0 Fuchsian group of the first
kind unless otherwise mentioned explicitly . The algebraic structure of the
space of nearly and weakly holomorphic vvaf of G with respect to an admissible
multiplier ρ of any arbitrary rank d is discussed .
In section 5.1 we will prove that the space N (c)
w (ρ) of nearly holomorphic
vvaf of G with respect to any cusp c ∈ CG
and of any even integer weight w ,
which is defined shortly, is a free module over the ring of nearly holomorphic
scalar-valued automorphic functions of G denoted by N (c)
0 (1) .
In section 5.2 it is shown that the spaceM!w(ρ) of all weakly holomorphic
vvaf of G and any arbitrary weight w ∈ 2Z is also a free module over the ring
RG
of scalar-valued automorphic functions of G , defined in chapter 2 . This is
the main result of this chapter .
We end this chapter giving some conjectures including a very brief sketch
and plan about how to extend the work of this chapter to classify the space of
all holomorphic vvaf, denoted by H(ρ) and the space of all vector-valued cusp
forms (vvcf) , denoted by S(ρ) . These spaces are also free module of rank d
over the ring of scalar-valued holomorphic automorphic forms of G.
76
5.1 Nearly holomorphic automorphic forms
Although more interesting cases to study are weakly holomorphic , holomor-
phic vvaf and vector-valued cusp forms , the simplest and most fundamental
is what we call nearly holomorphic vvaf . The other cases can be handled
through this . Therefore we begin our analysis with the classification of nearly
holomorphic vvaf .
Definition 5.1.1. Let G be any genus-0 Fuchsian group of the first kind with
∞ ∈ CG
. Let ρ : G → GLd(C) be an admissible multiplier of G and c ∈ CG
.
Then a nearly holomorphic vvaf of weight w ∈ 2Z and G with respect
to the cusp c is a weakly holomorphic vvaf X(τ) of weight w such that X(τ)
have poles only at the cusps G-equivalent to cusp c , i.e. X(τ) is holomorphic
everywhere on H∗G\G · c and meromorphic only at the orbit G · c . The space
of all nearly holomorphic vvaf of G and weight w with respect to the cusp c
and multiplier ρ is denoted by N (c)
w (ρ) .
As mentioned in chapter 2 , all hauptmoduls of G with respect to any
c ∈ CG
are examples of nearly holomorphic scalar-valued automorphic func-
tions . Note that whenever the word automorphic function is used in place
of automorphic form , it always means an automorphic form of weight zero .
The reason for studying first nearly holomorphic vvaf is that the correspond-
ing ring of scalar-valued nearly holomorphic automorphic functions is always
a polynomial algebra , i.e. N (c)
0 (1) = C[J(c)
G(τ)] . This information is recorded
in the form of following
Lemma 5.1.2. Let G be any genus-0 Fuchsian group of the first kind . Then
the space of scalar-valued nearly holomorphic automorphic functions with re-
spect to the cusp c ∈ CG
, denoted by N (c)
0 (1) is a polynomial algebra . More
precisely , N (c)
0 (1) = C[J(c)
G(τ)] where J
(c)
G(τ) is the normalized hauptmodul of G
with respect to the cusp c .
77
Proof. By definition of weight 0 nearly holomorphic svaf clearly J(c)
(τ) ∈
N (c)
0 (1) therefore C[J(c)
] ⊆ N (c)
0 (1) . To show N (c)
0 (1) ⊆ C[J(c)
] suppose it is
not true and let f(τ) be a nontrivial element of N (c)
0 (1)\C[J(c)
] . Since any
scalar-valued automorphic function is a rational function in J(c)
(τ) then we
can write f(τ) = p(J(c)
(τ))
q(J(c)
(τ)). Without loss of generality we may assume that
gcd(p(J(c)
(τ)) , q(J(c)
(τ))) = 1 . If deg(q(J(c)
(τ))) > 0 then q(J(c)
(τ)) has at
least one root in C , say z0 . Then J(c)
(τ0) = z0 for some τ0 ∈ H∗G
because
J(c)
(τ) identifies H∗G
with the Riemann sphere P1(C) = C ∪ ∞ . Clearly
τ0 /∈ G · c since J(c)
(c) =∞(6= z0) . Then f(τ0) =∞ and f(τ) has a pole at τ0
thus f(τ) is not holomorphic at τ0 which is contrary to the definition of nearly
holomorphicity of g(τ) .
As an obvious consequence of the above , we write an immediate
Corollary 5.1.3. N (c)
w (ρ) is a module over the polynomial ring N (c)
0 (1) .
Examples 5.1.4.
• If G = Γ(1) . Then c ∈ ∞ and with respect to c = ∞, N (∞)
0 (1) = C[J(τ)] ,
where J(τ) is the normalized hauptmodul of G with respect to the cusp ∞ ,
defined in example 2.2.14 .
• If G = Γ(2) then c ∈ 0, 1,∞ and N (∞)
0 (1) = C[J(τ)] , N (1)
0 (1) = C[J(1)
(τ)]
and N (0)
0 (1) = C[J(0)
(τ)] where J(τ) = 16 · θ43(τ)/θ4
2(τ) − 8 , J(1)
(τ) = 16J(τ)+8
and J(0)
(τ) = 168−J(τ)
are the normalized hauptmoduls of Γ(2) with respect to
the cusps ∞ , 1 and 0 whereas the space M!0(1) = C[J(τ) , J
(0)(τ) , J
(1)(τ)] .
As seen from the above example that in case of weakly holomorphic vvaf ,
this is not always true , i.e. M!0(1) is not necessarily a polynomial ring. Also ,
from now on the subscript G will be dropped as long as there is no confusion .
In addition we will usually drop the variable τ from the hauptmodul J(τ)
whenever it will be used to denote a variable for the polynomial ring C[J] . We
write the following obvious
78
Lemma 5.1.5. Let G be any genus-0 Fuchsian group of the first kind with
∞ ∈ CG
and ρ be an admissible multiplier of G of rank d then for any w ∈ 2Z
and cusp c ∈ CG
, N (c)
w (ρ) is naturally isomorphic to N (c)
0 (ρ⊗ ν−w) as C[J(c)
]-
module .
Proof. Proof follows from Lemma 3.1.1 where the natural isomorphism is de-
fined by X(τ) 7→ ∆−w/2L
GX(τ) .
One of the important tool used in this chapter is the Mittag-Leffler map
referred here as principal part map which is defined in the following
Definition 5.1.6 (Principal part map). Let G be a genus 0 Fuchsian group
of the first kind and ρ : G −→ GLd(C) be any admissible multiplier . Let
X(τ) ∈ Mw(ρ) be a weight w meromorphic vvaf of G for ρ and c ∈ CG.
Choose any exponent λc of G with respect to a cusp c and recall that X(τ) has
an expansion
X(τ) = qλ∞∞
∞∑n=−m1
X∞[n]qn
∞ , if c =∞ ,
and
X(τ) = Pc qλc
cP−1
c (τ − c)−w∞∑
n=−m2
X(c)
[n]qn
c, if c 6=∞ ,
for some m1,m2 ∈ Z . Then , for any exponent λc , the λc-principal part map
Pλc
: N (c)
w (ρ) → Cd[q−1
c] is the map which sends X(τ) to the finite sum of
terms with nonpositive powers in the Fourier expansion of q−λc
c(τ−c)wPcX(τ)
and we write
Pλc (X) =∑n≤0
X(c)
[n]qn
c,
similarly with respect to the cusp ∞ we define the λ∞-principal part map
Pλ∞ : N (∞)
w (ρ)→ Cd[q−1
∞ ] by
Pλ∞
(X) =∑n≤0
X[n]qn
∞ .
79
Example 5.1.7. Recall from example 2 , in case of Γ(1) we have
J(τ) = q−1 + 196884q + · · · ,
E4(τ) = 1 + 240(q + 9q2 + 10q3 + 73q4 + · · · ), and
E6(τ) = 1− 504(q + 33q2 + 244q3 + 1057q5 + · · · ) .
1. In case of d = 1 we can define various principal part maps Pλ : N (∞)
w (1) −→
C[q−1] . Clearly J(τ),E4(τ) ,E6(τ) ∈ N (∞)
w (1) respectively for w = 0 , 4 and 6
therefore
(a) for λ = −1, P−1(J) = 1 , P−1(E4) = 0 and P−1(E6) = 0 .
(b) for λ = 0, P0(J) = q−1 , P0(E4) = 1 and P0(E6) = 1 .
(c) for λ = 1, P1(J) = q−2 +196884 , P1(E4) = q−1 +240 and P1(E6) = q−1 +504 .
2. In case of d = 2 let ρ : Γ(1) −→ GL2(C) be an admissible representation 1⊕
1
and consider X(τ) =
(E
3
4(τ) ,E2
6(τ)
)t
∈ N (∞)
12 (ρ) then
(a) for λ =(
10
0−1
)Pλ(X) =
(q−1 + 720 , 0
)t.
(b) for λ =(
10
00
)Pλ(X) =
(q−1 + 720 , 1
)t.
Let us denote the restriction of Pλc
to any C[J(c)
]-submodule N of N (c)
w (ρ)
by P(N,λc )
.
Lemma 5.1.8. For any cusp c and any exponent λc , the λc-principal part
map Pλc is a linear map over C .
Definition 5.1.9. Let N be any C[J(c)
]-submodule of N (c)
w (ρ) at any cusp c
of CG
. We define the set of all exponents λ of N by E (N) for which the
λ-principal part map P(N,λ)
: N → Cd[q−1
c] is a C-vector space isomorphism
where d is the rank of ρ . Any such λ ∈ E (N (c)
w (ρ)) is called a bijective
exponent
80
Remark 5.1.10. The key to this chapter is our proof below that for any cusp
c ∈ CG
, the set E (N (c)
w (ρ)) is nonempty .
To prove the main result , first we show that for any cusp c ∈ CG
, N (c)
w (ρ)
is a free C[J(c)
]-module of rank d . To show this it is enough to show that
N (∞)
0 (ρ) is a free C[J(∞)]-module of rank d and the proof for the other cusps
will follow similarly using the Lemma 2.2.4 . For a notational purpose from
now on we can always restrict ourself to weight 0 and cusp∞ . We will denote
the cusp width of the cusp ∞ by h and the normalized hauptmodul J(∞)
G(τ)
simply by J . To prove the result , the following difficult lemma proved in [5]
is needed . In what follows eξ denote the column vector of order d × 1 which
consists 1 at the ξ-entry and 0 elsewhere .
Lemma 5.1.11. Let G be any genus-0 Fuchsian group of the first kind and
ρ be a rank d representation of G . Then there exist d linearly independent
nearly holomorphic vvaf Y1,Y2, · · · ,Yd in N (∞)
0 (ρ) over C and an exponent
λ0 such that Pλ0
(Yξ) = eξ for every 1 ≤ ξ ≤ d .
Proof. To prove the existence of rank d vvaf Y1,Y2, · · · ,Yd in N (∞)
0 (ρ) is
similar to the proof of existence of rank 2 vvaf of triangle groups shown in
chapter 4 . For more details see [5].
Corollary 5.1.12. Let N0 be the C[J]-span of Yξ’s and the exponent λ0 of
Lemma 5.1.11 such that Pλ0(Yξ) = eξ , then P(N0 ,λ0) : N0 → Cd[q−1
] is a
surjective vector space homomorphism over C.
Proof. N0 is a C[J]-submodule of N (∞)
0 (ρ). Construct a d×d-matrix Ξ0 whose
columns are Yξ. N0 is a free C[J]-submodule of N (∞)
0 (ρ) of rank d , i.e. Yξ’s
are linearly independent over C[J] , this follows since determinant det(Ξ0) has
leading term qtrλ0 and so does not vanish identically for all τ ∈ H since q = 0
only at τ =∞ .
81
Now, we show that P(λ0,N0) : N0 → Cd[q−1
] is surjective. Choose any
polynomial p(q−1
) ∈ Cd[q−1
] , i.e.
p(q−1
) =
p1(q
−1)
p2(q−1
)...
pd(q−1
)
=
a10 + a11q
−1+ a12q
−2+ · · ·
a20 + a21q−1
+ a22q−2
+ · · ·...
ad0 + ad1q−1
+ ad2q−2
+ · · ·
.
Suppose the maximal degree of the components of p is np . Write aηnp
for the corresponding coefficient (0 is also a possible value) of the component
pη for every 1 ≤ η ≤ d , i.e. pη(q−1
) = aηnp q−np
+ · · · . Rearranging the
components of p(q−1
) we get
p(q−1
) =
a1np q
−np+ a1np−1q
−np+1+ a1np−2q
−np+2+ · · ·+ a11q
−1+ a10
a2np q−np
+ a2np−1q−np+1
+ a2np−2q−np+2
+ · · ·+ a21q−1
+ a20...
adnp q−np
+ adnp−1q−np+1
+ adnp−2q−np+2
+ · · ·+ ad1q−1
+ ad0
.
Define p(1)(q−1
) = p(q−1
)−∑
η aηnpJnp q
−λ0Yη , which is equivalent to write
p(1)(q−1
) =
a1np q−np
+ a1np−1q−np+1
+ · · ·+ a10
a2np q−np
+ a2np−1q−np+1
+ · · ·+ a20
...
adnp q−np
+ adnp−1q−np+1
+ · · ·+ ad0
−Jnp
a1np + α11q + α12q2
+ · · ·a2np + α21q + α22q
2+ · · ·
...
adnp + αd1q + αd2q2
+ · · ·
where αξη = aξη
∑dη=1 aηnp . Then the leading power of the components of
p(1)(q−1
) will be ≤ np − 1. Now, suppose the leading power of the compo-
nents of p(1)(q−1
) is np − 1 and say the corresponding coefficients are a(1)
ηnp−1 .
Consider the polynomial
p(2)(q−1
) = p(1)(q−1
)−∑η
a(1)
ηnp−1Jnp−1q−λ0Yη
= p(q−1
)−∑η
aηnpJnp q
−λ0Yη −∑η
a(1)
ηnp−1Jnp−1q−λ0Yη
⇒ p(2)(q−1
) = p(q−1
)−∑η
(aηnpJnp + a
(1)
ηnp−1Jnp−1)q−λ0Yη
then by the same arguments as for p(1)(q−1
) , the leading power of the compo-
nents of p(2)(q−1
) is ≤ np − 2 . Recursively repeating this process for finitely
82
many times and more precisely ≤ np times we get the expression
p(np)(q−1
) = p(q−1
)−∑η
(aηnpJnp +a
(1)
ηnp−1Jnp−1 + · · ·+a(np−1)
η1 J+a(np)
η0 )q−λ0Yη
such that the leading power of the components of p(np) is< 0 i.e. no components
in the expression of p(np)(q−1
) will carry the nonpositive powers of q . Now,
write
X(τ) =∑η
(aηnpJnp + a
(1)
ηnp−1Jnp−1 + · · ·+ a(np−1)
η1 J + a(np)
η0 )Yη.
Clearly X(τ) ∈ N0 since N0 is the C[J]-span of Yξ : 1 ≤ ξ ≤ d , and
q−λ0X(τ) = p(q
−1) − p(np)(q
−1) implies that P(N0,λ0)(X) = p(q
−1) . This
establishes the surjectivity of the λ0-principal part map of submodule N0 .
As far as no confusion arises from now on Pλ will be used in place of P(N,λ) .
We begin with the following
Definition 5.1.13. Let X(τ) ∈M!0(ρ) be a vvaf which is not identically zero
in H∗G
. Then the leading power of X(τ) is defined to be the smallest power of
q appears in the series expansion of any component Xη of vvaf X(τ) , denoted
by l.p.(Xη) i.e. l.p.(X) = minηl.p.(Xη) . When the powers of q are in C then
l.p.(Xη) will be taken that power of q whose real part is the smallest .
Now , we state an important lemma proved in [5] .
Lemma 5.1.14. For any c ∈ CG
, there exists a constant C = C(ρ, w) such
that for every non constant X(τ) ∈ N (c)
w (ρ) , minηl.p.(Xη) ≤ C.
Proof. For complete proof see [5] .
Lemma 5.1.15. There exists an exponent Λ such that the Λ-principal part
map PΛ : N (∞)
0 (ρ) → Cd[q−1
] is an injective vector space homomorphism
over C .
83
Proof. Proof follows immediately by choosing any exponent Λ withmin(Λξξ) >
C and Lemma 5.1.14 .
Note 5.1.16. For complex numbers z, w we define z ≥ w if Re(z) ≥ Re(w) .
Similarly in case of Λ ,Ω are d× d diagonal matrices then we define Λ ≥ Ω if
Re(Λii) ≥ Re(Ωii) for every 1 ≤ i ≤ d .
Corollary 5.1.17. For any exponent λ and any C[J]-submodule N of N (∞)
0 (ρ),
ker(P(λ,N)) is a finite dimensional vector space over C.
Proof. It is known from Lemma 5.1.15 that there exists an exponent Λ such
that ker(PΛ) = 0. Also, for any exponent λ ≥ Λ it is clear that ker(Pλ) = 0
i.e. ∀ λ ≥ Λ, ker(Pλ) is a finite dimensional vector space. If λ 6= Λ, let Λξξ =
maxΛξξ , λξξ and then the exponent Λ′ = Diag(Λ′11 , · · · ,Λ′dd) can be chosen
for which ker(PΛ′) = 0 and λ ≤ Λ′ . So, without loss of generality for any
exponent λ we can always find an exponent Λ such that λ ≤ Λ . This implies
that λξξ < Λξξ for all 1 ≤ ξ ≤ d . Consider Λ−λ = λ′= Diag(m1,m2, · · · ,md)
with mi ∈ Z . Let X(τ) be any nonzero element of kerPλ then this implies that
q−λX(τ) has no nonpositive expression , i.e.
q−λX(τ) =
a11q + a12q
2 + · · ·a21q + a22q
2 + · · ·...
ad1q + ad2q2 + · · ·
⇒ q−ΛX(τ) = q
−λ′
a11q + a12q
2 + · · ·a21q + a22q
2 + · · ·...
ad1q + ad2q2 + · · ·
, since Λ− λ′ = λ
⇒ q−ΛX(τ) =
a11q
−m1+1+ · · ·+ a1m1−1q
−1+ a1m1 + a1m1+1q + · · ·
a21q−m2+1
+ · · ·+ a2m2−1q−1
+ a2m2 + a2m2+1q + · · ·...
ad1q−md+1
+ · · ·+ admd−1q−1
+ admd + admd+1q + · · ·
84
⇒ PΛ(X) =
a11q
−m1+1+ · · ·+ a1m1−1q
−1+ a1m1
a21q−m2+1
+ · · ·+ a2m2−1q−1
+ a2m2
...
ad1q−md+1
+ · · ·+ admd−1q−1
+ admd
and since ker(PΛ) = 0 and as per our choice X(τ) 6= 0 , there must be at least
one nonzero coefficient aξr , where 1 ≤ r ≤ mξ for every 1 ≤ ξ ≤ d . This
implies that every nonzero element of ker(Pλ) has principal part of the above
form with respect to the injective exponent Λ and we can take the basis of
ker(Pλ) ⊆Yξj ∈ N
(∞)
0 (ρ) | PΛ(Yξj) = q
−mξ+j
eξ, 1 ≤ ξ ≤ d, 1 ≤ j ≤ mξ
.
Clearly , dimension of ker(Pλ) ≤ m1 + m2 + · · · + md = tr(λ′) = tr(Λ − λ)
i.e. ker(Pλ) ≤ tr(Λ − λ) . Hence, for any exponent λ , ker(Pλ) is a finite
dimensional vector space .
Theorem 5.1.18. The set E (N (∞)
0 (ρ)) is nonempty, i.e. there is an exponent
Λ such that PΛ : N (∞)
0 (ρ)→ Cd[q−1
] defines an isomorphism.
Proof. To prove the theorem, we wish to find a bijective exponent Λ for
N (∞)
0 (ρ) . Following Corollary 5.1.12 the map P(N (∞)
0 ,λ0)is surjective . For nota-
tional convenience λ0 will be denoted by λ(0) . Let us choose X(τ) ∈ ker(Pλ(0)) .
This implies that q−λ(0)
X(τ) has no nonpositive terms i.e.
X(τ) = qλ(0)
a11q + a12q
2 + · · ·a21q + a22q
2 + · · ·...
ad1q + ad2q2 + · · ·
.
This implies that for all η (1 ≤ η ≤ d) , Xη(τ) = qλ
(0)ηη ·
(aη1q + aη2q
2 + . . .)
and therefore for all η, l.p(Xη) ≥ λ(0)ηη +1. Let us define µη := l.p(Xη)−λ(0)
ηη −1,
85
for all η. Then X(τ) is rewritten as follows
X(τ) = qλ(0)
aµ11q
µ1+1+ aµ12q
µ1+2+ . . .
aµ21qµ2+1
+ aµ22qµ2+2
+ . . ....
aµd1qµd+1
+ aµd2qµd+2
+ . . .
Now, choose any η for which µη is minimal and set λ(1) = λ(0) +(µη +1)Eη.
Here Eη is the elementary matrix which has 1 at ηη position and 0 elsewhere.
So λ(1) = Diag(λ(0)11 , λ
(0)22 , · · · , λ
(0)ηη + µη + 1, · · · , λ(0)
dd ). This implies that
q−λ(1)
X(τ) = q−λ(0)−(µη+1)EηX(τ) = q
−(µη+1)Eηq−λ(0)
X(τ)
⇒ q−λ(1)
X(τ) = q−(µη+1)Eη
aµ11q
µ1+1+ aµ12q
µ1+2+ · · ·
...
aµη1qµη+1
+ aµη2qµη+2
+ · · ·...
aµd1qµd+1
+ aµd2qµd+2
+ · · ·
⇒ q−λ(1)
X(τ) =
0 + aµ11q
µ1+1+ · · ·
...aµη1 + aµη2q + · · ·
...
0 + aµd1qµd+1
+ · · ·
This implies that P
(N (∞)0 ,λ(1))
(X) = aµη1eη, therefore X(τ) /∈ kerPλ(1) and fol-
lowing the method of the proof of Corollary 5.1.12 we show that P(N (∞)
0 ,λ(1))
remains surjective . Therefore P(N (∞)
0 ,λ(1))is surjective for λ(1) ≥ λ(0) . Hence,
an exponent λ(1) is found for N (∞)
0 such that P(N (∞)
0 ,λ(1))is still surjective but
ker(P
(N (∞)0 ,λ(1))
)6= ker
(P
(N (∞)0 ,λ(0))
), as X(τ) ∈ ker
(P
(N (∞)0 ,λ(0))
)but X(τ) /∈
ker(P
(N (∞)0 ,λ(1))
). Repeating this process recursively an exponent λ(i+1) ≥ λ(i) ,
for i ≥ 0 will be found such that each P(N (∞)
0 ,λ(i+1))is still surjective but
ker(P
(N (∞)0 ,λ(i+1))
)6= ker
(P
(N (∞)0 ,λ(i))
). Since ker
(P
(N (∞)0 ,λ(0))
)is finite dimen-
sional vector space therefore this process will terminate after finitely many
steps . Also note that ker(P
(N (∞)0 ,λ(i+1))
)( ker
(P
(N (∞)0 ,λ(i))
)therefore this gives
a finite sequence of subspaces
86
0 ( ker(P
(N (∞)
0 ,λ(i+1))
)( ker
(P
(N (∞)
0 ,λ(i))
)( · · · ( ker
(P(N∞0 ,λ(1))
)( ker
(P(N∞0 ,λ(0))
).
The last of these exponents λ(i) such that ker(P(N∞0 ,λ(i))
)= 0 is the required
bijective exponent Λ for N (∞)
0 . This proves that the set of bijective exponents
for C[J]-module N (∞)
0 (ρ) is nonempty . Hence there exists an exponent Λ such
that P(N (∞)
0 ,Λ): N (∞)
0 → Cd[q−1
] is a vector-space isomorphism over C i.e. as
a vector spaces N (∞)
0 (ρ) ≈ Cd[q−1
] .
Before moving further, let’s see some examples of injective, surjective, bi-
jective exponents and their corresponding principal part map in the case of
G = Γ(1).
Example 5.1.19. Consider d = 1 and ρ be the trivial multiplier of G then
M!0(ρ) = C[J] where the normalized hauptmodul J(τ) = q−1 + 196884q + . . .
and q = exp(2πiτ).
• Let exponent λ = (0) . Then P(0) : C[J] → C[q−1] implies that P(0)(J) = q−1
and ker(P(0)) = C . So, in this case P(0) is not injective .
• Now, let exponent λ = (1) . Then P(1) : C[J]→ C[q−1] implies that P(1)(J) =
q−2 + 196884 and ker(P(1)) = 0 . Therefore in this case P(1) is injective .
• More precisely, it can be observed that for every λ ≥ 1 Pλ is injective and for
every λ ≤ 1 Pλ is surjective . Therefore it is easy to observe that E(N (∞)
0 (1)) =
1 .
Corollary 5.1.20. N (∞)
0 (ρ) is a free C[J]-module of rank d and therefore any
C[J]-submodule N of N (∞)
0 (ρ) is also a free module of rank ≤ d .
Proof. Clearly N (∞)
0 (ρ) is a C[J]-module and from Theorem 5.1.18 there exists
a bijective exponent Λ such that the map PΛ : N (∞)
0 (ρ) → Cd[q−1] is an
isomorphism . This implies that for every eξ, 1 ≤ ξ ≤ d there exists a nearly
holomorphic vvaf Yξ such that PΛ(Yξ) = eξ . These Yξ : 1 ≤ ξ ≤ d
87
generate N (∞)
0 (ρ) over C . Now, consider the d×d matrix Ξ(τ) whose columns
are these Yξ’s then we see that determinant det(Ξ) has leading term qtrΛ
and so does not vanish identically for all τ ∈ H . Let N be the field of all
meromorphic functions on H then all the d2 entries of Ξ(τ) will lie in this field
N . Since, det(Ξ) is nonzero on H therefore these d-column vectors Yξ’s are
linearly independent over N . Clearly, C[J] is lying inside N therefore these
d nearly holomorphic vvaf Yξ’s are also linearly independent over C[J] . Now
we show that the C[J]-span of Yξ : 1 ≤ ξ ≤ d is N (∞)
0 (ρ) . Let X(τ) be any
arbitrary element of N (∞)
0 (ρ). Suppose
PΛ(X) =
α1m1 q
−m1 + α1m1−1q−m1+1
+ · · ·+ α11q−1
+ α10
α2m2 q−m2 + α2m2−1q
−m2+1+ · · ·+ α21q
−1+ α20
...
αdmd q−md + αdmd−1q
−md+1+ · · ·+ αd1q
−1+ αd0
.
Now write the exponents m1,m2, · · · ,md in their increasing order . Let us
write the exponents as µ1 ≤ µ2 ≤ · · · ≤ µd where ∀j µj ∈ m1, · · · ,md .
With respect to this order of the exponents we rewrite the PΛ(X) as follows :
PΛ(X) =
α1µd q
−µd + α1µd−1qµd−1
+ · · ·+ α11q−1
+ α10
α2µd q−µd + α2µd−1q
µd−1+ · · ·+ α21q
−1+ α20
...
αdµd q−µd + αdµd−1q
µd−1+ · · ·+ αd1q
−1+ αd0
then there exist polynomials
pξ(J) = αξµdJµd + αξµd−1J
µd−1
+ · · ·+ αξ1J + αξ0
of degree at most µd for each ξ, 1 ≤ ξ ≤ d such that
PΛ
( d∑ξ=1
pξ(J)Yξ
)= PΛ(X) .
Therefore X(τ) =∑d
ξ=1 pξ(J)Yξ . Since Ξ(τ) is invertible in H therefore we
can conclude that these polynomials pξ(J),∀ 1 ≤ ξ ≤ d are unique polynomials
in C[J] such that X(τ) = Ξ(τ) · p(J) where p(J) =(p1(J), p2(J), · · · , pd(J)
)t ∈Cd[J] .
88
Corollary 5.1.21. For any Λ ∈ E (N (∞)
0 ) there exists a fundamental matrix
Ξ(τ) = Ξ(N (∞)
0 ,Λ)(τ) obeying the limit
q1−Λξξ
Ξ(τ)ξη
= δξη +O(q) as q → 0 .
Proof. This is an immediate corollary of Theorem 5.1.18 . Ξ(τ) is a d × d
matrix whose columns are from the set Yξ : 1 ≤ ξ ≤ d . Observe that
Ξ(τ)ξη = Yηξ(τ) and therefore for fixed η and for all ξ
q−ΛξξYη
ξ = δξη + αηξ1q + αηξ2q2 + · · ·
⇒ q−ΛξξYη
ξ = δξη + αηξ1q + αηξ2q2
+ · · ·
⇒ q−ΛξξYη
ξ = δξη +O(q), as q −→ 0
⇒ q−Λξξ
Ξ(τ)ξη = δξη +O(q), as q −→ 0
Definition 5.1.22. The matrix Ξ(τ) := Ξ(∞)
(τ) corresponding to the bijective
exponent Λ is called the fundamental matrix of N (∞)
0 (ρ). This is expressed
as
Ξ(τ) = qΛ
(I +
∞∑n=1
Ξ[n]qn
),
where Ξ1 is denoted by X and called characteristic matrix of N (∞)
0 (ρ) .
5.2 Weakly holomorphic automorphic forms
This section gives the proof of the main result of this chapter which is achieved
by observing that the space M!w(ρ) is the C-span of all N (c)
w (ρ) at all cusps
c ∈ CG
. As mentioned earlier to show this for any w ∈ 2Z is equivalent to
show it for w = 0, we begin with the following :
Let x ∈ CG\∞ be any finite cusp of G and Ξ(τ) be the fundamental
matrix of N (∞)
0 (ρ) then the expansion of Ξ(τ) around x is of the following
89
form
Ξ(τ) =∞∑n=0
Ξ〈n〉qn
(x)
Lemma 5.2.1. Ξ〈0〉 is invertible .
Proof. If it is invertible then nothing to show . Suppose not then 0 is one of
its eigenvalue . Let v be the eigenvector of Ξ〈0〉 for 0 then Ξ〈0〉 · v = 0 . This
implies that J(x)
(τ)Ξ(τ) · v ∈ N (∞)
0 (ρ) but
J(x)
(τ)Ξ(τ) · v 6= Ξ(τ) · p(J(τ))
for any p(J(τ)) ∈ C[J(τ)] .
Lemma 5.2.2. Let x ∈ CG\∞ be any finite cusp of G then for any p(q
−1
x) ∈
Cd[q−1
x] there exists X(τ) ∈ M!
0(ρ) such that X(τ) = Ξ(τ) · P for some P ∈
Cd[J(τ) , J(x)
(τ)] and
PΛhol
(x)
(X(τ)) = p(q−1
x) , and
PΛhol
(c)
(X(τ)) = 0 , ∀c ∈ CG\∞ , x . (5.2.1)
Proof. Write KG
= Cd[q−1c1, · · · , q−1
cn] then observe that with respect to every
c ∈ CG
there exists a C-linear map PΛhol
(c)
: M!0(ρ) −→ K
G. Let x ∈ C
G\∞
and p(q−1
x) ∈ Cd[q
−1
x] then the result is proved by applying induction on the
maximal degree of p(q−1
x) . Suppose the maximal degree of p(q
−1
x) is 0 then
p(q−1
x) = p ∈ Cd and
X(τ) = Πc∈CG\∞,x
(J(τ)− J(c)
J(x)− J(c)
)· Ξ(τ) · Ξ −1
〈0〉 · p ∈M!0(ρ)
satisfies the condition (5.2.1) . Now suppose result is true for all p(q−1
x) of
maximal degree less than N . Let
p(q−1
x) = p[N ]q
−N
x+ p
(1)
(q−1
x) (5.2.2)
90
be the polynomial of maximal degree N where p[N ] ∈ Cd and maximal de-
gree of p(1)
(q−1
x) is N − 1 . Then by induction there exists Y(τ) ∈ M!
0(ρ)
satisfying (5.2.1) . Consider
X(τ) = Πc∈CG\∞,x
(J(τ)− J(c)
J(x)− J(c)
)· Ξ(τ) · Ξ −1
〈0〉 · p[N ]J(x)
(τ)N ∈M!0(ρ) .
Since p[N ]J(x)
(τ)N = p[N ]q−N
x+p
(2)(q−1
x) where the maximal degree of p
(2)(q−1
x)
is N − 1 . Write
Y(τ) = Πc∈CG\∞,x
(J(τ)− J(c)
J(x)− J(c)
)· Ξ(τ) · Ξ −1
〈0〉 · p(2)
(q−1
x) ∈M!
0(ρ)
and Y(τ) satisfy (5.2.1) . Now observe that X(τ) − Y(τ) satisfy (5.2.1) for
p[N ]q−N
x. Hence , X(τ) = X(τ) − Y(τ) + Y(τ) ∈ M!
0(ρ) will be the desired
candidate for the polynomial (5.2.2) satisfying (5.2.1) .
Theorem 5.2.3. M!0(ρ) is a free R
G-module of rank d.
Proof. Let X(τ) ∈ M!0(ρ) . For each x ∈ C
G\∞ let p(q
−1
x) = P
Λholx
(X) and
X(x)(τ) be the vvaf found in Lemma 5.2.2 i.e. P
Λholc
(X(x)(τ)) = 0 for c 6= x,∞
and PΛholx
(X(x)(τ)) = p(q
−1
x) , and X(x)
(τ) = Ξ(τ)P for some P ∈ Cd[J , J(x)
] .
Let Λ be the bijective exponent of N (∞)
0 (ρ) with respect to the fundamental
matrix Ξ(τ) then there exists a vvaf X(∞)(τ) such that
PΛ(X(∞)
) = PΛ(X)−∑x6=∞
PΛ(X(x)
) .
Now since Λ is a bijective exponent therefore
⇒ X(∞)
+∑x6=∞
X(x) − X = 0
⇒ X(τ) = X(∞)
(τ) +∑x6=∞
X(x)
(τ)
⇒ X(τ) = Ξ(τ) · P
where P ∈ Cd[J(τ) , J(x)
(τ)]x∈CG\∞ = Cd[J
c1 (τ) , · · · , Jcn(τ)] with c1 =∞ .
91
Definition 5.2.4. The matrix Ξ(τ) corresponding to the bijective exponent
Λ, is the fundamental matrix ofM!w(ρ), which has the following type of series
expression depending on the cusp c of G.
Ξ(τ) = qΛ
(I + X q +
∞∑n=2
Ξnqn
), when c =∞ ,
Ξ(c)
(τ) = P−1
c qΛcPc(τ − c)−w
(I + X (c)
qc +∞∑n=2
Ξ(c)
n q nc
), when c 6=∞,
(5.2.3)
in all the cases, Ξ(c)
1 called the characteristic matrix of G with respect to the
cusp c, denoted by X (c), of order d.
5.3 Differential operators
An important ingredient of this theory is the differential operator . Let G be a
genus-0 Fuchsian group of the first kind and z ∈ EG∪ C
G. Then the derivation
is defined by
Dw := D(w,z)
= D(z)
0− w
kzE
G
(2,z)(τ) (5.3.1)
maps M!w(ρ) to M!
w+2(ρ) where kz denotes the weight of the Ramanujan G-
Delta form with respect to z and EG
(2,z)(τ) is a quasi-automorphic form of G
with respect to the point z which is a solution of the equation
D(z)
0·∆(z)
G(τ) = ∆
(z)
G(τ) · EG
2 (τ) (5.3.2)
where
D(z)
0=
hc τ2
2πiddτ
= qz · ddqz
, if z = c ∈ CG
(ω−ω)n
ddτ
= q (1− 1n
)z
· (1− q 1n
z)2 · d
dqz, if z = ω ∈ E
G
,
here n is the order of the elliptic fixed point w .
92
Theorem 5.3.1. Let G be any Fuchsian group of the first kind and ρ be any
rank d admissible multiplier of G. Let c ∈ CG
such that ρ(tc) := Tc is a
diagonalizable matrix with Λ(c)
be the bijective exponent of c. Let Ξ(c)
(τ) be the
fundamental matrix of G with respect to the exponent Λ(c)
of the form given
in (5.2.3). Then
∇wΞ(c)
(τ) = Ξ(τ)
((J + ac)Λ
(c)
+ bc
)(5.3.3)
where
1. ∇w = f(τ) ·D(c)
w for nonzero f(τ) ∈ N (c)
−2(1) ,
2. ac is the constant term of f(τ) and bc = X (c)
w + [Λ(c)
w ,X(c)
w ] ,
3. Λ(c)
w = Λ(c)
0 − w2LId, X
(c)
w = X (c)
0 + 2wI .
Proof. For similar reasons it is enough to show this at the level of c =∞ and
w = 0 . Let ||EG∪ C
G|| = l + n and J(τ) be the normalized hauptmodul of G
with respect to cusp ∞ , then define
f(τ) =1
J′·
∏z∈E
G∪C
G\∞
J(τ)− J(z)
where J′ = D0J . Clearly ,
a = −∑
z∈EG∪C
G\∞
J(z) .
In this case (5.3.3) becomes
∇Ξ(τ) = Ξ(τ)
((J + a)Λ + b
). (5.3.4)
Observe that ∇ : N0(ρ) −→ N0(ρ) therefore each column of Ξ(τ) will get
mapped to some element in N0(ρ) . This implies that there exists a matrix D
such that
∇Ξ(τ) = Ξ(τ)D .
93
Now by applying PΛ on both sides and comparing the coefficients will give the
desired D in (5.3.4) . For any w ∈ 2Z the value of Λw,Xw in part 3 is a direct
consequence of Lemma 3.1.1 .
Theorem 5.3.2. Assume that w = 0 then with respect to the regularized
hauptmodul z(c)
G(τ) := z(τ) at cusp d of G , the fundamental matrix satisfies the
following Fuchsian differential equation
d
dzΞ
(c)
(z) = Ξ(c)
(z)
( ∑c∈C
G\d
Λ(c)
z(τ)− z(c)
)(5.3.5)
where
Λ(d)
+∑
c∈CG\d
Λ(c)
= 0 .
Proof. For notational convenience consider G be a triangle group of type
(` ,m ,∞) and give the details of the proof when c =∞ . Let ρ : G −→ GLd(C)
be an admissible multiplier with exponent matrices A ,B ,Λ respectively at
points ζ1 , ζ2 , ζ3 . Consider a regularized hauptmodul
z(τ) =∞∑
n=−1
an qn
3
of G with respect to cusp ∞ where q is the local variable in the punctured
disc follows from equation (4.1.5) . Normalize this hauptmodul in such a way
that its expansion is of the form
J(τ) = q−1
+∞∑n=1
bn qn
.
Write the values of regularized and normalized hauptmoduls in the following
table
τ J(τ) z(τ)
ζ3 ∞ ∞
ζ2 α2 1
ζ1 α1 0
94
where
z(τ) =α1 − J(τ)
α1 − α2
=J(τ)− α1
α2 − α1
. (5.3.6)
Define the ∇ := f(τ) ·D0 and f(τ) is the generator of free module N (ζ3)
−2 (1) of
rank 1 over C[J] which is defined as
f(τ) = −(J− α2)(J− α1)
J′= (α1 − α2)
z(z− 1)
z′. (5.3.7)
where J′= D0(J) and z
′= D0(z) . From Theorem 5.3.1 there exists a bijective
exponent Λ and a fundamental matrix Ξ(z) satisfying (5.3.3) i.e.
∇Ξ(τ) = Ξ(τ)((J + a)Λ + b
)(5.3.8)
where a = −(α1 + α2) is a constant term of f(τ) and b = X + [Λ,X ].
∇Ξ(z) = f(τ) qd
dqΞ(z) . (5.3.9)
By using chain rule this can be written
d
dzΞ(z) =
d
dqΞ(z) · d q
dz
⇒ d
dzΞ(z) =
1
f(τ)· 1
q· ∇Ξ(z) · dq
dz, from (5.3.9)
⇒ d
dzΞ(z) = Ξ(z)
((J + a)Λ + b
)· 1
f(τ) · q dzdq
⇒ d
dzΞ(z) = Ξ(z)
(J + a)Λ + b
(α1 − α2)z(z− 1),
since ∇z = f(τ) · q dzdq
= f(τ) · z′ = (α1 − α2)z(z− 1) . This implies that
d
dzΞ(z) = Ξ(z)
(Az
+B
z− 1
)(5.3.10)
where
A =α2
(α1 − α2)Λ− b
(α1 − α2),
B = − α1
(α1 − α2)Λ +
b
(α1 − α2). (5.3.11)
Clearly,
A+ B + Λ = 0 and α1A+ α2B = −b . (5.3.12)
95
5.4 Concluding remarks and conjectures
As for a fixed ρ : G −→ C× the set of all rank 1 nearly holomorphic modular
functions with respect to cusp ∞ for Γ(1) is a free module of rank 1 over the
polynomial ring C[J] where J is the normalized hauptmodul of the group Γ(1)
with respect to the cusp ∞ and the set of all rank 1 holomorphic modular
forms for Γ(1) is a free module of rank 1 over the polynomial ring C[E4, E6],
here E4, E6 are Eisenstein series of weight 4 and 6 respectively .
Now, from this chapter we know that in case of any genus 0 Fuchsian
group G of the first kind and for an admissible multiplier ρ of rank d the
above statement generalizes as well i.e. the set N (ζ)
0 (ρ) of all rank d nearly
holomorphic vector-valued automorphic functions with respect to ζ ∈ EG∪ C
G
is a free module of rank d over the polynomial ring C[J(ζ)
G(τ)] . Similarly
it is also true for the set M!0(ρ) of all rank d weakly holomorphic vector-
valued automorphic functions is a free module of rank d over the ring RG
=
C[J(ω1)
, · · · , J(ω`) , J(c1)
, · · · , J(cn)] of all weakly holomorphic scalar-valued au-
tomorphic functions of G where ` = ||EG|| and n = ||C
G|| . In the case of
triangle groups this generalization is discussed in the next chapter where we
have extended the definition of nearly holomorphic vvaf not only with respect
to cusps but also to the elliptic fixed points of triangle groups .
However the similar structural problem has not been answered yet in this
thesis in case of rank d holomorphic vvaf of G . Hence this goal is recorded
in the form of following
Conjecture 5.4.1. Let G be any genus-0 Fuchsian group of the first kind . Let
ρ : G −→ Cd be any representation of rank d . Then the set of all holomorphic
vvaf H(ρ) =⊕
w∈RHw(ρ) is a free module of rank d over the ring H(1) =⊕w∈RHw(1) of holomorphic scalar-valued automorphic forms .
The above conjecture has been proved for G = Γ(1) in [35] and in [17] .
96
Also in the current setting of structural questions one challenging open
problem could be the following
Problem. Find such Fuchsian group G (of positive genus) for which the
M!w(ρ) will not be a free module with respect to some multiplier ρ of G and
weight w ∈ C .
In this section, the idea how to classify the space H≥λ(ρ) of holomorphic
automorphic vvaf and the space S≥λ(ρ) of vector-valued cusp forms (vvcf) for
any admissible multiplier ρ from the approach used to classify the space of
weakly holomorphic vvaf of G is sketched .
From the definition 2.3.1 we know ρ(t∞) = exp(2πiλ) where λ ∈Md(C) is
a diagonal matrix . Note that
H≥λ(ρ) =⊕w∈2Z
H≥λw (ρ) and S≥λ(ρ) =⊕w∈2Z
S≥λw (ρ) .
Recall the definition 2.3.7. Let’s first classify the spaces of λ-cusp forms and
λ-holomorphic forms for a fixed weight w, then λ-cusp forms are defined as
S≥λw (ρ) = ker(Pλ) . This simply means that for any X ∈ S≥λw (ρ), q−λX(τ) =∑
n≥1 X[n]qn
Similarly, Y(τ) ∈ H≥λw (ρ) simply means that
q−λY(τ) =
∑n≥0
Y[n]qn
. (5.4.1)
Observe that q−(λ−I)Y(τ) = q
Iq−λX(τ) then from (5.4.1) we have q
−(λ−I)Y(τ) =∑n≥1 Y[n]q
n. This implies that Y(τ) ∈ ker(P(λ−I)) = S≥(λ−I)
w (ρ). So to clas-
sify the λ-holomorphic forms of weight w is equivalent to classify the (λ− I)-
cusp form of weight w. Therefore for any exponent λ,
Hλw(ρ) = S(λ−I)
w (ρ)
So it is enough to classify one of these spaces with respect to exponent λ
and weight w. Recall that the map Pλ(ρ) : M!w(ρ) −→ K
Gwhere K
G=
97
Cd[q−1ζ1, · · · , q−1
ζl+n] for ζi ∈ EG
∪ CG
and 1 ≤ i ≤ l + n . Consider the following
diagram at the level of fixed weight w
kerPλ// kerP(λ−I)
Sλw(ρ)
∆−w2L
//Hλw(ρ)
∆−w2L
//M!w(ρ)
Pλ //
∆−w2L
KG
M!0
− w2L
+1
(ρ⊗ ν−w) //M!0
− w2L (ρ⊗ ν−w) //M!
0(ρ⊗ ν−w)Pλ // K
G
kerP(λ− w2LI)(ρ⊗ ν−w) // kerP(λ−( w
2L+1)I)(ρ⊗ ν−w)
Following from Corollary 5.1.17 for any exponent λ, kerPλ is a finite dimen-
sional vector space with dim(kerPλ) ≤ max(0, tr[Λw−λ]) where Λ is a bijective
exponent of M!w(ρ) . As one of the future project we would like to show that
for any exponent λ there exists a minimal weight w0 such that H≥λw (ρ) = 0 for
all w < w0 and on the basis of various examples we conjecture the following
Conjecture 5.4.2. w0 = 2Ld
tr(λ) + 1− d . Moreover, dimC(H≥λw0(ρ)) = 1.
We rewrite the space of all holomorphic vvaf of G for ρ denoted by H≥λ(ρ)
then
H≥λ(ρ) =⊕`≥0
H≥λw` (ρ), w` = w0 + 2` .
In short we write ∀`,H≥λw` (ρ) = H` . An important question is to find the
dimension of H≥λ` with respect to any exponent λ .
98
Chapter 6
Vector-Valued AutomorphicForms of Triangle groups - II
In this chapter the rank 2 even integer weight nearly and weakly holomorphic
vvaf of any triangle group G with respect to any rank 2 admissible multiplier
are explicitly classified while building an abstract connection with hyperge-
ometric and Riemann’s differential equations . Unless otherwise mentioned
explicitly , throughout this chapter a vvaf X(τ) denotes the nearly or weakly
holomorphic vvaf of rank 2 and the representation ρ : G −→ GL2(C) always
stands for a rank 2 admissible representation of G . There is an obvious gen-
eralization of this approach to classify the higher rank vvaf of any weight
w ∈ 2Z .
The strategy is that rank d vvaf for any genus-g Fuchsian group G of
the first kind are equivalent to an order d ordinary differential equation with
regular singular points corresponding to the points in the set EG∪ C
Gon a
compact surface of genus-g . Thus, vvaf of the triangle group G correspond to
a Fuchsian differential equation on a sphere with three regular singular points .
We show that in the case of classifying rank 2 vvaf of triangle groups, this is
equivalent to the study of the hypergeometric differential equation using the
Riemann schemes .
99
6.1 Fuchsian differential equations and hyper-
geometric functions
In this part, we are interested in Fuchsian differential equations on the sphere
with three regular singular points. Its standard form is
d
dzW = W
(Az
+B
z − 1
)(6.1.1)
whereW (z) is a d×dmatrix valued function of z andA and B are d×d constant
matrices and we call equation (6.1.1) a rank d first order Fuchsian differential
equation . The singular points of this equation are 0, 1 and ∞. The solution
W (z) will be a many valued function on P1(C) holomorphic everywhere except
at 0, 1,∞, with monodromy group π1(P1(C)−0, 1,∞) ∼= F2, the free group
in two generators. The monodromy along small circles about 0, 1,∞ will be
matrices conjugate to exp(2πiA), exp(2πiB), exp(2πi(−A−B)
)respectively.
The point will be made below is that the theory of such rank 2 first order
differential equations is equivalent to the theory of rank 2 vvaf for triangle
groups .
However , we strongly believe that this theory will easily generalize to the
theory of higher rank vvaf of triangle groups and ultimately to genus-0 Fuch-
sian groups of the first kind . This deep connection between the theory of rank
d vvaf of genus-0 Fuchsian groups of the first kind and the theory of such rank
d first order Fuchsian differential equations has been achieved in [5] .
To date , the computation of the fundamental matrix in the higher rank
case whether it is of triangle groups or more generally to genus-0 Fuchsian
groups of the first kind remains an open challenge . Although as evident from
the rank 2 case it is clear on the grounds of few examples that the theory of
rank d vvaf for triangle groups is equivalent to the theory of rank d first order
Fuchsian differential equations and possibly to the generalized hypergeometric
differential equations of order d . It is an expectation that the free basis may
100
be formed by the generalized hypergeometric functions . For more details on
Fuchsian differential equations see [24, 47] .
Theorem 6.1.1. Let A,B,Λ ∈ M2(C) such that Λ is a diagonal matrix and
A+ B + Λ = 0 then the Fuchsian differential equation (6.1.1) of rank 2 gives
a second order Riemann’s differential equation at each component with their
regular singularities at 0, 1 and ∞ .
Proof. Let d = 2 and write A =(a11
a21
a12
a22
), B =
(b11
b21
b12
b22
)and Λ =
(Λ1
00
Λ2
)Then, consider the expanded form of the Fuchsian differential equation (6.1.1)
in rank 2 as follows
d
dz
(W11 W12
W21 W22
)=
(W11 W12
W21 W22
)(a11
z+ b11
z−1a12
z+ b12
z−1a21
z+ b21
z−1a22
z+ b22
z−1
).
Since A + B + Λ = 0, therefore a12 = −b12 and a21 = −b21. This implies
that
d
dz
(W11 W12
W21 W22
)=
(W11 W12
W21 W22
)(a11
z+ b11
z−1a12
z− a12
z−1a21
z− a21
z−1a22
z+ b22
z−1
).
Hence
d
dzW11 = W11
(a11
z+
b11
z − 1
)+W12 a21
(1
z− 1
z − 1
)(6.1.2)
d
dzW12 = W11 a12
(1
z− 1
z − 1
)+W12
(a22
z+
b22
z − 1
)(6.1.3)
d
dzW21 = W21
(a11
z+
b11
z − 1
)+W22 a21
(1
z− 1
z − 1
)(6.1.4)
d
dzW22 = W21 a12
(1
z− 1
z − 1
)+W22
(a22
z+
b22
z − 1
)(6.1.5)
Taking the derivative of equation (6.1.2) with respect to z we get
d2
dz2W11 = d
dzW11
(a11
z+ b11
z−1
)+W11
(− a11
z2− b11
(z−1)2
)+ ddzW12 a21
(1z− 1
z−1
)+W12 a21
(− 1
z2+ 1
(z−1)2
)(6.1.6)
101
by using equation (6.1.3) , (6.1.2) replace ddzW12 and W12 in the above sec-
ond order differential equation (6.1.6) and collecting alike terms this gives the
following second order differential equation
d2
dz2W11 + ddzW11
(1−a11−a22
z+ 1−b11−b22
z−1
)+W11
(a11a22−a12a21
z2 + b11b22−a12a21
(z−1)2 + a11b22+a22b11+2a12a21−a11−b11
z(z−1)
)= 0 .
(6.1.7)
Sincea11a22 − a12a21
z2=a11a22 − a12a21
z(z − 1)− a11a22 − a12a21
z2(z − 1)
andb11b22 − a12a21
(z − 1)2=b11b22 − a12a21
z(z − 1)+b11b22 − a12a21
z(z − 1)2.
Using these relations in the second order differential equation (6.1.7) the fol-
lowing equation is obtained.
d2
dz2W11 + ddzW11
(1−a11−a22
z+ 1−b11−b22
z−1
)+W11
(− a11a22−a12a21
z2(z−1)
+ b11b22−a12a21
z(z−1)2 + a11b22+a22b11+a11a22+b11b22−a11−b11
z(z−1)
)= 0 .
(6.1.8)
Let Λ1 ,Λ2 , Λ3 ,Λ4 , Λ5 ,Λ6 be the eigenvalues of Λ,A,B respectively.
Now using a11 + a22 = trA = Λ3 + Λ4 , b11 + b22 = trB = Λ5 + Λ6 , a11a22 −
a12a21 = detA = Λ3Λ4 , b11b22 − a12a21 = detB = Λ5Λ6 along with a11 + b11 +
Λ1 = 0 , a22 + b22 + Λ2 = 0 , obtained from A+ B + Λ = 0 , in (6.1.8) gives
d2
dz2W11 + ddzW11
(1−Λ3−Λ4
z+ 1−Λ5−Λ6
z−1
)+W11
(− Λ3Λ4
z2(z−1)+ Λ5Λ6
z(z−1)2 + Λ1(Λ2+1)z(z−1)
)= 0 .
(6.1.9)
This above differential equation (6.1.9) can be transformed into the following
Riemann’s form of hypergeometric differential equation
z2(z − 1)2 d2
dz2W11 + ddzW11(1− Λ3 − Λ4)z(z − 1)2 + (1− Λ5 − Λ6)z2(z − 1)
+W11Λ3Λ4(1− z) + Λ5Λ6z + Λ1(Λ2 + 1)z(z − 1) = 0(6.1.10)
102
The equation (6.1.9) and (6.1.10) have regular singularities at z = 0, 1,∞ with
the exponents Λ3,Λ4 at z = 0, Λ5,Λ6 at z = 1 and Λ1,Λ2 +1 at z =∞ .
Similarly, taking the derivative of equation (6.1.4) with respect to z and
using equations (6.1.5) and (6.1.4) replace the values of ddzW22 and W22 in the
second derivative of equation (6.1.4) gives exactly the same Riemann’s form of
differential equation as equation (6.1.10) with same regular singularities and
exponents. In other words, W11 and W21 satisfies the equation (6.1.10).
Similarly, W12 and W22 satisfy the following differential equation
d2Y
dz2+dY
dz
(1− Λ3 − Λ4
z+
1− Λ5 − Λ6
z − 1
)+Y
(− Λ3Λ4
z2(z − 1)+
Λ5Λ6
z(z − 1)2+
(Λ1 + 1)Λ2
z(z − 1)
)= 0
(6.1.11)
and this above differential equation (6.1.11) can be transformed into the
following Riemann’s form of hypergeometric differential equation
z2(z − 1)2 d2
dz2Y + ddzY (1− Λ3 − Λ4)z(z − 1)2 + (1− Λ5 − Λ6)z2(z − 1)
+Y Λ3Λ4(1− z) + Λ5Λ6z + (Λ1 + 1)Λ2z(z − 1) = 0 .(6.1.12)
The equation (6.1.11) and (6.1.12) have regular singularities at z = 0, 1,∞
with the exponents Λ3,Λ4 at z = 0, Λ5,Λ6 at z = 1 and Λ1 + 1,Λ2 at
z =∞ .
Corollary 6.1.2. The solution W =(W11
W21
W12
W22
)∈ M2(C[z]) of Fuchsian dif-
ferential equation (6.1.1) can be described as follows
1. the components W11 and W21 are solutions of equation (6.1.10) , and
2. the components W12 and W22 are solutions of equation (6.1.12) .
Proof. Solutions of equation (6.1.10) can be described by the following Rie-
mann Scheme
P
0 1 ∞
Λ3 Λ5 Λ1 ; zΛ4 Λ6 Λ2 + 1
103
and using the equations (4.3.11) , (4.3.12) and (4.3.13) . Hence , at z = 0 with
respect to the exponent Λ3 and Λ4 the solutions are spanned by
W01(z) = zΛ3(1− z)Λ52F1(Λ1 + Λ3 + Λ5 ,Λ2 + Λ3 + Λ5 + 1 , 1 + Λ3 − Λ4 ; z) ,
W02(z) = zΛ4(1− z)Λ52F1(Λ1 + Λ4 + Λ5,Λ2 + Λ4 + Λ5 + 1, 1 + Λ4 − Λ3; z) ,
(6.1.13)
at z =∞ with respect to the exponent Λ1 and Λ2 +1 the solutions are spanned
by
W∞1(z) = z−Λ1
(1z− 1
)Λ5
2F1
(Λ1 + Λ3 + Λ5,Λ1 + Λ4 + Λ5; Λ1 − Λ2; 1
z
),
W∞2(z) = z−(1+Λ2)
(1z− 1
)Λ5
×
2F1
(Λ2 + Λ3 + Λ5 + 1,Λ2 + Λ4 + Λ5 + 1; 2− Λ1 + Λ2; 1
z
),
(6.1.14)
and at point z = 1 with respect to the exponent Λ5 and Λ6 the solutions are
spanned by
W11(z) = zΛ3(1− z)Λ5×2F1
(Λ1 + Λ3 + Λ5,Λ2 + Λ3 + Λ5 + 1; 1 + Λ5 − Λ6; 1− z
),
W12(z) = zΛ3(1− z)Λ6×2F1
(1− Λ1 − Λ4 − Λ5 ,−Λ1 − Λ4 − Λ5 , 1 + Λ6 − Λ5 ; 1− z
).
(6.1.15)
Similarly , W12 and W22 satisfied the Riemann’s differential equation (6.1.12)
and therefore the solutions can be described by the following Riemann Scheme
P
0 1 ∞
Λ3 Λ5 Λ1 + 1 ; zΛ4 Λ6 Λ2
.
As before the solutions can be written by using the equations (4.3.11) , (4.3.12)
and (4.3.13) . Hence , at point z = 0 , 1 ,∞ respectively with respect to the
exponent Λ3 ,Λ4 , Λ5 ,Λ6 ,Λ1 +1 ,Λ2 the solutions are spanned by respec-
tively
Y01(z) = zΛ3(1− z)Λ52F1(1 + Λ1 + Λ3 + Λ5,Λ2 + Λ3 + Λ5; 1 + Λ3 − Λ4; z)
104
Y02(z) = zΛ4(1− z)Λ52F1(1 + Λ1 + Λ4,Λ2 + Λ3 + Λ5; 1 + Λ4 − Λ3; z) ,
(6.1.16)
Y∞1(z) = z−(1+Λ1)
(1z− 1
)Λ5
×
2F1
(1 + Λ1 + Λ3, 1 + Λ1 + Λ4; 2 + Λ1 − Λ2 − Λ5; 1
z
),
Y∞2(z) = z−Λ2
(1z− 1
)Λ5
×
2F1
(Λ2 + Λ3 + Λ5,Λ2 + Λ4 + Λ5; Λ2 − Λ1 + Λ5; 1
z
),
(6.1.17)
Y11(z) = zΛ3(1− z)Λ52F1(1 + Λ1 + Λ3,Λ2 + Λ3 + Λ5; 1− Λ6; 1− z)
Y12(z) = zΛ3(1− z)Λ62F1(−Λ1 − Λ4, 1− Λ2 − Λ4 − Λ5; 1 + Λ6; 1− z) .
(6.1.18)
6.2 Rank 2 vector-valued automorphic forms
Let G be the triangle group of type (`,m, n) defined by equation (4.1.1) . Let
ζ1 , ζ2 , ζ3 be the corners of the Fundamental triangle in H∗G
with respect to a
fixed realization of G such that tj ·ζj = ζj for j = 1, 2, 3 and tj is the generator
of the stabilizer group Gj of ζj where
Gj∼=
Z` if j = 1Zm if j = 2Zn if j = 3
.
Let ρ be a rank 2 admissible multiplier of G with ρ(tj) = Tj as defined by
equation (4.2.3) for j = 1, 2, 3 . As in subsection 4.2.2 , let λ3 , λ4 be eigenvalues
of T1 , λ5 , λ6 be eigenvalues of T2 and T3 =(λ1
00λ2
). Without loss of generality,
we can consider weight w = 0 and begin the classification of rank 2 vvaf (nearly
and weakly holomorphic) . Following Corolloary 5.1.20 with respect to any ζi
the space N (i)
0 (ρ) of nearly holomorphic vvaf is a free module of rank 2 over the
105
ring C[z(i)
(τ)] and following Theorem 5.2.3 the space M!0(ρ) is a free module
of rank 2 over the ring C[z(1)
(τ) , z(2)
(τ) , z(3)
(τ)] . Therefore to make it more
precise , by classifying the rank 2 nearly and weakly holomorphic vvaf of G
with respect to any rank 2 admissible multiplier we mean to compute the
fundamental matrices i.e. Ξ(i)
(ρ) in case of N (i)
0 and Ξ(ρ) in case of M!0(ρ) .
Following Theorem 5.2.3 it is enough to compute Ξ(i)
(ρ) for any i = 1, 2, 3 as
Ξ(ρ) = Ξ(i)
(ρ) for every i . Hence , the fundamental matrix Ξ(i)
(ρ) for i = 1, 2, 3
can be computed .
Note 6.2.1. For notational convenience in this chapter we will frequently
make use of the following short notations qi
= qζi,N (i)
w = N (ζi)
w (ρ) , z(i)
(τ) =
z(ζi)(τ) ,Ξ
(i)= Ξ
(ζi) ,X (i)= X (ζi) etc. for i = 1, 2, 3 .
6.2.1 Nearly holomorphic vvaf with respect to ζ3
Let N (3)
0 (ρ) denote the space of weight 0 nearly holomorphic vvaf with respect
to ζ3 of G with respect to the rank 2 admissible multiplier ρ . Choose the
unique Λi ∈ C for 3 ≤ i ≤ 6 such that λi = exp(2πiΛi) and 0 ≤ Re(Λi) < 1 .
Choose any Λ1 ,Λ2 ∈ C such that
λ1 = exp(2πiΛ1) , λ2 = exp(2πiΛ2) and6∑i=1
Λi = 0 .
For example if ` < ∞ then Λ3,Λ4 ∈
0`, 1`, 2`, · · · , `−1
`
whereas if n < ∞
then Λ1 ,Λ2 ∈ 1nZ . Consider a regularized hauptmodul
z(3)
G(τ) := z(τ) =
∞∑n=−1
an qn
3
of G with respect to ζ3 where q3 is the local variable in the punctured disc
with respect to ζ3 depending on ζ3 ∈ CG
or in EG
follows from equation (4.1.5) .
This hauptmodul can be normalized in such a way that its expansion is of the
form
J(3)
G(τ) := J(τ) = q
−1
3+∞∑n=1
bn qn
3.
106
Write the values of regularized and normalized hauptmoduls at the cusps in
the following table
τ J(τ) z(τ)
ζ3 α(3)
3:=∞ ∞
ζ2 α(3)
21
ζ1 α(3)
10
where
z(τ) =α1 − J(τ)
α1 − α2
=J(τ)− α1
α2 − α1
. (6.2.1)
As long as there is no confusion we will drop the superscript (3) from the
values α2 and α1 . Define the ∇ := f(τ) ·D(ζ3)
0where
D(ζ3)
0=
hc τ2
2πiddτ
= q3
ddq3
, if n =∞ i.e. ζ3 = c ∈ CG
(ω−ω)n
ddτ
= q(1− 1
n)
3 (1− q1n
3 )2 ddqn
, if n <∞ i.e. ζ3 = ω ∈ EG
,
and f(τ) is the generator of free module N (ζ3)
−2 (1) of rank 1 over C[J] which is
defined as
f(τ) = −(J− α2)(J− α1)
J′= (α1 − α2)
z(z− 1)
z′. (6.2.2)
where J′= D
(ζ3)
0(J) and z
′= D
(ζ3)
0(z) . From Theorem 5.3.1, there is a bijective
exponent Λ and a fundamental matrix Ξ(z) such that
∇Ξ(τ) = Ξ(τ)((J + a)Λ + b
)(6.2.3)
where a = −(α1 + α2) is a constant term of f(τ) and soon to be determined
b = X + [Λ,X ]. By using Theorem 5.3.2 this is equivalent to the following
rank 2 Fuchsian differential equation
d
dzΞ(z) = Ξ(z)
(Az
+B
z− 1
)(6.2.4)
where A,B are same as defined in (5.3.11) .
107
Λ = diag(Λ1,Λ2) , and Λ3,Λ4 and Λ5,Λ6 be the eigenvalues of A and
B respectively . Write for (G, ρ) the characteristic matrix
X (3)
=
(X (3)
11
X (3)
21
X (3)
12
X (3)
22
).
Then
b(3)
= X (3)
+ [Λ,X (3)
] =
(X (3)
11
X (3)
21 (1 + Λ2 − Λ1)
X (3)
12 (1 + Λ1 − Λ2)
X (3)
22
).
This gives
X (3)
11 + X (3)
22 = α2(Λ1 + Λ2)− (α1 − α2)(Λ3 + Λ4) . (6.2.5)
Solving (A − Λ3I)(A − Λ4I) = 0 and (B − Λ5I)(B − Λ6I) = 0 gives the
following two relations
X (3)
12 X(3)
21 =
[X (3)
11 − α2Λ1 + (α1 − α2)Λ3
][X (3)
11 − α2Λ1 + (α1 − α2)Λ4
](Λ1 − Λ2)2 − 1
,
X (3)
12 X(3)
21 =
[X (3)
11 − α1Λ1 − (α1 − α2)Λ3
][X (3)
11 − α1Λ1 − (α1 − α2)Λ4
](Λ1 − Λ2)2 − 1
.
(6.2.6)
Solving the two relations in equation (6.2.6) gives
X (3)
11 =(α1 + α2)Λ2
1 + (α1 − α2)(Λ5Λ6 − Λ3Λ4) + Λ1[α2(Λ3 + Λ4) + α1(Λ5 + Λ6)]
(Λ1 − Λ2),
(6.2.7)
and using equation (6.2.7) and (6.2.5)
X (3)
22 =(α1 − α2)(Λ3Λ4 − Λ5Λ6) + Λ2[α1(Λ1 + Λ3 + Λ4) + α2(Λ1 + Λ5 + Λ6)]
(Λ1 − Λ2).
(6.2.8)
Using equation (6.2.6) for some nonzero x(3) ∈ C
X (3)
12 =X11 − α2Λ1 + (α1 − α2)Λ3
(Λ1 − Λ2 + 1)· x(3)
X (3)
21 =X11 − α2Λ1 + (α1 − α2)Λ4
(Λ1 − Λ2 − 1)· 1
x(3)(6.2.9)
108
and we read off the fundamental matrix
Ξ(ζ3)
(z) := Ξ(3)
(z) =
(Ξ
(3)
11
Ξ(3)
21
Ξ(3)
12
Ξ(3)
22
)(6.2.10)
by using the solutions of Riemann’s differential equation with respect to ζ3
from Corollary 6.1.2 where
Ξ(3)
11 (z) =((α1 − α2)z
)−Λ1
(1z− 1
)Λ5
×
2F1
(Λ1 + Λ3 + Λ5,Λ1 + Λ4 + Λ5; Λ1 − Λ2; 1
z
),
Ξ(3)
21 (z) = X21
((α1 − α2)z
)−(Λ2+1)(
1z− 1
)Λ5
×
2F1
(Λ2 + Λ3 + Λ5 + 1,Λ2 + Λ4 + Λ5 + 1; 2− Λ1 + Λ2; 1
z
),
Ξ(3)
12 (z) = X12
((α1 − α2)z
)−(1+Λ1)(
1z− 1
)Λ5
×
2F1
(1 + Λ1 + Λ3, 1 + Λ1 + Λ4; 2 + Λ1 − Λ2 − Λ5; 1
z
),
Ξ(3)
22 (z) =((α1 − α2)z
)−Λ2
(1z− 1
)Λ5
×
2F1
(Λ2 + Λ3 + Λ5,Λ2 + Λ4 + Λ5; Λ2 − Λ1 + Λ5; 1
z
).
(6.2.11)
The parameter x in X (3)and the matrix T1 are related by analytic continuation
of Ξξη(z) for all 1 ≤ ξ, η ≤ 2 from z =∞ to z = 0. Ξ(γτ) = ρ(γ)Ξ(τ) for every
γ ∈ G and τ ∈ H∗G . Since t1 · ζ1 = ζ1 and Ξ(t1τ) = T1 · Ξ(τ) therefore taking
τ = ζ1 gives Ξ(ζ1) = T1 · Ξ(ζ1) . The value of fundamental matrix
Ξ(ζ1) =
(Ξ11(ζ1)
Ξ21(ζ1)
Ξ12(ζ1)
Ξ22(ζ1)
)can be computed by finding the value of each component Ξξη(ζ1) . Each com-
ponent can be computed by analytic continuation of the solutions of the Rie-
mann’s differential equation from z(τ) =∞ to z(τ) = 0 . Therefore(W∞1(z)
W∞2(z)
)= B ·
(W01(z)
W02(z)
)where B =
(B1
B3
B2
B4
)109
is defined by (4.4.2) for a = Λ1 +Λ3 +Λ5 , b = Λ2 +Λ3 +Λ5 +1 , c = 1+Λ3−Λ4 .
Now following the similar process described in subsection 4.5.2 , at z(τ) = 0
Ξ11(ζ1) = (α1 − α2)−Λ1B1
Ξ21(ζ1) = X21(α1 − α2)−(Λ2+1)B3 . (6.2.12)
Similarly (Y∞1(z)
Y∞2(z)
)= B′ ·
(W01(z)
W02(z)
)where B′ =
(B′1B′3
B′2B′4
)is defined similarly by (4.4.2) for the values of c = 1 + Λ3 − Λ4 , a = 1 +
Λ1 + Λ3 + Λ5 , b = Λ2 + Λ3 + Λ5 and following the similar process described in
subsection 4.5.2 , at τ = ζ1
Ξ12(ζ1) = X12 (α1 − α2)−Λ1B′1
Ξ21(ζ1) = (α1 − α2)−Λ2B′3 . (6.2.13)
Therefore
Ξ(ζ1) =
((α1 − α2)−Λ1B1
X21(α1 − α2)−(Λ2+1)B3
X12 (α1 − α2)−Λ1B′1(α1 − α2)−Λ2B′3
).
Then solving Ξ(ζ1) = T1 Ξ(ζ1) gives
(α1 − α2)−Λ1B1 = a1 (α1 − α2)−Λ1B1 + b1 X21(α1 − α2)−(Λ2+1)B3
⇒ 1− a1
b1
= X21(α1 − α2)Λ1−Λ2−1B3
B1
(6.2.14)
Now using the value of X21 from (6.2.9) and the value of B3 and B1 from (4.4.2)
in the above (6.2.14) we find the value of nonzero x appeared in the charac-
teristic matrix X , i.e.
x = (α1 − α2)Λ1−Λ2−1 · b1
1− a1
· X11 − α2Λ1 + (α1 − α2)Λ4
(Λ1 − Λ2 − 1)· B3
B1
. (6.2.15)
Note 6.2.2. Following the similar process explained in this section classifica-
tion of nearly holomorphic vvaf at ζ1 , ζ2 can be achieved by interchanging with
110
their respective regularized and normalized hauptmoduls and their values at
the special points ζi for i = 1, 2, 3 in the values of fundamental , characteristic
matrices and the value x(3)
. However, this interchanging process of values will
require careful attention as the values of αi’s are not the same . So, in case of
ζ2 , ζ1 the value (α1 − α2) is replaced by (α3 − α1) , (α3 − α2) and eigenvalues
Λ1,Λ2 by Λ5,Λ6 and Λ3,Λ4 respectively . Also one important difference
among all these is the calculation of the parameter x(i)
appearing in the char-
acteristic matrix X (i). For pedagogical reasons and to serve the purpose of
this thesis better ample details of the classification of N (2)
0 (ρ) and N (1)
0 (ρ) to
the extent of reader’s comfort and future reference are supplied .
6.2.2 Nearly holomorphic vvaf with respect to ζ2
Let N (2)
0 (ρ) denote the space of weight 0 nearly holomorphic vvaf with respect
to τ = ζ2 of G with respect to rank 2 admissible multiplier ρ . Choose the
unique Λi ∈ C for i = 1, 2, 3, 4 such that λi = exp(2πiΛi) and 0 ≤ Re(Λi) < 1 .
Choose any Λ5 ,Λ6 ∈ C such that
λ5 = exp(2πiΛ5) , λ6 = exp(2πiΛ6) and6∑i=1
Λi = 0 .
Write the regularized hauptmodul
z(ζ1)
G(τ) := z(τ) =
∞∑n=−1
a(c)
n qn
2
where q2 is the local coordinate of ζ1 in G\H∗G
follows from the equation (4.1.5)
and write the normalized hauptmodul
J(ζ2)
G(τ) := J(τ) = q
−1
2+∞∑n=1
b(2)
n qn
2.
Write the values of regularized and normalized hauptmoduls at ζ1, ζ2 and ζ3
in the following table
111
τ J(τ) z(τ)
ζ3 α(2)
30
ζ2 α(2)
2:=∞ ∞
ζ1 α(2)
11
where
z(τ) =α3 − J(τ)
α3 − α1
=J(τ)− α3
α1 − α3
. (6.2.16)
As long as there is no confusion the superscript (2) will be dropped from the
values α3 and α1 . Define the
∇(ζ2)
:= f(ζ2)
(τ) ·D(ζ2)
0
where f(ζ2)
(τ) is the generator of free module N (ζ2)
−2 (1) of rank 1 over C[J]
which is defined as
f(ζ2)
(τ) = −(J− α1)(J− α3)
J′= (α3 − α1)
z(z− 1)
z′
where J′= D
(ζ2)
0 (J) and z′ = D(ζ2)
0 (z) .
From Theorem 5.3.1 there is a bijective exponent Λ(ζ2) := B and a funda-
mental matrix Ξ(ζ2)
= qPcBP−1
c
2
(I + X (ζ2)
q2 +∑∞
n=2 Ξ(ζ2)
n q n2
)such that
∇(ζ2)
Ξ(ζ2)
(τ) = Ξ(ζ2)
(τ)
((J + a)B + b
)where a = −(α1 +α3) is a constant term of f
(ζ2)(τ) and b = X (ζ2)
+ [B,X (ζ2)] .
By similar process described in subsection 6.2.1 , the equation (6.2.2) is equiv-
alent to the following rank 2 Fuchsian differential equation
d
dzΞ(z) = Ξ(z)
(Λ
z+A
z− 1
)where
A = − α3
(α3 − α1)B +
b
(α3 − α1)
Λ =α1
(α3 − α1)B − b
(α3 − α1)
112
Clearly ,
A+ B + Λ = 0 and α1A+ α3Λ = −b . (6.2.17)
Λ1,Λ2, Λ3,Λ4 and Λ5,Λ6 are eigenvalues of Λ,A and B respectively ,
write
X (ζ2)
=
(X11
X21
X12
X22
)then by using equation (6.2.17)
X11 + X22 = α3(Λ5 + Λ6)− (α1 − α3)(Λ3 + Λ4) . (6.2.18)
Solving (A−Λ3I)(A−Λ4I) = 0 and (Λ−Λ1I)(Λ−Λ2I) = 0 gives the following
two relations
X12X21 =(X11 − α3Λ5 + (α1 − α3)Λ3)(X11 − α3Λ5 + (α1 − α3)Λ4)
(Λ5 − Λ6)2 − 1,
X12X21 =(X11 − α1Λ5 − (α1 − α3)Λ3)(X11 − α1Λ5 − (α1 − α3)Λ4)
(Λ5 − Λ6)2 − 1, (6.2.19)
and solving the two relations in equation (6.2.19) gives
X11 =(α1 + α3)Λ2
5 + Λ5[α1(Λ1 + Λ2) + α3(Λ3 + Λ4)] + (α1 − α3)(Λ1Λ2 − Λ3Λ4)
(Λ5 − Λ6).
(6.2.20)
Now, using equation (6.2.18) and (6.2.20)
X22 =(α1 − α3)(Λ1Λ2 − Λ3Λ4)− Λ6[α1(Λ5 + Λ3 + Λ4) + α3(Λ5 + Λ1 + Λ2)]
Λ5 − Λ6
.
(6.2.21)
Using equation (6.2.19) for some nonzero x(2) ∈ C
X12 =X11 − α3Λ5 + (α1 − α3)Λ3
(Λ5 − Λ6 + 1)· x(2)
X21 =X11 − α3Λ5 + (α1 − α3)Λ4
(Λ5 − Λ6 − 1)· 1
x(2)(6.2.22)
and we read off the fundamental matrix
Ξ(2)
(z) =
(Ξ
(2)
11
Ξ(2)
21
Ξ(2)
12
Ξ(2)
22
)(6.2.23)
113
by using the solutions of Riemann’s differential equation at the point ∞ from
Corollary 6.1.2 along with the help of following Riemann schemes
P
0 1 ∞
Λ1 Λ3 Λ5 ; zΛ2 Λ4 Λ6 + 1
P
0 1 ∞
Λ1 Λ3 Λ5 + 1 ; zΛ2 Λ4 Λ6
.
Hence ,
Ξ11(z) =((α3 − α1)z
)−Λ5
(1z− 1
)Λ3
×
2F1
(Λ1 + Λ3 + Λ5,Λ2 + Λ3 + Λ5; Λ5 − Λ6; 1
z
),
Ξ21(z) = X21
((α3 − α1)z
)−(Λ6+1)(
1z− 1
)Λ3
×
2F1
(Λ1 + Λ3 + Λ6 + 1,Λ2 + Λ3 + Λ6 + 1; 2− Λ5 + Λ6; 1
z
),
Ξ12(z) = X12
((α3 − α1)z
)−(1+Λ5)(
1z− 1
)Λ3
×
2F1
(1 + Λ1 + Λ3 + Λ5, 1 + Λ2 + Λ3 + Λ5; 2 + Λ5 − Λ6; 1
z
),
Ξ22(z) =((α3 − α1)z
)−Λ6
(1z− 1
)Λ3
×
2F1
(Λ1 + Λ3 + Λ6,Λ2 + Λ3 + Λ6; Λ6 − Λ5; 1
z
).
(6.2.24)
The parameter x(2)
in X (2)and the matrix T1 are related by analytic con-
tinuation of Ξξη(z) for all 1 ≤ ξ, η ≤ 2 from z(τ) = ∞ to z(τ) = 1 . For
every γ ∈ G and τ ∈ H∗G we have Ξ(γτ) = ρ(γ)Ξ(τ) . At τ = ζ1 t1 · ζ1 = ζ1
and z(ζ1) = 1 therefore Ξ(ζ1) = T1 · Ξ(ζ1) . To find the value of Ξ(2)
(ζ1) we
need to find the value of each component Ξξη(ζ1) which can be computed by
analytic continuation of the solutions of the Riemann’s differential equation
from z(τ) =∞ to z(τ) = 1 . Therefore(W∞1(z)
W∞2(z)
)= C ·
(W11(z)
W12(z)
)where C =
(C1
C3
C2
C4
)is defined by (4.4.3) for values a = Λ5 + Λ1 + Λ3 , b = 1 + Λ6 + Λ1 + Λ3 , c =
1 + Λ1 − Λ2 . Following the similar process from subsection 4.5.2 , at z(τ) = 1
Ξ11(ζ1) = (α3 − α1)−Λ5B1
114
Ξ21(ζ1) = X21(α3 − α1)−(Λ6+1)B3 .
Similarly,(Y∞1(z)
Y∞2(z)
)= C ′ ·
(Y11(z)
Y12(z)
)where C ′ =
(C ′1C ′3
C ′2C ′4
)is defined by using the values a = 1 + Λ5 + Λ1 + Λ3 , b = Λ6 + Λ1 + Λ3 , c =
1 + Λ1 − λ2 in equation (4.4.3) and following the similar process as above , at
z = 1
Ξ12(ζ1) = X12 (α3 − α1)−Λ5C ′1
Ξ22(ζ1) = (α3 − α1)−Λ6C ′3 .
Therefore ,
Ξ(ζ1) =
((α3 − α1)−Λ5C1
X21(α3 − α1)−(Λ6+1)C3
X12 (α3 − α1)−Λ5C ′1(α3 − α1)−Λ6C ′3
).
Then solving Ξ(ζ1) = T1 Ξ(ζ1) gives
(α3 − α1)−Λ5C1 = a1 (α3 − α1)−Λ5C1 + b1 X21(α3 − α1)−(Λ6+1)C3
⇒ 1− a1
b1
= X21(α3 − α1)Λ5−Λ6−1C3
C1
Now using the value of X21 from equation (6.2.22) and the value of B3 and B1
from equation (4.4.3) in the equation (6.2.25) we find the value of nonzero x(2)
appeared in the characteristic matrix X (2), i.e.
x(2)
= (α3−α1)Λ5−Λ6−1 · b1
1− a1
· X11 − α3Λ5 + (α1 − α3)Λ4
(Λ5 − Λ6 − 1)·C3
C1
. (6.2.25)
6.2.3 Nearly holomorphic vvaf with respect to ζ1
Let N (ζ1)
0 (ρ) denote the space of weight 0 nearly holomorphic vvaf with respect
to τ = ζ1 of G with respect to rank 2 admissible multiplier ρ . Choose the
unique Λi ∈ C for i = 1, 2, 5, 6 such that λi = exp(2πiΛi) and 0 ≤ Re(Λi) < 1 .
Choose any Λ3 ,Λ4 ∈ C such that
λ3 = exp(2πiΛ3) , λ4 = exp(2πiΛ4) and6∑i=1
Λi = 0 .
115
Write the regularized hauptmodul
z(ζ1)
G(τ) := z(τ) =
∞∑n=−1
a(1)
n qn
1
where q1 is the local coordinate of ζ1 in G\H∗G
follows from the equation (4.1.5) .
This hauptmodul can be normalized in such a way that its expansion is of the
form
J(ζ1)
G(τ) := J(τ) = q
−1
1+∞∑n=1
b(1)
n qn
1.
Write the values of regularized and normalized hauptmoduls at ζ1, ζ2, ζ3 in the
following table :
τ J(τ) z(τ)
ζ3 α(1)
30
ζ2 α(1)
21
ζ1 α(1)
1:=∞ ∞
where
z(τ) =α3 − J(τ)
α3 − α2
=J(τ)− α3
α2 − α3
.
From now on we will drop the superscript (1) and (ζ1) wherever possible and
as long as there is no confusion. Define the
∇(ζ1)
:= f(ζ1)
(τ) ·D(ζ1)
0(6.2.26)
where f(ζ1)
(τ) is the generator of free module N (ζ1)
−2 (1) of rank 1 over C[J]
which is defined as
f(ζ1)
(τ) = −(J− α2)(J− α3)
J′= (α3 − α2)
z(z− 1)
z′
where J′= D
(ζ1)
0 (J) and z′ = D(ζ1)
0 (z) .
From Theorem 5.3.1 there is a bijective exponent Λ(ζ1) := A and a funda-
mental matrix Ξ(ζ1)
= qP1AP
−11
1
(I + X (ζ1)
q1 +∑∞
n=2 Ξ(ζ1)
n q n1
)such that
∇(ζ1)
Ξ(ζ1)
(τ) = Ξ(ζ1)
(τ)
((J + a
(1)
)A+ b(1)
)(6.2.27)
116
where a(1)
= −(α3+α2) is a constant term of f(1)
(τ) and b(1)
= X (1)+[A,X (1)
] .
Following Theorem 5.3.2 equation (6.2.27) is equivalent to
d
dzΞ = Ξ
(Λ
z+B
z − 1
)(6.2.28)
where
B = − α3
(α3 − α2)A− b
(α3 − α2),
Λ =α2
(α3 − α2)A− b
(α3 − α2). (6.2.29)
Clearly,
A+ B + Λ = 0 and α2B + α3Λ = −b . (6.2.30)
Λ1,Λ2, Λ3,Λ4 and Λ5,Λ6 are eigenvalues of Λ,A and B respectively ,
write
X (1)
=
(X11
X21
X12
X22
)then by using equation (6.2.30)
X11 + X22 = α2(Λ3 + Λ4)− (α3 − α2)(Λ1 + Λ2) . (6.2.31)
Solving (B−Λ5I)(B−Λ6I) = 0 and (Λ−Λ1I)(Λ−Λ2I) = 0 respectively gives
the following two relations
X12X21 =(X11 − α3Λ3 + (α2 − α3)Λ5)(X11 − α3Λ3 + (α2 − α3)Λ6)
(Λ3 − Λ4)2 − 1,
X12X21 =(X11 − α1Λ3 − (α2 − α3)Λ5)(X11 − α1Λ5 − (α2 − α3)Λ6)
(Λ3 − Λ4)2 − 1, (6.2.32)
and solving the two relations in equation (6.2.35) gives
X11 =(α2 + α3)Λ2
3 + Λ3[α2(Λ1 + Λ2) + α3(Λ5 + Λ6)] + (α2− α
3)(Λ1Λ2 − Λ5Λ6)
(Λ3 − Λ4).
(6.2.33)
Now, using equation (6.2.31) and (6.2.33)
X22 =(α2 − α3)(Λ1Λ2 − Λ5Λ6)− Λ4[α2(Λ3 + Λ5 + Λ6) + α3(Λ3 + Λ1 + Λ2)]
Λ3 − Λ4
.
(6.2.34)
117
Using equation (6.2.19) for some nonzero x(1) ∈ C
X12 =X11 − α3Λ3 + (α2 − α3)Λ5
(Λ3 − Λ4 + 1)· x(1)
X21 =X11 − α3Λ3 + (α2 − α3)Λ6
(Λ3 − Λ4 − 1)· 1
x(1)(6.2.35)
and we read off the fundamental matrix
Ξ(1)
(z) =
(Ξ
(1)
11
Ξ(1)
21
Ξ(1)
12
Ξ(1)
22
)(6.2.36)
by using the solutions of Riemann’s differential equation at the point ∞ from
Corollary 6.1.2 with the help of following Riemann schemes
P
0 1 ∞
Λ1 Λ5 Λ3 ; zΛ2 Λ6 Λ4 + 1
P
0 1 ∞
Λ1 Λ5 Λ3 + 1 ; zΛ2 Λ6 Λ4
Therefore ,
Ξ(1)
11 (z) =((α3 − α2)z
)−Λ3
(1z− 1
)Λ5
×
2F1
(Λ1 + Λ3 + Λ5,Λ2 + Λ3 + Λ5; Λ3 − Λ4; 1
z
),
Ξ(1)
21 (z) = X21
((α3 − α2)z
)−(Λ4+1)(
1z− 1
)Λ5
×
2F1
(Λ1 + Λ5 + Λ4 + 1,Λ2 + Λ5 + Λ4 + 1; 2− Λ3 + Λ4; 1
z
),
Ξ(1)
12 (z) = X12
((α3 − α1)z
)−(1+Λ3)(
1z− 1
)Λ5
×
2F1
(1 + Λ1 + Λ5 + Λ3, 1 + Λ2 + Λ5 + Λ3; 2 + Λ3 − Λ4; 1
z
),
Ξ(1)
22 (z) =((α3 − α2)z
)−Λ4
(1z− 1
)Λ5
×
2F1
(Λ1 + Λ5 + Λ4,Λ2 + Λ5 + Λ4; Λ4 − Λ3; 1
z
).
(6.2.37)
The parameter x(1)
in X (1)and the matrix T1 are related by analytic con-
tinuation of Ξξη(z) for all 1 ≤ ξ, η ≤ 2 from z = ∞ to z = 1 . We know
Ξ(γτ) = ρ(γ)Ξ(τ) for every γ ∈ G and τ ∈ H∗G
. At τ = ζ2 t2 ·ζ2 = ζ2 therefore
118
taking γ = t2 gives Ξ(ζ2) = T2·Ξ(ζ2) and we compute Ξ(ζ2) by finding the value
of each component Ξξη(ζ2) which can be computed by analytic continuation of
the solutions of the Riemann’s differential equation from z(τ) =∞ to z(τ) = 1 .
Therefore following the similar computations performed in subsection 6.2.2 by
taking the values of a = Λ3 + Λ1 + Λ5 , b = 1 + Λ4 + Λ1 + Λ5 , c = 1 + Λ1 − Λ2
and a = 1 + Λ3 + Λ1 + Λ5 , b = Λ4 + Λ1 + Λ5 , c = 1 + Λ1 − Λ2 we find C,C ′ .
Then
Ξ(ζ2) =
((α3 − α2)−Λ3C1
X21(α3 − α2)−(Λ4+1)C3
X12 (α3 − α2)−(Λ3+1)C ′1(α3 − α2)−Λ4C ′3
).
Then writing Ξ(ζ2) = T2 Ξ(ζ2) gives
(α3 − α2)−Λ3C1 = a2 (α3 − α2)−Λ3C1 + b2 X21(α3 − α2)−(Λ4+1)C3
⇒ 1− a2
b2
= X21(α3 − α2)Λ3−Λ4−1C3
C1
(6.2.38)
Now using the value of X21 from equation (6.2.35) and the value of C3 and
C1 from equation (4.4.3) in the above equation (6.2.25) we find the value of
nonzero x(1)
appeared in the characteristic matrix X (1), i.e.
x(1)
= (α3−α2)Λ3−Λ4−1 · b2
1− a2
· X11 − α3Λ3 + (α2 − α3)Λ6
(Λ3 − Λ4 − 1)·C3
C1
. (6.2.39)
6.3 Examples
This section is devoted to a discussion of the developed theory through exam-
ples . The vvaf of triangle groups of type (∞∞∞) and (2,∞,∞) are discussed
in the following sections . In case of G = Γ(1) which is a triangle group of type
(2, 3,∞) ., the classification of nearly holomorphic vvaf with respect to the
cusp ζ3 := ∞ has been discussed in full detail in [17] . Inspite of their treat-
ment, in this section the classification of nearly holomorphic vvaf with respect
to all the special points namely at ζ1 , ζ2 and ζ3 is given . Our object is to
classify rank 2 vvmf for these triangle group of any even integer weight w .
119
From section 6.2 it is clear that to classify the vvaf of any triangle group G it
is enough to know the hauptmoduls with respect to ζ1 , ζ2 , ζ3 and the admis-
sible representation ρ . We begin with the classification of the triangle group
Γ(2) which is one of the most common realization of triangle group of type
(∞,∞,∞) well know as the principal congruence subgroup of the modular
group Γ(1) .
6.3.1 Γ(2)
For a complete description of Γ(2) and its hauptmoduls see example 3.2.2
and 2.4 . Following subsection 4.1.1 the regularized hauptmodul with respect
to cusp ∞ is
z(∞)
(τ) =θ3(τ)4
θ2(τ)4=
1
16q−1
3+
1
2+
5
4q3 −
31
8q
3
3+ · · · ,
= 16q2 − 8q2
2+
11
4q
3
2+ · · · ,
= 1 + 16q1 + 8q2
1+
11
4q
3
1+ · · ·
Following the expansion of z(∞)
(τ) and observation that z(1)
(τ) = 1/z(∞)
(τ)
and z(0)
(τ) = 1/1 − z∞
(τ) the regularized hauptmoduls of G with respect to
the cusps 1 and 0 are written as follows
z(1)
(τ) =θ2(τ)4
θ3(τ)4=
1
16q−1
2+
1
32+
5
1024q2 + · · · ,
z(0)
(τ) = −θ2(τ)4
θ4(τ)4= − 1
16q−1
1+
1
32+
5
1024q2 + · · · ,
Therefore the normalized hauptmoduls of G with respect to the cusps
∞ , 0 , 1 can be written as follows
J(∞)
(τ) = 16z(∞)
(τ)− 8 = q−1
3+ 20q − 62q
3
+ · · · ,
J(1)
(τ) = 16z(∞)
(τ)− 1
2= q
−1
2+
5
64q2 + · · · ,
J(0)
(τ) = −16z(0)
(τ) +1
2= q
−1
1− 5
64q1 + · · · .
120
The values of regularized and normalized hauptmoduls of G at the cusps
∞ , 1 , 0 are recorded in the following table :
τ J(∞)
(τ) z(∞)
(τ) J(1)
(τ) z(1)
(τ) J(0)
(τ) z(0)
(τ)
∞ α(∞)
3=∞ ∞ α
(1)
3= −1/2 0 α
(0)
3= 1/2 0
1 α(∞)
2= 8 0 α
(1)
2=∞ ∞ α
(0)
2= −31/2 1
0 α(∞)
1= −8 1 α
(1)
1= 31/2 1 α
(0)
1=∞ ∞
Table 6.1: Values of regularized and normalized hautmoduls of Γ(2)
Let ρ : G → GL2(C) be any admissible multiplier of G such that ρ(t∞) =
T3 , ρ(t1) = T2 and ρ(t0) = T1 where T3 , T2 , T1 are defined in 4.2.2 by equa-
tion (4.2.3) . Let Λ ,B ,A be the exponent matrices of respectively cusps
∞ , 1 , 0 . Let Λ1 ,Λ2 , Λ3 ,Λ4 and Λ5 ,Λ6 be the eigenvalues of re-
spectively the exponent matrices Λ ,A and B . Now following the theory build
in subsections 6.2.1 , 6.2.3 and 6.2.3 the fundamental matrices of N (c)
0(ρ) for
c =∞ , 1 , 0 are described respectively by
Ξ(c)
(z) =
(Ξ
(c)
11(z)
Ξ(c)
12(z)
Ξ(c)
21(z)
Ξ(c)
22(z)
)
where z = z(c)
(τ) and the components Ξ(c)
ξη(z) for 1 ≤ ξ, η ≤ 2 can be described
by equations (6.2.11) and (6.2.24) for c = ∞ and 1 respectively with α1 −
α2 = −16 = α3 − α1 and for c = 0 the components can be described by
equation (6.2.37) with α3 − α2 = 16 . Following the Corollary 5.1.20 for c ∈
∞ , 1 , 0 , N (c)
0 (ρ) is a free module of rank 2 over the polynomial ring C[z(c)
(τ)]
and Theorem 5.2.3 the space M!0(ρ) is a free module of rank 2 over the ring
C[z(∞)
(τ) , z(1)
(τ) , z(0)
(τ)] .
6.3.2 Γ0(2)
Let G = Γ0(2) . It is a triangle of type (2,∞,∞) . It has two cusps namely 0
and ∞ and an elliptic fixed point ω = 1+i2
of order 2 . For a complete descrip-
121
tion of G and its hauptmoduls see example 3.2.2 . Following subsection 4.1.1
the regularized hauptmodul with respect to cusp ∞ is
z(τ) := z(∞)
(τ) = − 1
64q−1
3+
3
8− 69
16q3 + 32q
2
3− 5601
32q
3
3+ · · · ,
= 64q2 − 1536q2
2+ 19200q
3
2+ · · · ,
= 1 + a1q1 + a21a2q
2
1+ a3
1a3q3
1+ · · · ,
where a1 = 12·Γ(1/16)Γ(9/16)
, a2 = 12
, a3 = 41240
and q3 = q , q2 = exp(−πi/τ) , q1 = τ−ωτ−ω .
Following the expansion of z∞
(τ) and observation that z0(τ) = 1/z
∞(τ) and
z(ω)
(τ) = 1/1 − z∞
(τ) the regularized hauptmoduls of G with respect to the
cusp 0 and elliptic fixed point ω are written as follows
z(τ) := z(0)
(τ) =1
64q−1
0+
3
642 · 8− 291
644q0 + · · · ,
z(τ) := z(ω)
(τ) = − 1
a1
q−1
ω+a2
a21
− a22 − a3
a21
qω + · · · .
Following the expressions of regularized hauptmoduls the expressions of their
normalized hauptmoduls can be written as follows
J(∞)
(τ) = 24− z(∞)
(τ) , J(0)
(τ) = 64z(0)
(τ)− 3
512, J
(ω)
(τ) =a2
a1
− a1z(ω)
(τ) .
The values of these regularized and their corresponding normalized haupt-
modul at the cusps 0,∞ and the elliptic fixed point ω = 1+i2
are recorded in
the following table
τ J(τ) z(τ) J(0)
(τ) z(0)
(τ) J(ω)
(τ) z(ω)
(τ)
∞ α(∞)
3=∞ ∞ α
(0)
3= − 3
5120 α
(ω)
3= a2
a10
0 α(∞)
2= 24 0 α
(0)
2=∞ ∞ α
(ω)
2=
a2−a21
a11
ω α(∞)
1= −40 1 α
(0)
1= 32765
5121 α
(ω)
1=∞ ∞
Table 6.2: Values of normalized and regularized hauptmoduls of Γ0(2)
122
Moduli space of admissible multiplier
It is known that G is the free product Z2 ∗ Z . Let ρ be a rank 2 admissible
multiplier of G, i.e. ρ(t∞) = T∞ = exp(2πiΛ), ρ(t0) = T0 and ρ(tω) = Tω .
By definition of Γ0(2) , T 2ω = 1 and T∞T0Tω = 1 . Let λ1, λ2 be the eigen-
values of T∞, λ3, λ4 be the eigenvalues of Tω, λ5, λ6 be the eigenvalues of T0 .
Let Λ ,B ,A be the exponent matrices respectively of cusps ∞ , 0 and elliptic
fixed point ω . Let Λ1 ,Λ2 , Λ3 ,Λ4 and Λ5 ,Λ6 be the eigenvalues re-
spectively of Λ ,A and B . Clearly Λ3 ,Λ4 ∈ 0 , 1/2 . The moduli space of
1 dimensional admissible multiplier is trivially described by the pair (T∞ , Tω)
where T∞ can be any nonzero complex number and Tω = ±1 therefore the
moduli space has two connected components . However , we are interested in
the moduli space of 2 dimensional admissible multiplier . Since Tω is diago-
nalizable and its eigenvalues are ±1 therefore if its eigenvalues are both 1 or
−1 then clearly Tω is either I or −I . Since by the definition of admissible
multiplier T∞ = diag(λ1 , λ2) therefore T0 will also be a diagonal matrix and
we have a reducible ρ . This gives us two isomorphic 2 dimensional compo-
nents to our full moduli space which can be described by choosing Tω = I
or Tω = −I and any unordered pair of nonzero complex numbers (λ1 , λ2) .
The generic component of the moduli space of G corresponds to Tω having the
eigenvalues ±1 both with multiplicity 1 . Following equation (4.2.3) we have
Tω =(a1
c1
b1d1
)and since tr(Tω) = 0 and det(Tω) = −1 therefore Tω =
(a1
c1
b1−a1
)with a2
1 + b1c1 = 1 . There are two cases to consider now
• If λ1 = λ2, choose Tω diagonal and therefore this will give us completely
reducible admissible multiplier ρ of the form (λ1 , 1)⊕
(λ2 ,−1) .
• So it suffices to assume that λ1 6= λ2 . Any diagonalizing matrix Pω will
now commute with T∞ . If c1 = 0 = b1 then ρ is a direct sum . If c1 = 0
but b1 6= 0 then use Pω to make b1 = 1 and therefore our representation
123
will be a semidirect sum . Similarly for c1 = 1 but b1 = 0 . Otherwise both
c1, b1 are nonzero and we can use Pω to make them equal thus in this case
the matrix Tω is parametrized by the complex numbers which are solutions
to a21 + b2
1 = 1 and this is parametrized by the w in the plane punctured at
w = 0 : a1 = w−w−1
2iand b1 = w+w−1
2. Thus the generic point in this component
corresponds to a choice (λ1 , λ2 ,w) or equivalently to (λ2 , λ1 ,w−1) and is 3
dimensional . w = ±i recovers the direct sum and these are triple points .
Hence the moduli space will consist 3 connected components two of 2 dimen-
sional when tr(Tω) = ±1 and one of 3 dimensional when tr(Tω) = 0 .
Rank 2 vvaf of Γ0(2)
Now let ρ be the generic admissible multiplier of G . Following the theory
built in subsection 6.2.3 for the classification of nearly holomorphic vvaf with
respect to the cusp ∞ let us fix the choice of Λ3 = 0 and Λ4 = 1/2 and for
j = 5, 6 take Λj ∈ C such that 0 ≤ Re(Λj) < 1 , and choose the exponents
Λ1 ,Λ2 ∈ C which satisfy∑6
j=1 Λj = 0 . In other words choose the exponent
matrix Λ of cusp ∞ such that tr(Λ) = −1/2 − tr(B) . Now the fundamental
matrix of N (ζ)
0 (ρ) for ζ =∞ , 0 , ω is defined by
Ξ(ζ)
(G ,ρ)(z) := Ξ
(ζ)
(z) =
(Ξ
(ζ)
11
Ξ(ζ)
21
Ξ(ζ)
12
Ξ(ζ)
22
)
where z = z(ζ)
(τ) and the components of Ξ(ζ)
(z) are defined respectively by
using the equation (6.2.11) with α1 − α2 = −64 , (6.2.24) with α3 − α1 =
−64 and (6.2.37) with α3 − α2 = a1 . Following the Corollary 5.1.20 for
ζ ∈ ∞ , 0 , ω , N (ζ)
0 (ρ) is a free module of rank 2 over the polynomial ring
C[z(ζ)
(τ)] and Theorem 5.2.3 the space M!0(ρ) is a free module of rank 2 over
the ring C[z(∞)
(τ) , z(0)
(τ) , z(ω)
(τ)] .
124
Some interesting problems
The ease with which the rank 2 vvaf for triangle groups are classified, leads us
to consider various interesting problems following the approach developed in
this chapter i.e. building the abstract connection between vvaf and hypergeo-
metric differential equations . It seems likely there is an obvious generalization
of this theory for higher rank vvaf of triangle groups and perhaps for the Fuch-
sian groups of the first kind . A few of these (open) problems are mentioned
below .
Problem. Let G be any triangle group and ρ : G −→ GLd(C) be any admis-
sible multiplier of rank d ≥ 3 then classify the rank d vvaf of G .
We hope to find a connection between the higher rank vvaf and general-
ized hypergeometric functions . Another interesting problem to consider is the
following
Problem. Let G be the genus-0 Fuchsian group of the first kind with `+n > 3
then classify all rank 2 vvaf with respect to any rank 2 admissible multiplier
ρ of G .
Γ(3) will be the first basic example to begin the exploration . Higher rank
vvaf and ρ to be any multiplier will be the next problems to consider .
125
Chapter 7
Vector-Valued AutomorphicForms of Triangle groups -III
This chapter revisits the theory of rank 2 vvaf of triangle groups . Even though
the goal is the same as in chapter 6 , the path differs slightly . More precisely ,
we completely avoid the existence of a fundamental matrix associated with
the modules N (i)
w (ρ) for any i = 1, 2, 3 and M!w(ρ) of any G for any given ρ .
This was our main tool in the classification of the above modules in chapter 6 .
Thematically this chapter is a continuation of chapter 4 : there infinitely many
rank 2 vvaf of any G with respect to any admissible multiplier ρ of G were
constructed satisfying certain conditions for any i such that the regularized
hauptmodul z(i)
(τ) takes the value ∞ at τ = ζi ; whereas in this chapter we
take the integer k = 0 and choose the eigenvalues of the exponent matrices
with respect to i = 1, 2, 3 such that the exponents of the associated Riemann’s
differential equation respectively are as follows
u1 + u2 = Λ1 + Λ2 + 1 for i = 3
u3 + u4 = Λ3 + Λ4 + 1 for i = 1
u5 + u6 = Λ5 + Λ6 + 1 for i = 2
The possible candidates of the free basis of the modules N (i)
w (ρ) for any
i = 1, 2, 3 and M!w(ρ) are found . We also expect to provide a completely
126
independent proof of the freeness of the above modules in the near future .
Hence for the time being we heavily rely on an appropriate principal part map
and Corollary 5.1.20 and Theorem 5.2.3 .
In order to classify rank 2 holomorphic vvaf of any triangle group G by de-
veloping this method we give the obvious generalization of the method in [37] .
That method used the modular linear differential equations (MLDE) and mod-
ularity of the wronskian associated to the the solutions of MLDE . Note that
the case of order 2 MLDE coincides with Riemann’s differential equations .
However , in comparison with their method our method gives explicit basis
vectors of the free module of rank 2 holomorphic vvaf of any G with respect
to admissible multiplier . Their basis vectors are given up to some unknown
constant χ whereas in rank 2 case we compute explicitly this constant χ .
7.1 Nearly holomorphic functions at z =∞ on
Riemann sphere
Let G be the triangle group of type (` ,m , n) . Let z(τ) := z(∞)
(τ) and J(τ) :=
J(∞)
G(τ) be respectively the regularized and normalized hauptmodul of G with
respect to τ = ζi for some 1 ≤ i ≤ 3 such that z(τ) = ∞ and J(τ) = ∞ .
Without loss of generality let us assume that i = 3 . We write the values
of these regularized and normalized hauptmoduls at the points ζ1 , ζ2 , ζ3 in
the Table 7.1 . Let ρ : G −→ GL2(C) be a rank 2 admissible multiplier and
τ J(τ) z(τ)
ζ3 α1 :=∞ ∞
ζ2 α2 1
ζ1 α1 0
Table 7.1: Values of regularized and normalized hauptmoduls of G
choose the unique Λi ∈ C for 3 ≤ i ≤ 6 such that λi = exp(2πiΛi) and
127
0 ≤ Re(Λi) < 1 . Choose any Λ1 ,Λ2 ∈ C such that
λ1 = exp(2πiΛ1) , λ2 = exp(2πiΛ2) and6∑i=1
Λi = 0 .
Now, consider the following two differential equations with regular singular
points z = 0, 1,∞
d2W
dz2+dW
dz
(1− Λ3 − Λ4
z+
1− Λ5 − Λ6
z− 1
)+W
(− Λ3Λ4
z2(z− 1)+
Λ5Λ6
z(z− 1)2+
Λ1(Λ2 + 1)
z(z− 1)
)= 0
(7.1.1)
and
d2Y
dz2+dY
dz
(1− Λ3 − Λ4
z+
1− Λ5 − Λ6
z− 1
)+Y
(− Λ3Λ4
z2(z− 1)+
Λ5Λ6
z(z− 1)2+
(Λ1 + 1)Λ2
z(z− 1)
)= 0
(7.1.2)
The solutions of these two equations can be described respectively by using
the following two Riemann scheme
P
0 1 ∞
Λ3 Λ5 Λ1 ; zΛ4 Λ6 Λ2 + 1
, P
0 1 ∞
Λ3 Λ5 Λ1 + 1 ; zΛ4 Λ6 Λ2
. (7.1.3)
The solutions can be written explicitly by using Corollary 6.1.2 . This implies
that at point z =∞ with respect to the exponent Λ1 and Λ2 + 1 the solutions
are spanned by W∞1(z) ,W∞2(z) which are defined by equations (4.3.12) by
replacing u1 = Λ1 and u2 = Λ2 + 1 and similarly with respect to the exponent
Λ1 + 1 and Λ2 the solutions are spanned by Y∞1(z) , Y∞2(z) which are also
defined by equation (4.3.12) by replacing u1 = Λ1 + 1 and u2 = Λ2 .
Note 7.1.1. Any solution to equations (7.1.1) and (7.1.2) is automatically
holomorphic at z 6= 0, 1,∞ and has branch points at 0, 1,∞ . The growth of
any solutions W (z) and Y (z) to equations (7.1.1) respectively (7.1.2) as z tends
to 0, 1 or ∞ is :
• |W (z)| < C∞|z|max[−Re(Λ1) ,−1−Re(Λ2)] , |Y (z)| < C ′∞|z|max[−1−Re(Λ1) ,−Re(Λ2)] for
|z| > 2 ,
128
• |W (z)| < C0|z|max[Re(Λ3) ,Re(Λ4)] , |Y (z)| < C ′0|z|max[Re(Λ3) ,Re(Λ4)] for |z| < 1/2 ,
and
• |W (z)| < C1|1− z|max[Re(Λ5) ,Re(Λ6)] , |Y (z)| < C ′1|z|max[Re(Λ5) ,Re(Λ6)] for |1− z| <
1/2 .
7.2 Nearly holomorphic automorphic forms at
τ = ζ3 on H∗G
Now, let z(τ) be the regularized hauptmodul of G as defined above . Using z(τ)
we can lift any solution W (z) respectively Y (z) of equations (7.1.1) respectively
(7.1.2) to H∗G
. Now we claim that the space N(3)
0 (ρ) of nearly holomorphic
vector-valued automorphic functions with respect to ζ3 is a free C[z(τ)]-module
with the basis
B =
Y1(τ) =
(W 1(τ)
W 2(τ)
),Y2(τ) =
(Y 1(τ)
Y 2(τ)
)(7.2.1)
where the components of Y1(τ) respectively Y2 is obtained by rescaling ap-
propriately the solutions W∞1 ,W∞2 respectively Y∞1 , Y∞2 as follows :
W 1(τ) =((α1 − α2)z(τ)
)−Λ1
(1z− 1
)Λ5
×
2F1
(Λ1 + Λ3 + Λ5,Λ1 + Λ4 + Λ5; Λ1 − Λ2; 1
z(τ)
),
W 2(τ) = χ21
((α1 − α2)z(τ)
)−(1+Λ2)(
1z− 1
)Λ5
×
2F1
(Λ2 + Λ3 + Λ5 + 1,Λ2 + Λ4 + Λ5 + 1; 2 + Λ2 − Λ1; 1
z(τ)
),
(7.2.2)
129
Y 1(τ) = χ12
((α1 − α2)z(τ)
)−(1+Λ1)(
1z− 1
)Λ5
×
2F1
(1 + Λ1 + Λ3 + Λ5, 1 + Λ1 + Λ4 + Λ5; 2 + Λ1 − Λ2; 1
z(τ)
),
Y 2(τ) =((α1 − α2)z(τ)
)−Λ2
(1z− 1
)Λ5
×
2F1
(Λ2 + Λ3 + Λ5,Λ2 + Λ4 + Λ5; Λ2 − Λ1; 1
z(τ)
).
(7.2.3)
for some as yet undetermined constants χ12 , χ21 .
The reason for rescaling the solutions W∞1(τ),W∞2(τ) and Y∞1(τ) , Y∞2(τ)
by the factor (α1−α2) is so that we have the following desired series expansion
of Y1 and Y2. More precisely
Y1 = qΛ1
(1 + · · ·0 + · · ·
)and Y2 = q
Λ2
(0 + · · ·1 + · · ·
).
The reason for the constants χ12 , χ21 is to make sure that these vectors trans-
form correctly with respect to t1 . Now we show that Y1(τ) and Y2(τ) are
in N (3)
0 (ρ) . Firstly , from the Note 7.1.1 that W (z(τ)) and Y (z(τ)) will be
automatically holomorphic at any non elliptic fixed point in H . If say ` <∞
then in that case ζ1 is an elliptic fixed point and Λ3 ,Λ4 ∈
0`, 1`, 2`, · · · , `−1
`
and z(τ) has a Taylor expansion in q1 =
(τ−ζ1τ−ζ1
)`, so W (z(τ)) and Y (z(τ))
have Taylor expansion in (τ − ζ1) and hence they are holomorphic at τ = ζ1 .
If say m = ∞ then z(τ) has a Taylor expansion in q2 and from this it is con-
cluded that both W (z(τ)) and Y (z(τ)) are holomorphic at ζ2 . Finally , both
W (z(τ)) and Y (z(τ)) have moderate growth at ζ3 . Thus any lift of W (z(τ))
and Y (z(τ)) is nearly holomorphic at ζ3 . Therefore Y1(τ) ,Y2(τ) are rank 2
nearly holomorphic vector-valued function at ζ3 . The functional behaviour of
Y1(τ) and Y2(τ) follows from computing the monodromy of Riemann’s differ-
ential equations (7.1.1) and (7.1.2) with respect to the solution basis U , V fol-
lowing the process described in subsection 4.5.1 where U =⟨W 1(τ) ,W 2(τ)
⟩C
130
and V =⟨Y 1(τ) , Y 2(τ)
⟩C
. Hence , the fundamental matrix
Ξ(3)
(z(τ)) =
(W∞1
W∞2
Y ∞1
Y ∞2
)of the module N (3)
0 (ρ) is determined upto the constants χ1 and χ2 and using
the principal part map PΛ
: N (3)
0 (ρ) −→ C2[q−1
3] for Λ = diag(Λ1 ,Λ2) and
the method developed in chapter 5 following from Corollary 5.1.20 gives that
the C[z(3)
(τ)]-module N (3)
0 (ρ) is free of rank 2 . To complete the construction
of fundamental matrix the constants χ12 respectively χ21 are computed in a
similar manner as the constants χ(i)
12and χ
(i)
21for any i = 1, 2, 3 were computed
in subsection 6.2.1 .
Similarly the fundamental matrix of N (2)
0 (ρ) and N (1)
0 (ρ) can also be com-
puted and shown that these are free module of rank 2 over C[z(2)
(τ)] respec-
tively C[z(1)
(τ)] . Also following from Theorem 5.2.3 M!0(ρ) is also a free
module of rank 2 over C[z(1)
(τ) , z(2)
(τ) , z(3)
(τ)] .
Generally the constants χ12 and χ21 are determined by analyzing the an-
alytic continuation of the components of the fundamental matrix Ξ(i)
(z(i)
(τ))
or Ξ(z(τ)) but, in case of Hecke triangle groups there is a straightforward ap-
proach to compute these constants which is explained in the following section .
7.3 Hecke triangle groups and the matrix X
By definition a Hecke triangle group G is a triangle group of type (` ,m ,∞)
with ` = 2 and 2 < m ≤ ∞ . Let z(τ) := z(3)
(τ) be the regularized hauptmodul
of G such that z(ζ1) = 1 , z(ζ2) = 0 , z(ζ3) =∞ . Write
Ξ(3)
(z(τ)) =
(Ξ11(z(τ))
Ξ21(z(τ))
Ξ12(z(τ))
Ξ22(z(τ))
)and X (3)
=
(χ11
χ21
χ12
χ22
)are the fundamental and characteristic matrices of G with respect to any
admissible irrep ρ mentioned in the beginning of section 7.1 .
131
Note that t1 · ζ1 = ζ1 and for any X(τ) ∈ N (3)
0 (ρ), X(t1τ) = ρ(t1)X(τ),
∀τ ∈ H therefore in particular for τ = ζ1, X(t1 · ζ1) = ρ(t1)X(ζ1) and this
implies that X(ζ1) = T1X(ζ1) and since T2
1 = 1 therefore 1 is an eigenvalue of T1
(otherwise T1 = −I and hence ρ will be reducible) . Clearly (T1− 1)X(ζ1) = 0
implies that X(ζ1) is the eigenvector of T1 with respect to the eigenvalue 1
of T1. Since, T1 =(a1
c1
b1d1
)therefore X(ζ1) is proportional to
(b1
1−a1
). Also
observe that the eigenvalues of the exponent matrix A of T1 will be 0 and 1/2 .
Therefore in that case Λ5 = 0 and Λ6 = 1/2 . Then from the previous section
clearly
Ξ(3)
(z(τ)) = [Y1(τ) Y2(τ)]
where from (7.2.1)
Y1(z(τ)) =
(((α1 − α2)z)−Λ1
2F1(Λ1 + Λ3 ,Λ1 + Λ4; Λ1 − Λ2; 1z )
χ21((α1 − α2)z)−(1+Λ2)2F1(Λ2 + Λ3 + 1,Λ2 + Λ4 + 1; 2 + Λ2 − Λ1; 1
z )
)
Y2(z(τ)) =
(χ12((α1 − α2)z)−(1+Λ1)
2F1(1 + Λ1 + Λ3, 1 + Λ1 + Λ4; 2 + Λ1 − Λ2; 1z )
((α1 − α2)z)−Λ22F1(Λ2 + Λ3,Λ2 + Λ4; Λ2 − Λ1; 1z )
).
Now, consider X(τ) = Y1(τ) and X(τ) = Y2(τ) and following above analy-
sis on these nearly holomorphic vector-valued automorphic functions with the
fact that as τ → ζ1, z(τ)→ 1 and that for some k1, k2 ∈ C×
k1 · Y1(ζ1) =
(b1
1− a1
)= k2 · Y2(ζ1) ,
gives
(b1
1− a1
)= k1 ·
((α1 − α2)−Λ1F (Λ1 + Λ3,Λ1 + Λ4; Λ1 − Λ2; 1)
χ21
(α1 − α2)−(1+Λ2)F (Λ2 + Λ3 + 1,Λ2 + Λ4 + 1; 2 + Λ2 − Λ1; 1)
)and(
b11− a1
)= k2 ·
(χ
12(α1 − α2)−(1+Λ1)F (1 + Λ1 + Λ3, 1 + Λ1 + Λ4; 2 + Λ1 − Λ2; 1)
(α1 − α2)−Λ2F (Λ2 + Λ3,Λ2 + Λ4; Λ2 − Λ1; 1)
).
132
This implies that
χ12 =1
k2
b1(α1 − α2)(1+Λ1)
F (1 + Λ1 + Λ3, 1 + Λ1 + Λ4; 2 + Λ1 − Λ2; 1),
χ21 =1
k1
(1− a1)(α1 − α2)(1+Λ2)
F (Λ2 + Λ3 + 1,Λ2 + Λ4 + 1; 2 + Λ2 − Λ1; 1),
(1− a1) = k2(α1 − α2)−Λ2F (Λ2 + Λ3,Λ2 + Λ4; Λ2 − Λ1; 1) ,
b1 = k1(α1 − α2)−Λ1F (Λ1 + Λ3,Λ1 + Λ4; Λ1 − Λ2; 1) .
(7.3.1)
Now, writing x = k1/k2 and by using the identity
F (1 + a, 1 + b; 2 + c; 1) =c(1 + c)
(c− a)(c− b)F (a, b; c; 1) (7.3.2)
implies
χ12 = (α1 − α2)(Λ2 + Λ3)(Λ2 + Λ4)
(Λ1 − Λ2)(1 + Λ1 − Λ2)· x
χ21 = (α1 − α2)(Λ1 + Λ3)(Λ1 + Λ4)
(Λ2 − Λ1)(1 + Λ2 − Λ1)· 1
x. (7.3.3)
Similarly χii
for i = 1, 2 can be computed by expanding the Ξii(z(τ)) and
therefore
χ11 = (α1 − α2)−Λ1(Λ1 + Λ3)(Λ1 + Λ4)
(Λ1 − Λ2),
χ22 = (α1 − α2)−Λ2(Λ2 + Λ3)(Λ2 + Λ4)
(Λ2 − Λ1). (7.3.4)
The parameter x appearing in χ12 and χ21 can be computed by similar
process the parameters x(i)
’s have been computed in chapter 6 . More precisely
by using the analytic continuation of Y1(τ) or Y2(τ) from z(τ) =∞ to z(τ) = 1
along with the fact that Yξ(ζ1) = T1Yξ(ζ1) for 1 ≤ ξ ≤ 2 .
7.4 An Explicit Example
Let G = Γ0(2) and H = Γ(2) for details about G,H see subsection 3.2.2 . Let
σ : H −→ C× be a character, i.e. a one dimensional representation of H defined
133
as follows :
σ(t∞) = exp
(2πia
N
), σ(t−1
0 ) = exp
(2πib
N
); 0 ≤ a < b ≤ N − 1 . (7.4.1)
Consider ρ = IndG
H(σ) : G −→ GL2(C) be the induced representation of G
from H of character σ. So, by definition of the induced representation
ρ(γ) =
(σ(IγI−1)
σ(tγt−1)
σ(Iγt−1)
σ(tγt−1)
); ∀γ ∈ G .
This implies that
ρ(t) =
(σ(t)
σ(t2)
σ(I)
σ(t)
)=
(0
σ(t2)
σ(I)
0
)=
(0
exp(2πiaN
)
1
0
)= T∞
ρ(t0) =
(σ(t0)
σ(tt0)
σ(t0t−1)
σ(tt0t−1)
)=
(σ(t0)
0
0
σ(t1)
)=
(e−
2πibN
0
0
e2πi(b−a)
N
)= T0
and T0 = exp(2πiB
)implies that the exponent matrix B =
(− bN0
0b−aN
).
Write u5 = − bN, u6 = b−a
Nand x = πia
N, y = πib
N. Consider the change of basis
matrix P =(
exp(−x)1
− exp(−x)1
)for ρ . Then P−1T∞P = exp(x) ·
(10
0−1
)and
P−1T0P =1
2
(exp(2(y − x)) + exp(−2y)
exp(2(y − x))− exp(−2(y + x))
exp(2(y − x))− exp(−2y)
exp(2(y − x)) + exp(−2(y + x))
)
where P−1 = 12
(exp(x)− exp(x)
11
). This means that Λ =
(a
2N0
0a
2N− 1
2
)and write
u1 = a2N
and u2 = a2N− 1
2. Since T 2
ω = 1 and Tω ∼ exp(2πiA) therefore
A =(
00
012
)or A =
(120
00
). Write u3 = 0 and u4 = 1/2 then notice that∑6
i=1 ui = 0 i.e. tr(A + B + Λ) = 0. This gives an equivalent representation
ρ′ : Γ0(2) −→ GL2(C) of induced representation ρ, defined by
ρ′(t) = T ′∞ = exp
(2πi
(a
2N
0
0a
2N− 1
2
))
ρ′(t0) = T ′0 ∼ exp
(2πi
(− bN
0
0b−aN
)), ρ′(tω) = T ′ω ∼ exp
(2πi
(0
0
012
))We restrict the choice on the eigenvalues of the exponent matrices of
tω , t1 , t∞ depending on which space of nearly holomorphic vvaf we want to
134
classify . For example in case of N (∞)
w(ρ) the eigenvalues of the exponent ma-
trices A and B can be chosen such that Λ3,Λ4,Λ5,Λ6 ∈ [0, 1). With this
restriction and the trace condition together implies that −52< tr(Λ) < −1
2.
In that case u5 = − bN< 0 so eigenvalue Λ5 = − b
N+ 1 of matrix B is chosen
and this doesn’t change the representation . With this choice observe that
tr(Λ) = −12− tr(B) = −3
2+ a
Nand therefore
Λ =
(a
2N− 1
0
0a
2N− 1
2
)=
(Λ1
0
0
Λ2
),
B =
(1− b
N
0
0b−aN
)=
(Λ5
0
0
Λ6
),
A =
(0
0
012
)=
(Λ3
0
0
Λ4
)(7.4.2)
can be chosen .
7.4.1 Explicit data for a = 2, b = 3 and N = 5 for Γ0(2)
The data for fixed choices of a = 2, b = 3 and N = 5 is produced. These
choices on a, b,N satisfies the above conditions given in the equation (7.4.1).
For the regularized and normalized hauptmoduls of Γ0(2) see subsection 6.3.2 .
Now following (7.4.2) , write
Λ1 = −4
5, Λ2 = − 3
10, Λ3 = 0 , Λ4 =
1
2, Λ5 =
2
5, Λ6 =
1
5.
Now by interchanging the role of Λ3,Λ4 by Λ5,Λ6 respectively in the funda-
mental matrix Ξ(3)
(z) obtained in section 7.2 . This gives
Ξ(∞)
(z) =
(W∞1
W∞2
Y ∞1
Y ∞2
)which form a free basis of the rank 2 C[z
(∞)(τ)]−module N (∞)
0 (ρ), where
W∞1(z) = (−64z)45 2F1
(− 2
5,−3
5;−1
2;1
z
),
W∞2(z) = χ21(−64z)−710 2F1
(11
10,
9
10;5
2;1
z
),
135
Y ∞1(z) = χ12(−64z)−15 2F1
(3
5,2
5;3
2;1
z
),
Y ∞2(z) = (−64z)310 2F1
(1
10,− 1
10;1
2;1
z
)and
χ12 = −64
25, and χ21 = −512
25.
Similarly,
Ξ(0)
(z(0)
(τ)) := Ξ(z) =
(W 01
W 02
Y 01
Y 02
).
where
W 01 = (−64z)−2/52F1
(− 2
5,
1
10;1
5;1
z
),
W 02 = χ(0)
21
(− 64z
)−6/52F1
(2
5,
9
10;
9
10;1
z
),
Y 01 = χ(0)
12
(− 64z
)−7/52F1
(3
5,11
10;11
10;1
z
),
Y 02 =(− 64z
)−1/52F1
(− 3
5,− 1
10;−1
5;1
z
)with χ
(0)
12= 16 = χ
(0)
21and at ζ1
Ξ(ω)
(z(ω)
(τ)) := Ξ() =
(W ω1
W ω2
Y ω1
Y ω2
).
where
W ω1 = 2F1
(− 2
5,
1
10;−1
2;1
),
W ω2 = χ(ω)
21
(− a1
)−3/22F1
(11
10,8
5;5
2;1
),
Y ω1 = χ(ω)
12
(− a1
)−12F1
(3
5,11
10;3
2;1
),
Y ω2 =(− a1
)−1/22F1
(1
10,3
5;1
2;1
)with χ
(ω)
12= 6
25a1 , χ
(ω)
21= 4
75a1 .
136
Chapter 8
Bounded Vs. UnboundedDenominators
In this chapter, the behaviour of Fourier coefficients of vvaf is studied . In
particular , let X(τ) =(
X1
X2
)be a nontrivial rank 2 vector-valued modular
form (vvmf) of a triangle group G with respect to multiplier ρ : G→ GL2(C) .
It is demonstrated that the components X1(τ) and X2(τ) have integral Fourier
coefficients , only when the kernel of ρ is a congruence group (see section 8.1 for
details and qualifications) . To accomplish this the theory of hypergeometric
differential equations is used .
8.1 Introduction and historical background
In 1971, Atkin & Swinnerton-Dyer [4] noticed that the Fourier coefficients
of modular forms for subgroups of Γ(1) := PSL2(Z) only have bounded de-
nominator when they are modular forms for a congruence subgroup. This
suggests the generalisation: if all Fourier coefficients of a vvmf X(τ) of Γ(1)
have bounded denominator , then X(τ) is a vvmf for a multiplier ρ with kernel
a congruence subgroup . This can be regarded as one of the most important
(and difficult) conjectures in the theory of vvmf . Following the suggestion of
Selberg [45] to develop the theory of vvmf , it is quite natural to consider vvmf
as a tool to study the growth of Fourier coefficients of scalar-valued modular
137
forms of nonconguence groups . One of the most striking observations found
in [4] , was the unbounded denominator (ubd) property of the noncongruence
modular forms . By nonconguence groups we mean the following :
Definition 8.1.1. Let H be any subgroup of PSL2(R) (not necessarily a Fuch-
sian group of the first kind) then H is said to be a noncongruence group if
H ∩ Γ(1) does not contain any principal congruence subgroup Γ(N) for any
integer N ≥ 1 .
More precisely , let Γ be any finite index subgroup of Γ(1) and f(τ) =∑n anq
n/h be any modular form of Γ of weight k with Fourier coefficients an
lying in some number field (we are interested in the case Q) . We say that f(τ)
satisfies the ubd property , if there does not exist any integer M > 0 such
that Mf(τ) has all algebraic integer coefficients . Otherwise , we will say that
f(τ) satisfies the bounded denominator (bd) property . Γc is defined to be the
congruence closure of Γ in Γ(1) if Γc is the intersection of all congruence
subgroups of Γ(1) containing Γ . f(τ) is said to be a true modular form of
Γ if it is a modular form of Γ but not of Γc . We now phrase the following well
known conjecture :
Conjecture 8.1.2 (UBD-conjecture). Let Γ be a finite index subgroup of Γ(1)
and f(τ) be any weight k weakly holomorphic true modular form of Γ with
algebraic Fourier coefficients . Then f(τ) satisfies the ubd property if Γ is
noncongruence .
The converse of the conjecture 8.1.2 is a classical result . Some special cases
of conjecture 8.1.2 are known . For example: Li & Long [32] have verified it
for weight k subgroups Γ and cusp forms f(τ) such that the space Sk(Γ) of
cusp forms is Cf(τ) . For detailed exposition on noncongruence modular forms
see [31, 30] .
We say that an algebraic number α has denominator n ∈ Z>0 if n is the
138
smallest positive integer such that nα is an algebraic integer . There are two
ways unbounded denominators can arise :
• A prime p occurs in denominators to arbitrarily high order . In other words
there exists a prime p whose power in the denominator grows monotonically .
Such a prime is called a ubd prime .
• Infinitely many distinct primes appear in denominators.
The former would be expected to happen with only finitely many primes
appearing in denominators when the kernel of ρ is finite index in Γ(1) but
noncongruence (according to Scholl [44]) , and we say that f(τ) satisfies the
p-ubd property or p is an ubd prime for f(τ) . At this stage it is less clear
about what happens when the kernel of ρ is infinite index .
Let G be any Fuchsian group of the first kind , k , d ∈ Z and H denote the
upper half plane . Let ρ : G → GLd(C) be any representation of G . Recall
from the definition 2.3.8 that X(τ) : H→ Cd is a weakly holomorphic vvaf of
weight k and rank d with respect to multiplier ρ if X(τ) is holomorphic on H
and poles allowed only at the cusps of G have certain functional and cuspidal
behaviour . For more details on Fuchsian groups see [46 , 25] and on the theory
of vvmf of the modular group see [17 , 7] .
Definition 8.1.3. Let X(τ) a weakly holomorphic vvaf of arbitrary weight
and rank with respect to multiplier ρ of any Fuchsian group of the first kind
G with a cusp∞ . Then we say that X(τ) satisfies the ubd property if at least
one of its components satisfies the ubd property . In addition , p is said to be
an ubd prime of X(τ) if it is a ubd prime of one of its components .
An obvious generalization of conjecture 8.1.2 can be phrased as follows :
Conjecture 8.1.4. Let G be any Fuchsian group of the first kind and ρ : G→
GLd(C) be any representation . Let X(τ) be any weakly holomorphic vvaf with
139
algebraic Fourier coefficients of arbitrary weight with respect to multiplier ρ .
Then X(τ) satisfies the ubd property if ker(ρ) is a noncongruence group .
Note 8.1.5. ker(ρ) in Conjecture 8.1.4 is not necessarily of finite index in G .
In this chapter , the ubd property for rank 2 vvmf X(τ) of a modular
triangle group G of type (`,m,∞) with respect to its 2-dimensional admissible
irrep ρ is studied . Recall from chapter 4 that a triangle group G of type
(`,m,∞) with 2 ≤ ` ≤ m ≤ ∞ is a genus-0 Fuchsian group of the first kind
isomorphic to Z` ∗ Zm where we define Z∞ := Z . Then (`,m) 6= (2, 2) and
G will be unique up to conjugation in PSL2(R) with number of elliptic fixed
points and parabolic points exactly 3 . Its representation is 〈t1, t2, t3|t`1 = tm2 =
1 = t1t2t3〉 . A fundamental domain of G which depends on the realization of
G, denoted by FG
is naturally a quadrilateral and exactly double the hyperbolic
triangle with vertices ζ1 = − exp(− πi
`
), ζ2 = exp(πi
m) , ζ3 =∞ . These vertices
form a set of representatives for the orbits of the elliptic and parabolic points
of extended upper half plane H∗G
(defined in chapter 2) under the action of G .
A triangle group G is modular if it is commensurable with Γ(1) , i.e.
G∩Γ(1) has finite index in G and Γ(1). More generally , a triangle group G is
arithmetic if it is a PSL2(R)-conjugate of some modular triangle group .An
irrep ρ : G → GLd(C) is admissible if ρ(ti), i = 1, 2, 3 are diagonalizable
matrices. Almost every irrep is admissible. We show the following :
Theorem 8.1.6. Let G be any modular triangle group of type (`,m,∞) and
ρ : G → GL2(C) be any rank 2 admissible irrep of G. Then ker(ρ) is a
noncongruence group if ρ admits an ubd prime.
The above is discussed in subsection 8.4.2 . We say ρ admits an ubd prime
p if any nonzero X(τ) ∈M!k(ρ) , the space of all weaky holomorphic vvmf of G
of weight k with respect to multiplier ρ , has a ubd prime p , i.e. X(τ) satisfies
the ubd property with respect to prime p .
140
For what happens to triangle groups which are not modular see section 8.2.1 .
The 2-dimensional case of Γ(1) has been studied in detail by Mason, see
[36, 38, 39]. In particular he has shown that the ubd property is satisfied
for all but approximately 175 of the 2-dimensional irreps of Γ(1). More pre-
cisely he found an ubd prime p in all these cases. In this 2 dimensional case,
the ker(ρ) has finite index if and only if ker(ρ) is congruence (there are 27 ρ of
this type). When ker(ρ) is a noncongruence group , Franc-Mason [15] showed
that there are infinitely many primes appearing in the denominators . Marks
found in [34] an ubd prime for any nontrivial vvmf of all but finitely many
3-dimensional irreps of Γ(1).
To our knowledge this is the first time when integrality has been considered
in detail for Fuchsian groups which are not the subgroups of modular group .
One novelty of this approach is that to have the q-series expansion depend
nontrivially on the group : e.g. for two groups we need q = i√3
exp(πiτ) . This
will certainly be necessary for more general Fuchsian groups .
8.2 Sufficiently integral Fuchsian groups
In this chapter , all the Fuchsian groups are assumed to be of the first kind
with at least one cusp (i.e. non cocompact ones) . For definitions of Fuchsian
groups , cusps etc. see chapter 2 . A cusp is necessary for q-series expansions .
Without loss of generality we can and will assume that a cusp is at ∞ .
Definition 8.2.1. Call a pair (G, κ) sufficiently integral if G < PSL2(R) is a
Fuchsian group of the first kind and κ is a nonzero complex number, such that
(i) ∞ is a cusp of G , say with cusp-width h > 0 ;
(ii) there exists a holomorphic modular form ∆G
(τ) of some weight K , with
no zeros in H , such that ∆G
(τ) = q δG
(1 + qGZ[[q
G]]) for some δ ∈ Z>0, where
qG
= κ exp(2πiτ/h) .
141
For notational convenience, from now on we will drop the subscript G
from qG
. Recall from chapter 2 that the cusp width of G at ∞ is the least
positive integer h such that ±(
10h1
)is contained in G . Let C
G⊂ R ∪ ∞
denote the set of all cusps of G . If (G, κ) is sufficiently integral , then by the
extended upper half-plane H∗G
we mean the union of H and CG
. For example
in case of G = Γ(1), κ = 1, h = 1, CG
= Q ∪ ∞ therefore qG
= exp(2πiτ)
and H∗G
= H ∪ Q ∪ ∞. From now on as long as there is no confusion the
subscript G will be dropped wherever possible .
Define θ = h2πi
ddτ
= q ddq
and
EG
2 (τ) =1
∆G
(τ)· θ∆
G(τ) ∈ δ + qZ[[q]] . (8.2.1)
Then EG
2 (τ) is holomorphic everywhere , and quasi-modular of weight 2 and
depth 1 :
EG
2
(aτ + b
cτ + d
)=Khc
2πi(cτ + d) + (cτ + d)2E
G
2 (τ) . (8.2.2)
This permits us to define the weight-k modular derivative
Dk = θ − k
KE
G
2 (τ) (8.2.3)
and D2k = Dk+2 Dk etc ; the differential operator Dj
k sends weight k modular
forms to weight k + 2j modular forms .
There are lots of examples of sufficiently integral pairs . For example, (G, 1)
for any finite-index subgroup G of
Γ+
0 (N) = 〈Γ0(N),Wp | p|N〉 =
1√e
(ae
cf
b
de
) ∣∣ a, b, c, d ∈ Z, e|f,
for any square-free N , where Wp := 1√p
(apcf
bdp
)for any integers a, b, c, d with
adp2− cfb = p (square-free N behave a little simpler, and suffice as explained
shortly). For those subgroups, one can take [8] ∆G
(τ) =∏
d|N η(dτ)24 and
hence EG
2 (τ) =∑
d|N E2(dτ). The group Γ+
0 (N) is the normaliser of Γ0(N)
in PSL2(R), but what makes it, and hence the subgroups G, important is
142
Helling’s theorem [22] that any Fuchsian group commensurable with Γ(1) is
conjugate in PSL2(R) to a subgroup of some Γ+
0 (N) for N square-free. Because
(ii) requires that ∆G
(τ) has bounded denominator, conjecturally this should
mean G contains a congruence subgroup, and hence Helling’s Theorem would
apply.
The reason for including the factor κ is because we can, and it increases
significantly the generality. In particular, by Scholl’s Theorem ([43], Prop 5.2),
for each finite index subgroup G of Γ(1) there is an associated integer M such
that if there is a basis of the space of weight k modular forms for G with
rational Fourier coefficients, then there exists a basis such that every prime
factor occurring in the denominators of the basis coefficients is a factor of M .
In this case, choosing κ to be a sufficiently large power of M will take care of
any denominators. Conjecturally, κ 6= 1 will be necessary whenever G is non-
arithmetic, i.e. no PSL2(R)-conjugate of G contains a congruence subgroup of
Γ(1).
8.2.1 Modular triangle groups
Two groups G1 and G2 are said to be commensurable if and only if G1∩G2 has
finite index in G1 and G2. A triangle group with at least one cusp is arithmetic
if and only if it is conjugate to a modular triangle group in PSL2(R). By a
modular triangle group we mean a triangle group which is commensurable
with the modular group Γ(1) . As we will see shortly that modular triangle
groups are sufficiently integral . The class of arithmetic triangle groups is
bigger than the class of modular triangle groups. An arithmetic triangle group
is not necessarily a modular triangle group. For example Γ(1) is an arithmetic
and a modular triangle group both but gΓ(1)g−1 for g =(π0
01
), is an arithmetic
but not a modular triangle group .
For any triangle group G of type (2,m,∞) , we define G2 to be the subgroup
143
generated by the squares of all the elements as well as the order m elements
of G . This implies G2 is an index 2 triangle subgroup of type (m,m,∞) of
G . Consequently, G2 is a triangle group of type (m,m,∞) and a subgroup of
G of index 2. For a given type there may be different modular representatives
which will be conjugate in PSL2(R) but not necessarily in Γ(1). For example,
Γ0(4) and Γ(2) are both modular triangle groups of type (∞,∞,∞) which
are conjuates to each other in PSL2(R) but not in Γ(1). For more details see
[9, 49] .
8.2.2 Nine Modular Triangle Groups
Modular triangle groups have exactly 9 distinct types: namely (2, 3,∞), (2, 4,∞),
(2, 6,∞), (2,∞,∞), (∞,∞,∞), (3,∞,∞), (3, 3,∞), (4, 4,∞) and (6, 6,∞) .
Modular triangle group of type (2, 3,∞)
Let G be the triangle group of type (2, 3,∞) . Following equation (4.1.2) one
way to realize
G =⟨t1, t2, t3 ∈ PSL2(R) | t21 = t32 = t∞3 = 1 = t1t2t3
⟩is through the generators
t1 = ±(
0
−1
1
0
), t2 = ±
(0
−1
1
1
), t3 = ±
(1
0
1
1
). (8.2.4)
Hence , the corners of the fundamental triangle are
ζ1 = i, ζ2 = exp
(πi
3
)=
1 + i√
3
2, ζ3 =∞ . (8.2.5)
ζ1, ζ2 are elliptic fixed points of order 2 and 3 respectively, and ζ3 is a cusp
of cusp width 1 . There are infinitely many triangle groups of this type . One
such group is the modular group Γ(1) and
Γ(1) =⟨γ1, γ2, γ3 ∈ PSL2(Z)
∣∣γ21 = γ3
2 = 1 = γ1γ2γ3
⟩(8.2.6)
144
where γ1 = s, γ3 = t and γ2 = (ts)−1 . Since all triangle group of type
(`,m,∞) are conjugate in PSL2(R) therefore
γ1 = t1 = s, γ2 = t2 = (ts)−1, γ3 = t3 = t.
In this case, the group G is identical with Γ(1) or trivially conjugate in
PSL2(R), i.e. Γ(1) = gGg−1 with g = 1 = ±I.
Modular triangle group of type (2,∞,∞)
Let G be the triangle group of type (2,∞,∞). Following equation (4.1.2) one
way to realize
G =⟨t1, t2, t3 ∈ PSL2(R) | t21 = t∞2 = t∞3 = 1 = t1t2t3
⟩is through generators
t1 = ±(
0
−1
1
0
), t2 = ±
(0
−1
1
2
), t3 = ±
(1
0
2
1
)(8.2.7)
and the corners of the fundamental triangle of G are
ζ1 = i, ζ2 = 1, ζ3 =∞ . (8.2.8)
ζ1 is an elliptic fixed point of order 2 and ζ2, ζ3 are the two cusps of cusp
width 1 and 2 respectively . There are infinitely many triangle groups of this
type . One such modular triangle group is
Γ0(2) =⟨γ1, γ2, γ3 ∈ PSL2(Z)
∣∣γ21 = 1 = γ1γ2γ3
⟩where γ3 = t, γ2 = st2s−1 and γ1 = (γ2γ3)−1 = (st2s−1t)−1, i.e.
γ1 = ±(
1
−2
1
−1
), γ2 = ±
(1
−2
0
1
), γ3 = ±
(1
0
1
1
)(8.2.9)
and Γ0(2) = gGg−1 with g = 1√2
(10−1
2
)∈ PSL2(R) . The corners of the
fundamental traingle of Γ0(2) are
ω1 = g · ζ1 =−1 + i
2, ω2 = g · ζ2 = 0, ω3 = g · ζ3 =∞, (8.2.10)
i.e. ω1 is an elliptic fixed point of order 2 whereas ω2 and ω3 are the two cusps
of cusp width 2 and 1 respectively .
145
Modular triangle group of type (2, 4,∞)
Let G be the triangle group of type (2, 4,∞). Following equation (4.1.2) , one
way to realize this
G = 〈t1, t2, t3 ∈ PSL2(R) | t21 = t42 = t∞3 = 1 = t1t2t3〉
is through the generators
t1 = ±(
0
−1
1
0
), t2 = ±
(0
−1
1√2
), t3 = ±
(1
0
√2
1
). (8.2.11)
Hence , the corners of it’s fundamental triangle are
ζ1 = i, ζ2 = exp
(π
4
)=
1 + i√2, ζ3 =∞ . (8.2.12)
ζ1, ζ2 are elliptic fixed points of order 2 and 4 respectively , whereas ζ3 is
a cusp of cusp width 1. One such modular triangle group of this type is
Γ+
0 (2) = 〈Γ0(2),W2〉. The group presentation of
Γ+
0 (2) =⟨γ1, γ2, γ3 ∈ PSL2(Z)
∣∣γ21 = γ4
2 = 1 = γ1γ2γ3
⟩where
γ1 = W2, γ3 = t, γ2 = (tW2)−1 = ± 1√2
(0
−2
1
2
), (8.2.13)
and Γ+
0 (2) = gGg−1 with g = ±(
2−1/4
00
21/4
)∈ PSL2(R) with the corners of
the triangle group Γ+
0 (2) are
ω1 = g · i =i√2, ω2 = g · exp
(πi
4
)=
1 + i
2, ω3 = g · ∞ =∞, (8.2.14)
i.e. ω1, ω2 are elliptic fixed points of order 2 and 4 respectively, whereas ω3 is
a cusp of cusp width 1 .
146
Modular triangle group of type (3,∞,∞)
Let G be the triangle group of type (3,∞,∞) . Following equation (4.1.2)
G =⟨t1, t2, t3|t31 = t∞2 = t∞3 = 1 = t1t2t3
⟩is realized by
t1 = ±(
1
−1
1
0
), t2 = ±
(0
−1
1
2
), t3 = ±
(1
0
3
1
). (8.2.15)
The corners of it’s fundamental triangle are
ζ1 =−1 + i
√3
2, ζ2 = 1, ζ3 =∞ (8.2.16)
where ζ1 is an elliptic fixed point of order 3 and ζ2, ζ3 are two cusps of cusp
width 1 and 3 respectively . One such modular triangle group is
Γ0(3) =⟨γ1, γ2, γ3 ∈ PSL2(Z)
∣∣γ31 = 1 = γ1γ2γ3
⟩where γ3 = t, γ2 = st3s−1 and γ1 = −(γ2γ3)−1 = −(st3s−1t)−1, i.e.
γ1 = ±(
2
−3
1
−1
), γ2 = ±
(1
−3
0
1
), γ3 = ±
(1
0
1
1
). (8.2.17)
In this case, Γ0(3) = gGg−1 with g = ± 1√3
(10−1
3
)∈ PSL2(R) and the corners
of the fundamental triangle of Γ0(3) are
ω1 = g · ζ1 =−3 + i
√3
6, ω2 = g · ζ2 = 0, ω3 = g · ζ3 =∞ (8.2.18)
where ω1 is an elliptic fixed point of order 3 and ω2, ω3 are two cusps of cusp
width 3 and 1 respectively.
Modular triangle group of type (2, 6,∞)
Let G be the triangle group of type (2, 6,∞) . Following equation (4.1.2)
G = 〈t1, t2, t3 ∈ PSL2(R)∣∣t21 = t62 = t∞3 = 1 = t1t2t3〉
147
is realized through
t1 = ±(
0
−1
1
0
), t2 = ±
(0
−1
1√3
), t3 =
(1
0
√3
1
), (8.2.19)
and therefore the corners of its fundamental triangle are
ζ1 = i, ζ2 =
√3 + i
2, ζ3 =∞ (8.2.20)
where ζ1, ζ2 are elliptic fixed points of order 2 and 3 respectively, and ζ3 is the
cusp of cusp width√
3 . One such group is Γ+
0 (3) = 〈Γ0(3),W3〉 . The group
presentation of
Γ+
0 (3) =⟨γ1, γ2, γ3 ∈ PSL2(R)
∣∣γ31 = γ6
1 = 1 = γ1γ2γ3
⟩where γ1 = W3, γ3 = t and γ2 = −(γ3γ1)−1 = −(tW3)−1, i.e.
γ1 =1√3
(0
−3
1
0
), γ2 =
1√3
(0
−3
1
3
), γ3 =
(1
0
1
1
). (8.2.21)
In this case, Γ0(3) = gGg−1 with g =
(3− 1
4
0
0
314
)∈ PSL2(R) and the corners
of the fundamental triangle of Γ+
0 (3) are
ω1 = g · ζ1 =i√3, ω2 = g · ζ2 =
√3 + i
2√
3, ω3 = g · ζ3 =∞ (8.2.22)
where ω1, ω2 are elliptic fixed points of order 2 and 6 respectively, and ω3 is
the cusp of cusp width 1.
Modular triangle group of type (3, 3,∞)
Let G be the triangle group of type (3, 3,∞). Following equation (4.1.2) one
way
G = 〈t1, t2, t3 ∈ PSL2(R)∣∣ t31 = t32 = t∞3 = 1 = t1t2t3〉
can be realized is through the generators
t1 = ±(
1
−1
1
0
), t2 = ±
(0
−1
1
1
), t3 = ±
(1
0
2
1
). (8.2.23)
148
and the corners of its fundamental triangle are
ζ1 =−1 + i
√3
2, ζ2 =
1 + i√
3
2, ζ3 =∞ . (8.2.24)
ζ1, ζ2 are elliptic fixed points of both order 3, and ζ3 is the cusp of cusp width
2 . One such group is Γ(1)2 =⟨η, γ2
∣∣ γ, η ∈ Γ(1), η3 = 1⟩
. The group
presentation of Γ(1)2 =⟨γ1, γ2, γ3 ∈ PSL2(R)
∣∣ γ31 = γ3
2 = 1 = γ1γ2γ3
⟩where
γ3 = t2, γ1 = t−1s = t1, γ2 = st−1 = t2. In this case, Γ(1)2 = gGg−1 with
g = 1 = ±I .
Modular triangle group of type (4, 4,∞)
Let G be the triangle group of type (4, 4,∞) . Following equation (4.1.2) one
way
G =⟨t1, t2, t3 ∈ PSL2(R)
∣∣ t41 = t42 = t∞3 = 1 = t1t2t3⟩
can be realized is
t1 = ±
(√2
−1
1
0
), t2 = ±
(0
−1
1√2
), t3 = ±
(1
0
2√
2
1
). (8.2.25)
with the corners of it’s fundamental triangle are
ζ1 =−1 + i√
2, ζ2 =
1 + i√2, ζ3 =∞ (8.2.26)
where ζ1, ζ2 are elliptic fixed points of both order 4, and ζ3 is the cusp of cusp
width 2√
2. Γ+
0 (2)2 =⟨Γ0(2),W2
⟩2=⟨η, γ2
∣∣ γ, η ∈ Γ+
0 (2), η4 = 1⟩
is one such
group . The group presentation of
Γ+
0 (2)2 =⟨γ1, γ2, γ3 ∈ PSL2(R)
∣∣ γ41 = γ4
2 = 1 = γ1γ2γ3
⟩where γ1 = −(W2t)
−1, γ2 = −(tW2)−1, γ3 = t2, i.e.
γ1 = ± 1√2
(2
−2
1
0
), γ2 = ± 1√
2
(0
−2
1
2
), γ3 = ±
(1
0
2
1
). (8.2.27)
149
In this case, Γ+
0 (2)2 = gGg−1 with g = ±(
2−14
0
0
214
)∈ PSL2(R) and therefore
the corners of it’s fundamental triangle are
ω1 = g · ζ1 =−1 + i
2, ω2 = g · ζ2 =
1 + i
2, ω3 = g · ζ3 =∞, (8.2.28)
where ω1, ω2 are elliptic fixed points of both order 4 and ω3 is a cusp of cusp
width 2 .
Modular triangle group of type (6, 6,∞)
Let G be the triangle group of type (6, 6,∞). Following equation (4.1.2)
G =⟨t1, t2, t3
∣∣ t31 = t32 = t∞3 = 1 = t1t2t3⟩
is realized by
t1 = ±
(√3
−1
1
0
), t2 = ±
(0
1
1
−√
3
), t3 = ±
(1
0
2√
3
1
). (8.2.29)
Therefore the corners of its fundamental triangle are
ζ1 =−√
3 + i
2, ζ2 =
√3 + i
2, ζ3 =∞ (8.2.30)
where ζ1, ζ2 are elliptic fixed points of both order 6, and ζ3 is the cusp of
cusp width 2√
3 . One such modular triangle group of this type is Γ+
0 (3)2 =⟨γ0(3),W3
⟩2=⟨η, γ2
∣∣ γ, η ∈ Γ+
0 (3), η6 = 1⟩. The group presentation of
Γ+
0 (3)2 =⟨γ1, γ2, γ3 ∈ PSL2(R)
∣∣ γ61 = γ6
2 = 1 = γ1γ2γ3
⟩where γ1 =
−(W3t)−1, γ2 = −(tW3)−1, γ3 = t2, i.e.
γ1 = ± 1√3
(3
−3
1
0
), γ2 = ± 1√
2
(0
−3
1
3
), γ3 = ±
(1
0
2
1
). (8.2.31)
In this case, Γ+
0 (3)2 = gGg−1 with g = ±(
3− 1
4
0
0
314
)∈ PSL2(R) and the corners
of the fundamental triangle of Γ+
0 (3)2 are
ω1 = g · ζ1 =−3 + i
√3
6, ω2 = g · ζ2 =
3 + i√
3
6, ω3 = g · ζ3 =∞ , (8.2.32)
where ω1, ω2 are elliptic fixed points of both order 6 and ω3 is the cusp of cusp
width 2 .
150
Modular triangle group of type (∞,∞,∞)
Let G be the triangle group of type (∞,∞,∞) . Following equation (4.1.2)
G =⟨t1, t2, t3
∣∣ t∞1 = t∞2 = t∞3 = 1 = t1t2t3⟩
can be realized by
t1 = ±(
2
−1
1
0
), t2 = ±
(0
−1
1
2
), t3 = ±
(1
0
4
1
)(8.2.33)
and the corners of it’s fundamental triangle are
ζ1 = −1, ζ2 = 1, ζ3 =∞ (8.2.34)
where ζ1, ζ2, ζ3 are three cusps of cusp width 1, 1, 4 respectively.
There are infinitely many triangle groups of this type. A particular interest
is put on in the modular triangle group of this type up to conjugation in
PSL2(R) . Γ(2),Γ0(4) are two important modular triangle groups of this type
which are conjugate to G in PSL2(R). Γ(2),Γ0(4) are both subgroups of index
6 and 2 respectively of the modular group. As a group
Γ0(4) =⟨γ1, γ2, γ3 ∈ PSL2(Z)
∣∣ γ1γ2γ3 = 1⟩
where γ3 = t, γ2 = st4s−1 and γ1 = −(γ2γ3)−1 = −(st4s−1t)−1, i.e.
γ1 = ±(
3
−4
1
−1
), γ2 = ±
(1
−4
0
1
), γ3 = ±
(1
0
1
1
). (8.2.35)
In this case, Γ0(4) = gGg−1 with g = ±12
(10−1
4
)∈ PSL2(R) and the corners of
the fundamental triangle of Γ0(4) are
ω1 = g · ζ1 = −1
2, ω2 = g · ζ2 = 0, ω3 = g · ζ3 =∞ (8.2.36)
where ω1, ω2, ω3 are cusps of cusp width 1,4 and 1 respectively.
A group
Γ(2) =⟨γ1, γ2, γ3 ∈ PSL2(Z)
∣∣ γ1γ2γ3 = 1⟩
151
where γ3 = t2, γ2 = st2s−1 and γ1 = −(γ2γ3)−1 = −(st2s−1t2)−1, i.e.
γ1 = ±(
3
−2
2
−1
), γ2 = ±
(1
−2
0
1
), γ3 =
(1
0
2
1
). (8.2.37)
In this case, Γ(2) = gGg−1 with g = ± 1√2
(10−1
2
)∈ PSL2(R) and therefore
the corners of the triangle group Γ(2) are
ω1 = g · ζ1 = −1, ω2 = g · ζ2 = 0, ω3 = g · ζ3 =∞ , (8.2.38)
where ω1, ω2, ω3 are cusps of cusp width 2 .
8.3 Integrality of the hauptmodul
Let G be any modular triangle group of type (`,m,∞) . For each such G there
exists a hauptmodul of the form
z(τ) = q−1
+ b0 + b1q + b2q2
+ · · · (8.3.1)
with bj ∈ C for all j ≥ 0, which gives an uniformization z : G\H∗G→ P1(C)
such that z(ζ3) =∞, z(ζ2) = 0, z(ζ1) = c for some nonzero c that depends on
G . We call such hauptmodul of G semi-regularized .
Definition 8.3.1. For any modular triangle group G of type (`,m,∞) the
commensurability index m of G is defined to be the index of G∩Γ(1) in Γ(1),
i.e. m = [Γ(1) : G ∩ Γ(1)] .
In the table below we list all these 9 types along with the value of m, κ, h,
z(ζ1) = c, expression for q, their normalized hauptmodul z(τ), order of the zero
δG
of their Ramanujan G-Delta form ∆G
and one of their modular conjugates
G.
Now , an explanation is given why necessarily the hauptmodul of modular
triangle groups can have integer coefficients. The following useful Lemma
follows immediately from Lemma 3 of [16] .
152
Gt y
pe
mκ
hq
cδ G
z G(τ
)
Γ(1
)(2,3,∞
)1
11
exp(2π
iτ)
1728
1q−
1+
744
+19
6884q
+21
4937
60q
2+
8642
9997
0q3
+···
Γ0(2
)(2,∞
,∞)
31
1ex
p(2π
iτ)−
641
q−
1−
24+
276q−
2048q
2+
1120
2q3
+···
Γ0(3
)(3,∞
,∞)
21
1ex
p(2π
iτ)−
272
q−
1−
12+
54q−
76q
2−
243q
3+···
Γ(2
)(∞
,∞,∞
)6
12
exp(π
iτ)
161
q−
1+
8+
20q−
62q
3+
216q
5+···
Γ(1
)2(3,3,∞
)2
i √3
2i √3
exp(π
iτ)
144
1q−
1+
72+
1476q−
2033
10q
3+
9919
800q
5+···
Γ+ 0
(2)
(2,4,∞
)3
11
exp(2π
iτ)
256
1q−
1+
104
+43
72q
+96
256q
2+
1240
002q
3+···
Γ+ 0
(3)
(2,6,∞
)2
11
exp(2π
iτ)
108
2q−
1+
42+
1113q
+72
86q
2+
6213
9q3
+···
Γ+ 0
(2)2
(4,4,∞
)3
i2
i exp(π
iτ)
322
q−1
+16
+76q−
702q
3+
5224q5
+···
Γ+ 0
(3)2
(6,6,∞
)2
i √3
2i √3
exp(π
iτ)
364
q−
1+
18+
99q−
1377q
3+
1925
1q5
+···
Tab
le8.
1:S
emi-
regu
lari
zed
hau
ptm
odu
lof
G
153
Lemma 8.3.2. Let f(τ) =∑∞
`=0 fnqn be a modular form of weight k ∈ Z for
any modular triangle group G and w be the weight of the Ramanujan G-Delta
form ∆G
. If fn ∈ Z ,∀n ≤ k·[Γ(1):G∩Γ(1)]w
then fn ∈ Z for every n.
Proposition 8.3.3. Let G be any modular triangle group. Then the haupt-
modul of G, zG
lie in q−1 + Z[[q]] where q = κ exp(2πiτh
), κ and h depends on
G.
Proof. Let m = [G : G ∩ Γ(1)]. Let h be the cusp width of G with respect
to cusp ∞. From subsection 4.1.1 of chapter 4 , we see in the current setting
that the hauptmodul zG
(τ) = c · z∞G
(τ) where c = 1/z∞
[−1]where the left hand
side zG
(τ) represents the semi-regularized hauptmodul and right hand side one
represents the regularized hauptmodul of G . Define fG
(τ) = ∆G·z
G(τ) , where
∆G
= (−1)L
c
(DzG
)L
zLG−d L
me · (zG
− 1)L−dL`e
=
qδG +
∑∞n=1 αnq
δG
+2n
if ` = 4, 6
qδG +
∑∞n=1 αnq
δG
+n
otherwise.
is the Ramanujan G-Delta form ∆(τ) of weight w = 2L of G, where
L =
1 if ` =∞ = m` if m =∞
lcm(`,m) if ` 6=∞,m 6=∞
and δG
= L − dL`e − d L
me. Clearly, f
G(τ) = εq
δG−1
+∑∞
n=1 anqδG−1+n
, ε =
±1, is a holomorphic modular form of weight w = 2L of G . Now by using
Lemma 8.3.2, it is clear that if all the coefficients of fG
(τ) for 0 ≤ n ≤ m
are integers then fG
(τ) ∈ Z[[q]], which can be verified easily. Since 1/∆G∈
q−δ
G + Z[[q]], therefore fG/∆
G∈ q −1
+ Z[[q]] .
In the table 8.2 , we write the Ramanujan G-Delta forms ∆G
(q) and the
function fG
(q) , used in the proof of proposition 8.3.3 of all modular triangle
groups G of type (`,m,∞) .
154
G∆
G(q
)f G
(q)
Γ(1
)q−
24q
2+
252q
3−
1472q
4+···
1+
720q
+17
9280q
2+
1695
4560q
3+
3969
7416
0q4
+···
Γ0(2
)q
+8q
2+
28q
3+
64q
4+···
−1
+16q−
112q
2+
448q
3−
1136q
4+···
Γ0(3
)q
2+
6q3
+27q
4+
80q
5+···
−q
+6q
2−
9q3−
4q4
+···
Γ(2
)q
+4q
3+
6q5
+8q
7+···
1+
8q+
24q
2+
32q
3+
24q
4+
48q
5+
96q
6+···
Γ(1
)2q
+36q
3+
486q
5+
2376q
7+···
1+
72q
+15
12q
2+
2592q
3−
1496
88q
4+···
Γ+ 0
(2)
q−
8q2
+12q
3+
64q
4+···
1+
96q
+35
52q
2+
6259
2q3
+52
8864q
4+···
Γ+ 0
(3)
q2−
12q
3−
4566q
4+
8702
0q5
+···
q+
30q
2−
3957q
3−
1108
22q
4+
2060
130q
5+···
Γ+ 0
(2)2
q2
+8q
4+
12q
6−
64q
8+···
q+
16q
2+
84q
3+
128q
4−
82q
5+···
Γ+ 0
(3)2
q4
+36q
6+
486q
8+
2700q
10
+···
q3
+18q
4+
135q
5+
648q
6+···
Tab
le8.
2:R
aman
uja
nG
-Del
tafo
rms
∆G
(q)
and
the
fun
ctio
nf G
(q)
155
8.3.1 Why restrict to modular triangle groups?
It is easy to see that restricting to the modular triangle groups is very natural
when considering integer Fourier coefficients. For example:
Theorem 8.3.4. Let G be any triangle group with exactly one cusp . Let X(τ)
be a nonconstant weakly holomorphic vvmf of arbitrary weight and rank of G
whose components have all integer Fourier coefficients , then G is an arithmetic
triangle group .
Before giving the proof of Theorem 8.3.4 we introduce the notion of wron-
skian of a vvmf X(τ) which is needed in the proof . Let X(τ) be a nontrivial
element of M!k(ρ) with respect to multiplier of rank d. Then the wronskian
WX(τ) of X(τ) is the determinant of the d × d matrix whose d columns are
formed by X(τ), DkX(τ), D2kX(τ), · · · , Dd−1
k X(τ). It is easy to show thatWX(τ)
is a scalar valued modular form of G of weight d(k + d − 1) with multiplier
det(ρ) . For more details on wronskian of a vvmf , see [36] .
Proof of Theorem 8.3.4. Let f(τ) be any scalar modular form of G of weight
k ∈ Z with integer Fourier coefficients . Then we know [40] that G is an
arithmetic triangle group and the converse of this is a classical result reproved
by Atkin & Swinnerton-Dyer in [4].
Suppose for contradiction G is nonarithmetic triangle group and assume
that the components of X(τ) are linearly independent. Let d be the rank of the
multiplier ρ of X(τ). Consider the modular wronskian WX(τ) of X. Consider
f(τ) = WX(τ)`m
then f is a scalar modular form with trivial multiplier of even
weight and with integer coefficients. This is a contradiction of the converse
part of the above statement.
If instead the components of X(τ) are not linearly independent, then we
can choose a basis such that X(τ) = (X1,X2, · · · ,Xr, 0, · · · , 0)t for some r ≤ d,
one can always do that and moreover this can be done over Q. Hence, their
156
coefficients will be bounded. Now applying the previous technique to the
truncated vector Y (τ) = (X1, · · · ,Xr)t will give the desired result .
We expect the similar result follow in the more general case , namely for
any arithmetic triangle group of type (`,∞,∞) . Hence , we conjecture the
following :
Conjecture 8.3.5. If G is a triangle group with at least one cusp and X(τ)
is a nonconstant weakly holomorphic vvmf of arbitrary weight and rank with
integer Fourier coefficients, then G is an arithmetic triangle group.
8.4 Modular vs. hypergeometric differential
equations
Let (G, κ) be sufficiently integral and ρ : G → GL2(C) be an admissible
multiplier . Let M!k(ρ) denote the space of all weakly holomorphic vvmf of
weight k and multiplier ρ . In this chapter we are mainly interested in weakly
holomorphic vvmf of G with respect to cusp ∞ i.e. those vvmf which have a
pole at cusp ∞ . The point made below is that to classify such rank 2 vvmf is
equivalent to finding the solutions of certain type of second-order differential
equations which is called as second-order modular differential equations . From
chapter 4 we know that in case of G being a triangle group these second-order
modular differential equations are hypergeometric differential equations .
Since G is sufficiently integral therefore∞ ∈ CG
and the stabilizer subgroup
G∞ of the cusp∞ is a cyclic group generated by the matrix t∞ =(
10h1
)where
h is called the cusp width of∞ and by definition it is the smallest nonzero real
number such that(
10
11
)h ∈ G . Since ρ is admissible, ρ(t∞) is diagonalizable
and there exist a diagonal exponent matrix Λ such that
q−ΛX(τ) =
∑n∈Z
X[n]qn
157
where q = κ exp(2πiτh
) for some nonzero κ ∈ C . By definition of weakly
holomorphic vvmf the Fourier coefficients X[n] vanish for all but finitely many
negative n .
A second-order modular differential equation for G is a linear ordinary
differential equation of the form
L[f(τ)] = D2kf(τ) +Q2(τ)Dkf(τ) +Q4(τ)f(τ) = 0 , (8.4.1)
where k is an arbitrary integer, and Q2(τ), Q4(τ), are meromorphic modular
form of weight 2 and 4 respectively for G. Recall that Dk is covariant with
respect to the |k-action of G on the space of meromorphic functions f : H→ C,
so that Dk(f |kγ) = (Dkf)|k+2γ for each γ ∈ G. Because of this covariance, if
f is a solution of (8.4.1) then for each γ ∈ G we have
0 = L[f ]|k+4γ
= D2k(f |kγ) +Q2Dk(f |kγ) +Q4(f |kγ)
= L[f |kγ] ,
so f |kγ is again a solution of (8.4.1). This shows that G acts linearly on the
space V of solutions of (8.4.1), so that each equation has associated to it a
2-dimensional G-module and any basis X1(τ),X2(τ) for V keep candidates
for a rank 2 vvmf X(τ) = (X1(τ),X2(τ))t for some representation ρ . To
ensure that this indeed is the case, we require that (8.4.1) is Fuchsian in H∗G
,
in other words it is required that Q2(τ), Q4(τ) to be chosen so that (8.4.1)
has only regular singular points in H∗G
. This means that the Q2(τ), Q4(τ)
are meromorphic throughout H, with no pole there of order greater than 2
and 4 respectively, and are holomorphic at the cusps. It is also required that
Q2(τ) and Q4(τ) have bounded denominators, i.e. there is a (minimal) positive
integer M such that
Q2(τ) =∑n≥0
α2(n)
Mqn
, Q4(τ) =∑n≥0
α4(n)
Mqn
(8.4.2)
158
for some integers α2(n), α4(n) .
Using modular derivative (8.2.3), (8.4.1) looks like
q2 d2f
dq 2 + g1(τ)qdf
dq+ g0(τ)f(τ) = 0, (8.4.3)
where q = 0 (i.e. τ = i∞) is a regular singular point and define
g1(τ) = 1− 2k + 2
KE
G
2 (τ) +Q2(τ)
=
[1− (2k + 2)δ
K+α2(0)
M
]+
1
KM
∑n≥1
A1(n)qn
,
(8.4.4)
g0(τ) = − k
Kq
d
dqE
G
2 (τ) +k(k + 2)
K2E
G
2 (τ)2 − k
KE
G
2 (τ)Q2(τ) +Q4(τ)
=k(k + 2)δ2
K2− kα2(0)δ
KM+α4(0)
M+
1
K2M
∑n≥1
A0(n)qn
for some integers A0(n), A1(n) and EG
2 (τ) = δ + d1q + d2q2
+ · · · .
The explicit formulae
A1(n) = Kα2(n)− 2M(k + 1)dn , (8.4.5)
A0(1) = K2α4(1) +Mkd1(4δ + 2kδ −K)−Kkd1α2(0) (8.4.6)
are used in what follows. Assume that a solution of (8.4.1) has the form
f(q) = qλ +∑n≥1
a(n) qλ+n
. (8.4.7)
Then λ is a root of the indicial equation φ(z) = 0, where from (8.4.4)
φ(z) = z2 +
[α2(0)
M− (2k + 2)δ
K
]z +
[k(k + 2)δ2
K2− kα2(0)δ
KM+α4(0)
M
].
Setting a(0) = 1, the other coefficients of (8.4.7) are determined by the recur-
sive formula
a(n) = −∑n−1
j=0 Yn(j)a(j)
φ(λ+ n), n ≥ 1
where
Yn(j) = (λ+ j)A1(n− j)KM
+A0(n− j)K2M
.
159
We are interested in the cases where the indicial roots are rational numbers,
so assume now that λ = xN
is the exponent in (8.4.7), and the other indicial
root is yN
with gcd(x, y,N) = 1 . Then
φ(λ+ n) =n[nN + (x− y)]
N
and the recursive formula reads
a(n) = − 1
K2Mn[nN + (x− y)]
n−1∑j=0
Cn(j)a(j) , (8.4.8)
where for n ≥ 1, 0 ≤ j ≤ n− 1 we define the integers
Cn(j) = NA0(n− j) +K(x+ jN)A1(n− j). (8.4.9)
For a prime p we will write νp(m) for the p-adic valuation of a rational
number m, i.e. if m = pla ∈ Z with p - a then νp(m) = l, and if m = bc
then
νp(m) = νp(b)− νp(c). With this notation we have the elementary but useful
Proposition 8.4.1. Suppose p is a prime dividing M such that for all n ≥ 1
the condition
νp(M) > νp(Cn(n− 1)) = minνp(Cn(j))n−1j=0 (8.4.10)
is satisfied. Then νp(a(n)) is a negative, strictly decreasing function of n, and
(8.4.7) has p-unbounded coefficients.
Proof. This follows inductively on n from (8.4.8), (8.4.10), and the basic prop-
erties of the p-adic valutation νp .
8.4.1 Hypergeometric differential equations and trian-gle groups
Turning now to study of second order hypergeometric differential equations
corresponding to a special class of sufficiently integral pair (G, κ) where G is
any triangle group of type (`,m,∞) and κ 6= 0. Following chapters 4 , 6 and 7
160
we know that the theory of such differential equations is equivalent to the
theory of rank 2 vvmf of triangle groups. Thus , the idea now is to begin with
differential equations
d2f
dz2+ P1(z)
df
dz+ P2(z)f = 0 (8.4.11)
on the punctured sphere P1(C)\0, c,∞. Such an equation lifts to the cover
H∗G
of G\H∗G
, and a simple application of the chain rule yields an equation
(8.4.1) in weight k = 0, with coefficient functions
Q2(q) = P1(z) · θz− D2(θz)
θz, Q4(q) = P2(z) · (θz)2. (8.4.12)
We obtain from (8.3.1) the q-expansions
θz = −q −1
+ b1q + 2b2q2
+ · · · ,
θ2z = q−1
+ b1q + 4b2q2
+ · · · ,D2(θz)
θz=
θ2z
θz− 2
KE
G
2
= −[1 + 2b1q
2
+ 6b2q3
+ · · ·]− 2δ
K− 2
K
∞∑n=1
dnqn
,
(8.4.13)
1
z− α= q + (α− b0)q
2
+ ((α− b0)2 − b1)q3
+ · · · , α ∈ C (8.4.14)
which will feature in the computations below .
We are interested in the case where (8.4.11) is of the Fuchsian class, mean-
ing that every point of P1(C) is regular singular, and such that the set of
singularities is contained in the puncture set 0, c,∞. This case was inten-
sively studied by Riemann and the results obtained serve as the first example
of what is now called a Riemann-Hilbert correspondence. The correspondence,
in this setting, is between two-dimensional representations
ρ : π1
(P1(C)\0, c,∞
)→ GL(V) (8.4.15)
161
of the fundamental group of the punctured sphere and equivalence classes of
Fuchsian equations (8.4.11) whose singularities are at 0, c,∞. If z0 6= 0, c,∞
and V is the solution space of such an equation in a neighborhood of z0, then
(8.4.15) is simply the representation obtained by analytically continuing V
throughout the punctured sphere. The representations obtained in this way are
“rigid”, meaning that the indicial roots of (8.4.11) at the regular singular points
completely determine the associated representation (8.4.15). In particular, two
equations (8.4.11) correspond to the same monodromy representation (8.4.15)
if and only if the sets of indicial roots for the two equations are congruent
(mod Z), and this relation defines the equivalence class corresponding to each
monodromy representation.
This data is encoded in the Riemann scheme,
P
0 c ∞α1 β1 γ1 zα2 β2 γ2
, (8.4.16)
which lists in each column a regular singular point and its associated indicial
roots, with respect to the local variable z. The Riemann scheme (8.4.16)
defines a Riemann’s differential equation (8.4.11) with
P1(z) =1− α1 − α2
z+
1− β1 − β2
z− c(8.4.17)
P2(z) =β1β2c
z(z− c)2− α1α2c
z2(z− c)+
γ1γ2
z(z− c)
and the eigenvalues of the monodromy matrix for each singular point are the
exponentials of the corresponding indicial roots, taken according to any fixed
branch of C. Of course, since we are viewing the local variable z as a Haupt-
modul for a triangle subgroup G, as discussed above, so there will be restric-
tions on the monodromy of (8.4.11) around any singular points that correspond
to elliptic points on G\H∗G
; in particular, the indicial roots associated to these
points must be in 1`Z where ` is the order of the elliptic fixed point.
162
We are interested in the cases where all the indicial roots are rational, so
write αi = riL, βi = si
L, γi = ti
L, for i = 1, 2. We now get
r1 + r2 + s1 + s2 + t1 + t2 = L. (8.4.18)
In this notation, it follows from (8.4.14) and (8.4.17) that the coefficient func-
tions of (8.4.11) read
P1(z) =L− (r1 + r2)
L· 1
z+L− (s1 + s2)
L· 1
z− c(8.4.19)
P1(z) =L− (r1 + r2)
L[q − b0q
2
+ · · · ] +L− (s1 + s2)
L[q + (c− b0)q
2
+ · · · ],
P2(z) =s1s2c
L2· 1
z(z− c)2− r1r2c
L2· 1
z2(z− c)+t1t2L2· 1
z(z− c)(8.4.20)
P2(z) =s1s2c
L2[q
3
+ · · · ]− r1r2c
L2[q
3
+ · · · ] +t1t2L2
[q2
+ (c− 2b0)q3
+ · · · ],
Now, using (8.4.19), (8.4.20) and (8.4.13) in (8.4.12) , we get
Q2(q) =∑n≥0
α2(n)
LLqn
, Q4(q) =∑n≥0
α4(n)
L2qn
,
for some integers α2(n), α4(n) , where in particular
α2(1) = Lb0[L− (r1 + r2)] + L(b0 − c)[L− (s1 + s2)] + d1L,
α4(1) = (c− 2b0)t1t2 + c(s1s2 − r1r2) .
Setting now M = LL2, in the notation of (8.4.12) we have α2(n) = Lα2(n),
α4(n) = Lα4(n) for each n ≥ 0, and using k = 0 in (8.4.5) now yields
A1(1) = LL[2Lb0 − b0(s1 + s2 + r1 + r2) + c(s1 + s2 − L)],
A1(1) = 2L2L[2Lb0 − b0(L− (t1 + t2))− c(L− (s1 + s2))],
A0(1) = 4L3[(c− 2b0)t1t2 + c(s1s2 − r1r2)] .
163
There is a minimal positive N such that t1L
= xN
, t2L
= yN
are the indicial roots
of (8.4.1), as in Section 8.2; thus L = L0N for some integer L0 and we may
assume (permuting t1 and t2 if needed) that (x,N) = 1. This gives
Cn(n− 1) = NA0(1) +K[N(n− 1) + x]A1(1) = 4L3 ·N · C (8.4.21)
where
C = (c−2b0)t1t2+c(s1s2−r1r2)+L0[N(n−1)+x][L(b0−c)+b0(t1+t2)+c(s1+s2)] ,
(8.4.22)
By using the trace condition (8.4.18), we see that (8.4.22) is equivalent to
C = (c−2b0)t1t2+c(s1s2−r1r2)+L0[N(n−1)+x][Lb0+(b0−c)(t1+t2)−c(r1+r2)] .
(8.4.23)
8.4.2 Modular triangle groups and unbounded denom-inator property
We now turn our attention to those triangle groups G of type (`,m,∞) for
which zG
(τ) has all integer coefficients and it takes an integer value c at ζ1 .
More precisely , modular triangle groups G satisfy these requirements as ev-
ident from the table 1 in section 8.2.1 . Our aim is to probe and extend the
Atkin-Swinnerton-Dyer conjecture for such G with the help of the theory of
rank 2 vvmf as suggested by Selberg. This will be done by finding necessary
conditions on the exponent matrix of the generating element t∞ of G and the
hauptmodul zG
(τ). The point should be noted that similar arguments can
be made in case of considering the Fourier expansions with respect to other
cusps . This will require to considering a different expression of q , i.e. different
values of κ and h .
Following from (8.4.21), ker(ρ) will be a noncongruence group if (8.4.10)
of proposition 8.4.1 is satisfied. In other words,
νp(M) > νp(Cn(n− 1)) = minνp(Cn(j))n−1j=0
164
is a necessary condition for ker(ρ) to be a noncongruence group. Focus is now
put on eliminating those primes p which will satisfy this necessary condition.
As we show below these primes are highly dependent on the values of c, b0, `,m
and the eigenvalues of the exponent matrix of t∞ or equivalently on the indicial
roots of equation (8.4.11) . We begin by observing the following simple
Lemma 8.4.2. Let G be any modular triangle group of type (`,m,∞) and p
be any prime such that p - L then p - c .
Proof. Since L ∈ 1, 2, 3, 4, 6, only primes p which divides L are 2 and 3.
From table 8.1, it can be seen that c = 2a3b, 0 ≤ a ≤ 8, 0 ≤ b ≤ 3. Clearly,
p - c.
From (8.4.22), observe that νp(C) ≤ νp(c− 2b0) and from (8.4.21) write
νp(Cn(n− 1)) = νp(4) + 3 · νp(L) + νp(N) + νp(C) (8.4.24)
≤ νp(4) + 3 · νp(L) + νp(L) + νp(c− 2b0) .
Lemma 8.4.3. Let p > 2 be any prime such that p - L. Then νp(M) >
νp(Cn(n− 1)) if νp(c− 2b0) < νp(L).
Proof. Suppose that νp(L) = α, α ∈ Z≥0. Then νp(c − 2b0) < α. Since
M = LL2, νp(L) = 0, νp(4) = 0 and L = L0N therefore νp(M) = 2α, νp(M) >
νp(N) and νp(N) ≤ α. From (8.4.24) we write
νp(Cn(n− 1)) ≤ νp(4) + 3 · νp(L) + α + νp(c− 2b0)
< 2α .
Lemma 8.4.4. Let p > 2 be any prime such that p | L. Then νp(M) >
νp(Cn(n− 1)) if νp(c− 2b0) < νp(L)− 2νp(L).
165
Proof. Suppose that νp(L) = α, νp(L) = β, α, β ∈ Z≥0. Then νp(c − 2b0) <
α − 2β. Since M = LL2, νp(4) = 0 and L = L0N therefore νp(M) = 2α + β,
νp(M) > νp(N) and νp(N) ≤ α. From (8.4.24) write
νp(Cn(n− 1)) ≤ νp(4) + 3 · νp(L) + α + νp(c− 2b0)
< 2α + β .
Lemma 8.4.5. ν2(M) > ν2(Cn(n− 1)) if ν2(c− 2b0) < ν2(L)− 2(ν2(L) + 1).
Proof. Suppose that ν2(L) = α, ν2(L) = β, α, β ∈ Z≥0. Then ν2(c − 2b0) <
α−2(β+1). SinceM = LL2, ν2(4) = 2 and L = L0N therefore ν2(M) = 2α+β,
ν2(M) > ν2(N) and ν2(N) ≤ α. From (8.4.24) write
νp(Cn(n− 1)) ≤ νp(4) + 3 · νp(L) + α + νp(c− 2b0)
< 2α + β .
Lemma 8.4.6. νp(Cn(n−1)) = minνp(Cn(j))n−1j=0 if νp(Cn(n−1)) = νp(N).
Proof. Using the values of K = 2L, M = LL2 in (8.4.5)
A1(n) = 2LL[α2(n)− L(k + 1)dn] (8.4.25)
This shows that N divides A1(j) for all j, and this fact combined with (8.4.9)
implies that νp(Cn(j)) ≥ νp(N) for all 0 ≤ j ≤ n− 1. This implies that
minνp(Cn(j))n−1j=0 ≥ νp(N). (8.4.26)
If νp(Cn(n− 1)) = νp(N) then using (8.4.26)
νp(Cn(n− 1)) ≥ minνp(Cn(j))n−1j=0 ≥ νp(N) = νp(Cn(n− 1)).
Therefore νp(Cn(n− 1)) = minνp(Cn(j))n−1j=0 .
166
Lemma 8.4.7. Let G be any modular triangle group and p > 2 is a prime
such that p - L. Then νp(Cn(n− 1)) = νp(N) if and only if p - C.
Proof. Since νp(L) = 0 = νp(4), therefore from (8.4.24)
νp(Cn(n− 1)) = νp(N) + νp(C) (8.4.27)
and if p - C then νp(Cn(n− 1)) = νp(N) is obvious. Conversely, if νp(Cn(n−
1)) = νp(N) then using (8.4.27) we get νp(C) = 0 .
Theorem 8.4.8. Suppose p > 2 is a prime such that p|L and p - L . Then
(8.4.10) of proposition 8.4.1 satisfies if p - C .
Proof. Combining Lemma 8.4.7 and Lemma 8.4.6 , the desired result is ob-
tained .
Lemma 8.4.9. Let G be any modular triangle group and p > 2 is a prime
such that p - L . Then p - C if any of the following are satisfied :
1. p - r1r2
2. p - s1s2
3. p - t1t2 and p - c− 2b0 when c− 2b0 6= 0 .
Proof. If p - r1r2 or p - s1s2, its easy to see that p - C since p - c following from
lemma 8.4.2. Also in case of c− 2b0 = 0, from (8.4.22) and using c = 2b0
C = c(s1s2 − r1r2) + L0b0[N(n− 1) + x][(s1 + s2) + (r1 + r2)]
Clearly, if p - r1r2 or p - s1s2 then p - C since p - c. Now, we assume that
c− 2b0 6= 0 and p divides r1, r2 and s1, s2 then p - C if p - t1t2 and p - (c− 2b0)
follows from (8.4.22) .
Corollary 8.4.10. If G = Γ(1) then any prime p > 5 which divides L is an
ubd prime .
167
G c− 2b0 ubd prime
Γ(1) 24 · 3 · 5 p /∈ 2, 3, 5
Γ0(2) −24 p /∈ 2
Γ0(3) 3 p /∈ 2, 3
Γ(2) 0 p /∈ 2
Γ(1)2 0 p /∈ 2, 3
Γ+
0 (2) 24 · 3 p /∈ 2, 3
Γ+
0 (3) 23 · 3 p /∈ 2, 3
Γ+
0 (2)2 0 p /∈ 2
Γ+
0 (3)2 0 p /∈ 2, 3
Table 8.3: ubd primes of modular triangle groups
Proof. Since c = 1728, b0 = 744, so c − 2b0 = 240 = 24 · 3 · 5 6= 0. Let
p > 5 be any prime such that p | L then p does not divide (c − 2b0) and
p - L. Furthermore, since two of the three regular singular points of (8.4.11)
correspond to elliptic points on the modular curve H∗G
and p > 3, p must
divide r1, r2, s1, s2. Then the trace condition (8.4.18) imply that p also divides
r1 + r2, s1 + s2, t1 + t2, and since L is assumed to be minimal, t1t2 can not be
divisible by p (since otherwise p divides the numerator of every indicial root).
Thus the condition (3) of lemma 8.4.9 is satisfied. This imply that p - C.
Now, from theorem 8.4.8 we obtains that any prime p > 5 is an ubd prime for
Γ(1) .
Corollary 8.4.11. If G be any modular triangle group then any prime p > 5
which divides L is an ubd prime.
Proof. Follow immediately from the proof of Lemma 8.4.10 and the table 8.3 .
168
8.5 End on a high note with p-curvature
We end this chapter by giving a very brief introduction and speculations about
the p-curvature and its relation with unbounded denominator conjecture about
the Fourier coefficients of scalar-valued modular forms and its generalization
in the theory of vector-valued modular forms. Throughout this section G will
always denote a modular triangle group of type (` ,m ,∞) .
For any prime p, we define the local ring of Z at (p), i.e.
Z(p) =
m
n∈ Q
∣∣ p - n.Let S be any set of rational primes. We say a vvmf X(τ) is integral, rational
and S-integral if its Fourier coefficients X[n] lie in Zd,Qd and Zd(p)(∀p ∈ S)
respectively.
Let S be any set of rational primes. We say a ρ : Γ −→ GLd(C) is Fourier-
integral, Fourier-rational and Fourier-S-integral if for each k ∈ Z there exists
a basis over C of the space Hw(ρ) of holomorphic weight w vvmf for ρ such
that each basis vector is integral, rational and S-integral respectively.
A prime p is called ubd-prime of a rational vvmf X(τ) and Fourier-rational
ρ respectively, if X(τ) is not a p-integral vvmf and ρ is not a Fourier-p-
integral respectively.
Conjecture 8.5.1 (Generalized Atkin-Swinnerton-Dyer-Mason’s conjecture
of bounded denominator). Let G be any triangle group of type (`,m,∞) . If a
representation ρ : G −→ GLd(C) is Fourier-integral, then ker(ρ) is a congru-
ence subgroup G.
Conjecture 8.5.2. Let ρ : G −→ GL2(C) be any admissible irreducible rep-
resentation of any modular triangle group G . Let X(τ) be any nonzero vvmf
for ρ of weight w ∈ 2Z with rational coefficients . If ρ has infinite image (i.e.
ker(ρ) has infinite index in G), then X has infinitely many ubd-primes.
169
The following question is worth mentioning before moving further .
Question 8.5.3. Let ρ : G −→ GLd(C) be any admissible irreducible repre-
sentation of any modular triangle group G of rank d. For which values of d,
all ρ’s are rigid?
For example, in case of G = Γ(1), all ρ’s are rigid for d ≤ 5 . Let us
define first the rational Fuchsian differential equations and describe them in
the following
Definition 8.5.4. Letd
dzΞ(z) = Ξ(z)Ω(z) (8.5.1)
be a Fuchsian Differential Equation in the matrix form defined over P1(C)\0, 1,∞.
Ξ(z),Ω(z) are matrices of order d with entries in C[z−1, z]] and
Ξ(z) = zΛ
∞∑n=0
Ξnzn (8.5.2)
where Ξn ∈ Md(Q), Λ is a diagonal matrix of order d and Ξ0 is the identity
matrix of order d. The points 0, 1,∞ are the regular singular points of
equation (8.5.1).
We are interested in those rank d first order Fuchsian differential equation
where
Ω(z) =Az
+B
z− 1(8.5.3)
and A,B ∈ Md(Q). We get such Fuchsian system from the theory of rank d
vvaf for triangle groups. Let G be any triangle group and ρ : G −→ GLd(C) be
any admissible multiplier . Let X(τ) be any nonzero vvmf of G with respect to
multiplier ρ such that the components X1, · · · ,Xd are linearly independent over
C, then we construct the differential equation from the followng determinant :
det
f D0f · · · Dd
0fX1 D0X1 · · · Dd
0X1...
... · · · ...Xd D0Xd · · · Dd
0Xd
=d∑
k=0
gkDk0f = 0 (8.5.4)
170
where for k ∈ 2Z, Dk0 = D2k−2· · ·D2D0 , Dk = q d
dq− k
2LE
G
2(τ) is a modular
derivative of G ,q = exp(
2πiτh
)and h is the cusp width of the cusp ∞ , clearly
D0 = q ddq
is the modular derivative of G of weight 0 .
Lemma 8.5.1. The solution space of the (8.5.4) is the span of the components
of Xi of the vvmf X and its monodromy corresponds to the multiplier ρ.
Now, rewrite equation (8.5.4) using the chain rule and change of variable
with respect to z(τ). We may get order d Riemann’s differential equation with
regular singular points at 0, 1 and ∞ of the following form :
ddf
dzd+
d−1∑n=1
zn−1(anz− bn)
zd−1(1− z)
dnf
dzn+
a0f
zd−1(1− z)= 0 (8.5.5)
From chapter 5 we know that for any genus-0 Fuchsian group G of first kind
the space of vvaf M!k(ρ) and H(ρ) are free module of rank d over M!
0(1) and
H(1) respectively and more precisely in the case of H(ρ) the d free generators
are constructed by the existence of one vvmf X(τ). These can be arranged in
the form of fundamental matrix as follows :
Ξ(τ) =
X1 D0X1 · · · Dd
0X1
X2 D0X2 · · · Dd0X2
...... · · · ...
Xd D0Xd · · · Dd0Xd
(8.5.6)
and it is known that this Ξ(τ) satisfies the Fuchsian differential equation (8.5.1)
of rank d . More precisely, from chapter 6 in case of rank 2 and of any traingle
group G the following is true :
Lemma 8.5.2. Let ρ : G −→ GL2(C) be an admissible multiplier of any
triangle group G then Riemann’s differential equation of order 2 can be obtained
from rank 2 first order (8.5.1) .
Proof. Proof follows from Theorem 6.1.1 and Corollary 6.1.2 .
171
8.5.1 Modulo p Matrix Form
Let mn∈ Q and any prime p such that p - n then define an integer, denoted
by [mn
](p) a reduction of mn
mod p. For example, [35](7) is equivalent to 2. We
generalize this definition naturally to define reductions of the rational functions
in z modulo p. For example, [7z2+2z+1z2+z+3
](3) is equivalent to z+1z
since[7z2 + 2z + 1
z2 + z + 3
](3)
≡[z2 + 2z + 1
z2 + z
](3)
≡[
(z + 1)2
z(z + 1)
](3)
≡ z + 1
z.
Similarly, this notion of reduction modulo p can be extended to the formal
power series as well as to the matrices by reducing each and every entry modulo
p. Also, define [zr](p) := z`|0 ≤ ` < p, ` ≡ [r](p).
The ring of formal power series of reduction modulo p is denoted by Zp[[z]]
and its field of fractions is denoted by Lp[z]] := Zp[z−1, z]].
Define BP to be the set of finitely many primes appearing in any de-
nominator of any of the rational number entries of the matrices A and B of
equation (8.5.3) . This is called as the set of bad primes . We can easily reduce
the matrices A and B modulo p for all p /∈ BP , i.e. [A](p), [B](p) ∈ Md(Zp).
Therefore, define
[Ω(z)](p) =
[Az
+B
z− 1
](mod p) =
[A](p)z
+[B]pz− 1
and [F (z)](p) for the matrix of reduction modulo p, if p does not divide any of
the denominator of infinitely many coefficients of all the entries in F (z).
Note 8.5.5. There are other primes which are bad primes in this theory
namely those which appear in the denominator of the entries of exponent ma-
trix Λ of ρ(t∞) . These primes were handled in the first part of this chapter .
For any p /∈ BP , we say (8.5.1) has sufficiently many solutions modulo p,
if
172
1. there is a matrix F (z) =∑∞
n=0 Fnzn of order d with coefficients Fn ∈ Md(Z)
which satisfiesd
dz[F (z)](p) = [Ω(z)](p)[F (z)](p) (8.5.7)
2. and for which det [F (z)](p) 6= 0.
Lemma 8.5.3. Let X(z) be a vvmf of G for multiplier ρ satisfying the equation
d
dz[X(z)](p) ≡ [Ω(z)](p)[X(z)](p) (8.5.8)
then zmpX(z), for any m ∈ Z will also satisfy the equation (8.5.4).
Proof. Proof follows from the product rule of differentiation.
Corollary 8.5.4. For any fixed w ∈ 2Z, M!w(ρ) is a module over the subfield
Lp[zp]] of Lp[z]] and if Ξ(z) be the fundamental matrix of M!w(ρ) then for any
P (z) ∈Md(Lp[z]]), [Ξ(z)](p)[P (z)](p) also satisfies equation (8.5.7).
Proof. Proof follows from Lemma 8.5.3.
8.5.2 p-curvature
Let
Ly = an(z)dny
dzn+ an−1(z)
dn−1y
dzn−1+ · · ·+ a1(z)
dy
dz+ a0(z)y = 0 (8.5.9)
be the ordinary differential equation of n-th order with ai, (∀0 ≤ i ≤ n) are
being algebraic functions with an(z) 6= 0. The equation (8.5.9) can be written
into matrix form as follows :
d
dz
ydydz...
dn−1ydzn−1
=
0 1 0 · · · 00 0 1 · · · 0...
...... · · · ...
− a0
an− a1
an− a2
an· · · −an−1
an
ydydz...
dn−1ydzn−1
(8.5.10)
Just to record as an information we write the following well known conjecture
in the theory of p-curvature :
173
Conjecture 8.5.6 (Grothendieck’s conjecture on p-curvatures). The equation
Ly = 0 has a full set of algebraic solutions if (and only if ) for almost all primes
p ∈ Z the reduction modulo p of Ly = 0 has a full set of solutions in Fp(x).
Definition 8.5.7 (p-curvature). Define A(z)(1) = A(z) and
A(z)(k+1) =d
dz
(A(z)(k)
)+ A(z)(k)A(z) .
For any p /∈ BP , the p-curvature is the quantity [A(z)(p)](p) ∈ Md(Lp[zp]]).
We say the p-curvature vanishes if A(z)(p) ≡ 0(mod p).
Lemma 8.5.8. For any k ∈ Z>0, the equation (8.5.1) implies that
dk
dzkΞ(z) = A(z)(k)Ξ(z). (8.5.11)
Proof. Proof follows by using the product rule and applying induction on k .
An alternate approach to the p−curvature involves the differential operator
δ :=d
dz− [A(z)](p)
which acts on the vector space Lp[z]]d over the field of Laurent polynomials
Lp[z]].
Lemma 8.5.9. The following is true about the operator δ : Lp[zp]] −→ Lp[zp]],
• The map δ is a linear map over the subfield Lp[zp]].
• Since δ(zf(z)) = f(z) + zδ(f(z)), therefore δk(zf(z)) = kδk−1(f(z)) + zδk(f(z)).
• δp commutes with z and therefore this implies that δp is a Lp[z]]− linear map
on the space Lp[z]]d, i.e. δp is a differential operator of degree 0.
Proof. For complete proof see [18] .
174
8.5.3 Bounded vs. unbounded denominator
Let G be any triangle group of type (` ,m , n) and ρ : G −→ GLd(C) be any
rank d multiplier then we will call now on the pair (G , ρ) a system and in case
of ρ being admissible then (G , ρ) will be called as an admissible system and
similarly if ρ is an admissible irrep .
Definition 8.5.10 (p-integral). For any prime p, we call a rational number
x = mn
, p-integral if p doesn’t divide the denominator of x in its reduced form.
This implies that [x](p) is a finite integer.
Example 8.5.11. Let x = 268
then x is a p-integral for all prime p except 2.
For example [x](3) = 1, [x](7) = 5 etc. In this case the set BP = 2.
Similarly, we can define the notion of p-integrality for any formal rational
power series. Let f(z) =∑∞
n=0 fnzn ∈ Q[[z]] then we say f(z) is p-integral if
∀n, fn ∈ Q is p-integral. This implies that [f(z)](p) ∈ Z[[z]].
Definition 8.5.12 (p-bounded denominator property). Let p be any prime.
Let Ξ be the fundamental matrix of (G, ρ) then we know from our vvmf theory
that Ξ(z) is of the form 8.5.2. We say (G, ρ) satisfies the p-bounded denom-
inator property, if there exists some integer L such that the series pLΞ(z) is
p-integral. We abbreviate it by p-bd property .
Note 8.5.13. If (G, ρ) doesn’t satisfy p-bd property then we say that (G, ρ)
satisfies the p-unbounded denominator property (p-ubd). In other words, if
(G, ρ) satisfies p-ubd property then p is a ubd-prime of (G, ρ).
Definition 8.5.14 (Bounded Denominator Property). Let Ξ(z) be the
fundamental matrix of system (G, ρ). We say that the system (G, ρ) satisfies
the bounded denominator property if there exists some positive integer M such
that M(Ξn(z))ij are integer for all n. Here 1 ≤ i, j ≤ d.
175
Lemma 8.5.15. If the system (G, ρ) satisfies the bounded denominator prop-
erty then the system (G, ρ) has no ubd-primes .
Remark 8.5.16. Converse of the Lemma 8.5.15 is not true.
We state the following
Conjecture 8.5.17. Let (G, ρ) be an admissible system of rank d. If there
exists one vvmf X(z) of (G, ρ) with rational coefficients and whose components
are linearly independent such that it satisfies the bounded denominator property
then (G, ρ) has a fundamental matrix of rank d which satisfies the bounded
denominator property.
Conjecture 8.5.18.
(a) Suppose (G, ρ) satisfies the p−bd property for some fixed prime p. Then there
exists a fundamental matrix Ξ(z) of (G, ρ) satisfying the equation (8.5.1) is
p−integral and whose reduction modulo p matrix [Ξ(z)](p) is invertible over the
field Lp[z]].
(b) Suppose in addition the prime p in (a) does not lie in BP, and does not divide
any denominators of any entry in the exponent Λ. Then there are sufficiently
many solutions to equation (8.5.1). This is equivalent to say that p−curvature
vanishes. In case p−curvature does not vanish then p is a ubd-prime.
We state the following problems which we will be closely looking in near
future
Problem. Let (G1 , ρ1) and (G2 , ρ2) be two admissible system for two modular
triangle groups of type (` ,m ,∞) such that G2 = γG1γ−1 for some γ ∈ GL2(C)
then show that the following is true :
• ρ2 = γρ1γ−1 is a natural equivalence between ρ1 and ρ2 .
• ρ1 is Fourier-rational respectively Fourier-integral if and only if ρ2 is so .
176
• ρ1 admits an ubd-prime if and only if ρ2 admits as well . In other words ker(ρ1)
is congruence if and only if ker(ρ2) is so .
Problem. Let (G , ρ) be any admissible irrep system of modular triangle group
G . Let X(τ) be any weight w ∈ 2Z nonzero vvmf of (G , ρ) with rational
coefficients then show that if ρ has infinite image (i.e. ker(ρ) has infinite index
in G ) then X(τ) has infinitely many ubd-primes .
177
Chapter 9
Bilateral series andRamanujan’s Radial Limits
Ramanujan’s last letter to Hardy explored the asymptotic properties of mod-
ular forms, as well as those of certain interesting q-series which he called mock
theta functions. For his mock theta function f(q), he claimed that as q ap-
proaches an even order 2k root of unity ζ,
limq→ζ
(f(q)− (−1)k(1− q)(1− q3)(1− q5) · · · (1− 2q + 2q4 − · · · )
)= O(1),
and hinted at the existence of similar statements for his other mock theta
functions. Recent work of Folsom-Ono-Rhoades provides a closed formula for
the implied constant in this radial limit of f(q). Here, by different methods,
we prove similar results for all of Ramanujan’s 5th order mock theta functions.
Namely, we show that each 5th order mock theta function may be related to
a modular bilateral series, and exploit this connection to obtain our results.
We further explore other mock theta functions to which this method can be
applied.
9.1 Introduction
In his deathbed letter to Hardy in 1920, Ramanujan wrote down 17 curious q-
series which he dubbed mock theta functions. Due to work of Zwegers [55, 54],
178
Bringmann-Ono and others [53], we are now able to recognize Ramanujan’s
mock theta functions as holomorphic parts of weight 1/2 harmonic weak Maass
forms. Although this has been a catalyst for recent developments in numerous
areas of mathematics, here we will focus on Ramanujan’s original formulation.
Ramanujan analyzed q-hypergeometric series with asymptotics similar to
those of modular theta functions near roots of unity, but which were not
themselves modular. In his letter, he asked of such series:
“...The question is: - is the function taken the sum of two functions one of
which is an ordinary theta function and the other a (trivial) function which is
O(1) at all the points e2πim/n? ...I have constructed a number of examples in
which it is inconceivable to construct a ϑ-function to cut out the singularities
of the original function.”
Examples of this form are what he then referred to as mock theta functions.
Though Ramanujan did not prove his assertion, recent work by Griffin-Ono-
Rolen [21] has confirmed that no such theta functions exist for Ramanujan’s
examples. The only example Ramanujan offered details for is the function
f(q) := 1 +q
(1 + q)2+
q4
(1 + q)2(1 + q2)2+ · · ·
He claimed that as q approaches an even order 2k root of unity ζ radially
within the unit disk, we have that
limq→ζ
(f(q)− (−1)kb(q)
)= O(1), (9.1.1)
where b(q) := (1− q)(1− q3)(1− q5) · · · (1− 2q+ 2q4− · · · ) is a modular form.
Remark 9.1.1. Here and throughout this paper, we take q := e2πiτ for τ ∈ H.
Then, by modular form we will mean the function is modular, up to a rational
power of q, with respect to some character. The character for these modular
forms can be explicitly calculated using [41], for example.
Recently, Folsom, Ono, and Rhoades [13, 14] provided a closed formula for
the O(1) numbers in Ramanujan’s claim (9.1.1):
179
Theorem (Theorem 1.1 in [13, 14]). If ζ is a primitive even order 2k root of
unity, then, as q approaches ζ radially within the unit disk, we have that
limq→ζ
(f(q)− (−1)kb(q)
)= −4
k−1∑n=0
(1 + ζ)2(1 + ζ2)2 · · · (1 + ζn)2ζn+1.
To obtain their closed form, they utilized a bilateral series associated to
f(q) that is not a modular form, but does have similar asymptotics to b(q).
The goal of this paper is to show that a bilateral series naturally associated to a
given mock theta function can sometimes be used to not only provide a similar
closed formula, but can also be modular and play the role of b(q). When this
is the case, we obtain the closed formulas by using a different method of proof
than that employed in [13, 14].
We first consider Ramanujan’s 5th order mock theta functions:
f0(q) :=∑n≥0
qn2
(−q; q)n
ψ0(q) :=∑n≥0
q(n+1)(n+2)/2(−q; q)n
φ0(q) :=∑n≥0
qn2
(−q; q2)n
F0(q) :=∑n≥0
q2n2
(q; q2)n
χ0(q) :=∑n≥0
qn
(qn+1; q)n
f1(q) :=∑n≥0
qn(n+1)
(−q; q)n
ψ1(q) :=∑n≥0
qn(n+1)/2(−q; q)n
φ1(q) :=∑n≥0
q(n+1)2
(−q; q2)n
F1(q) :=∑n≥0
q2n(n+1)
(q; q2)n+1
χ1(q) :=∑n≥0
qn
(qn+1; q)n+1
where (a; q)n is the q-Pochhammer symbol defined as
(a; q)n =(a; q)∞
(aqn; q)∞,
where (a; q)∞ := (1−a)(1−aq)(1−aq2) · · · and n ∈ Z. From this formulation,
we see the well-known form (see [12], [19] for example)
(a; q)−n =(−a)−nqn(n+1)/2
(a−1q; q)n. (9.1.2)
180
For a mock theta function M(q) :=∑
n≥0 c(n; q), we define its associated
bilateral series by B(M ; q) :=∑
n∈Z c(n; q). For example, the bilateral series
B(f0; q) is given by
B(f0; q) :=∑n∈Z
qn2
(−q; q)n.
Surprisingly, the bilateral series associated to the 5th order mock theta
functions are in fact modular forms (with the exception ofB(χ0; q) andB(χ1; q)
which will be addressed in Section 9.4). Since a key component of Ramanu-
jan’s claim (9.1.1) is the fact that b(q) is a modular form, these bilateral series
beautifully lend themselves to similar radial limits. Moreover, these bilateral
series can be written as linear combinations of mock theta functions which
then reveal the following simple closed formulas similar to Theorem 1.1 in
[13].
Theorem 9.1.1. Let ζ be a primitive root of unity, k ∈ N, and suppose q → ζ
radially within the unit disk:
(a) If ζ has order 2k, then we have that
limq→ζ
(f0(q)−B(f0; q)
)= −2
k−1∑n=0
ζ(n+1)(n+2)/2(−ζ; ζ)n,
limq→ζ
(f1(q)−B(f1; q)
)= −2
k−1∑n=0
ζn(n+1)/2(−ζ; ζ)n,
where B(f0; q) and B(f1; q) are modular forms of weight 1/2 with level Γ1(20).
(b) If ζ has order 2k − 1, then we have that
limq→ζ
(F0(q)−B(F0; q)
)= 1−
k−1∑n=0
(−ζ)n2
(ζ; ζ2)n,
limq→ζ
(F1(q)−B(F1; q)
)= ζ−1
k−1∑n=0
(−ζ)(n+1)2
(ζ; ζ2)n,
where B(F0; q) and B(F1; q) are modular forms of weight 1/2 with level Γ1(10).
181
Theorem 9.1.2. Let ζ be a primitive root of unity, k ∈ N, and suppose q → ζ
radially within the unit disk:
(a) If ζ has order 2k − 1, then we have that
limq→ζ
(ψ0(q)−B(ψ0; q)
)= −1
2−
2k−1∑n=1
ζn2
(−ζ; ζ)n,
limq→ζ
(ψ1(q)−B(ψ1; q)
)= −1
2−
2k−1∑n=1
ζn(n+1)
(−ζ; ζ)n,
where B(ψ0; q) and B(ψ1; q) are modular forms of weight 1/2 with level Γ1(20).
(b) If ζ has order m, then we have that
limq→ζ
(φ0(q)−B(φ0; q)
)=
−2
2k−1∑n=1
ζ2n2
(−ζ; ζ2)nif m = 2k − 1,
−22k∑n=1
ζ2n2
(−ζ; ζ2)nif m = 4k,
limq→ζ
(φ1(q)−B(φ1; q)
)=
−2ζ
2k−1∑n=1
ζ2n(n−1)
(−ζ; ζ2)nif m = 2k − 1,
−2ζ2k∑n=1
ζ2n(n−1)
(−ζ; ζ2)nif m = 4k,
where B(φ0; q) and B(φ1; q) are modular forms of weight 1/2 with level Γ1(10).
Four remarks.
(a) These theorems cover all of the roots of unity where the mock theta functions
f0, f1, F0, F1, ψ0, ψ1, φ0, and φ1 have singularities.
(b) The hypotheses on the roots of unity in Theorem 9.1.2 ensure that the denom-
inators in the closed formulas do not cause singularities.
(c) The modular forms B(M ; q) are given explicitly in Section 9.2.3.
(d) These results are particularly elegant because only one modular form is needed
to cut out all of the singularities. This differs from Ramanujan’s claim (9.1.1)
where the modular form changes by a factor of (−1)k depending on which even
order 2k root of unity is being considered.
182
These theorems, coupled with the fact that the bilateral series are modular
forms, provide insight into why Ramanujan might have been so fascinated with
these functions. The above formulas embody the property that, although mock
theta functions are not themselves modular, they do have similar asymptotic
properties to modular forms. Additionally, the closed formulas for each of the
radial limits suggest further connections between mock theta functions and
quantum modular forms (see [13]).
In Section 9.2 we prove the modularity of the bilateral series in Theorems
9.1.1 and 9.1.2. In Section 9.3 we provide the proofs of these theorems. In
order to obtain results for χ0 and χ1, we use a different type of bilateral series,
which is covered in Section 9.4. Section 9.5 develops similar results for other
mock theta functions where this method using bilateral series can be applied.
Acknowledgement
The authors would like to thank Amanda Folsom, Ken Ono, and Robert
Rhoades for their advice and guidance. The authors would also like to thank
the organizers of the 2013 Arizona Winter School for providing the environ-
ment in which this project was developed.
9.2 Modularity of the bilateral series
As mentioned in the introduction, a key component in Theorems 9.1.1 and
9.1.2 is that the bilateral series are in fact modular forms. This section is
devoted to showing the modularity of these series, as is summarized in the
following lemma.
Lemma 9.2.1. The bilateral series B(M ; q) is a modular form of weight
1/2 with level Γ1(20) when M ∈ f0, f1, ψ0, ψ1 and level Γ1(10) when M ∈
φ0, φ1, F0, F1.
183
To prove this, we first rewrite our bilateral series as linear combinations of
mock theta functions. This then allows us to utilize the mock theta conjec-
tures, which relate these linear combinations to specific modular forms. Using
the work in Section 9.2.2, we are then able to determine weight and level.
9.2.1 Alternative forms of bilateral series
By using (9.1.2), we can express the bilateral series associated to each of the
5th order mock theta functions in Lemma 9.2.1 as follows:
B(f0; q) = f0(q) + 2ψ0(q) = 2B(ψ0; q), (9.2.1)
B(f1; q) = f1(q) + 2ψ1(q) = 2B(ψ1; q),
B(F0; q) = F0(q) + φ0(−q)− 1 = B(φ0;−q), (9.2.2)
B(F1; q) = F1(q)− q−1φ1(−q) = −q−1B(φ1;−q).
9.2.2 Reformulations of the Rogers-Ramanujan func-tions
In our proof of Lemma 9.2.1 we will make use of the Rogers-Ramanujan func-
tions,
G(q) :=∑n≥0
qn2
(q; q)n=
1
(q; q5)∞(q4; q5)∞and
H(q) :=∑n≥0
qn2+n
(q; q)n=
1
(q2; q5)∞(q3; q5)∞.
We provide a reformulation of G(q) and H(q) in terms of the Dedekind η
function η(τ) := q1/24(q; q)∞ and Klein forms as defined below. This will be
helpful in determining the weight and level of the bilateral series described in
the lemma. Though these functions have been studied in great detail, we were
184
unable to find these reformulations in the literature and therefore we include
them for completeness.
The Klein form t(N)(r,s) = t(r,s) for N ∈ N and (r, s) ∈ Z2 such that (r, s) 6≡
(0, 0) mod N ×N is a function on H defined as
t(r,s)(τ) := −ζs(r−N)
2N2
2πiqr(r−N)
2N2 (1− ζsNqrN )
∞∏n=1
(1− ζsNqn+ rN )(1− ζ−sN qn−
rN )
(1− qn)2,
(9.2.4)
where ζn := e2πi/n.
The Klein form has the following transformation law under γ = ( a bc d ) ∈
SL2(Z):
t(r,s)(γτ) = (cτ + d)−1t(r,s)γ(τ)
where (r, s)γ = (ra+ sc, rb+ sd). For more details on Klein forms see [29].
Lemma 9.2.2. Assuming the notation above,
G(q) = − ζ35
2πi
q160
η2(5τ)t(1,5)(5τ)and H(q) = − ζ
710
2πi
q−1160
η2(5τ)t(2,5)(5τ).
Proof. By taking r = 1, s = 5 and N = 5 in (9.2.4), we get that
t(1,5)(τ) = −ζ−25
2πiq−
225 (1− q
15 )∞∏n=1
(1− qn+ 15 )(1− qn− 1
5 )
(1− qn)2.
Then, letting τ 7→ 5τ , we see
t(1,5)(5τ) = −ζ35q
160
2πi
(q; q5)∞(q4; q5)∞η2(5τ)
.
This implies that
−ζ−35 (2πi)q−
160 t(1,5)(5τ)η2(5τ) = (q; q5)∞(q4; q5)∞. (9.2.5)
Now, by using the definition of G(q) and equation (9.2.5), we immediately
obtain the desired form for G(q). Similarly, by taking r = 2, s = 5, N = 5
and letting τ 7→ 5τ , we get the result for H(q).
Note that t(N)(r,N)(Nτ) is a modular form of weight −1 and level Γ1(N).
Therefore from Lemma 9.2.2, we immediately see that the Rogers-Ramanujan
functions G and H are modular forms of weight 0 and level Γ1(5).
185
9.2.3 Proof of Lemma 9.2.1
We start by proving the modularity of B(f0; q). For this, we use four of the
identities established by Watson [51] (reprinted as (2.13)R − (2.16)R in [2]),
C1(q) := f0(q) + 2F0(q2)− 2− ϑ4(0;−q)G(−q) = 0,
C2(q) := φ0(−q2) + ψ0(q)− ϑ4(0;−q)G(−q) = 0,
C3(q) := 2φ0(−q2)− f0(q)− ϑ4(0; q)G(q) = 0,
C4(q) := ψ0(q)− F0(q2) + 1− qK(q2)H(q4) = 0,
(9.2.6)
where
ϑ4(0; q) :=∑n∈Z
(−1)nqn2
=η(τ)2
η(2τ)and K(q) :=
∑n≥0
qn(n+1)/2 =η(2τ)2
q1/8η(τ).
Note that to obtain C1(q) and C2(q) in (9.2.6), we let q 7→ −q in (2.13)R and
(2.14)R, respectively, from [2]. In this form, it is not hard to see that
4C4(q)−2C2(q)+C3(q)+2C1(q) = f0(q)+2ψ0(q)−ϑ4(0; q)G(q)+4qK(q2)H(q4) = 0,
that is,
B(f0; q) = f0(q) + 2ψ0(q) = ϑ4(0; q)G(q) + 4qK(q2)H(q4).
We now make use of the well-known fact that, for N ∈ N, η(Nτ) is a weight
1/2 modular form on Γ0(N) ⊇ Γ1(N) (see, for example, [41]). Then, using
Lemma 9.2.2 and the transformation of the Klein forms, we see that B(f0; q) is
a weight 1/2 modular form on Γ1(20), as desired. Since B(f0; q) = 2B(ψ0; q),
we have shown that B(ψ0; q) is a modular form as well.
The modularity of the remaining bilateral series is obtained similarly: using
the mock theta conjectures in Section 2 of [2], also proved by Watson, we can
show that
B(f1; q) = 2B(ψ1; q) = −ϑ4(0; q)H(q) + 4K(q2)G(q4).
186
Further, using the mock theta conjectures in Section 3 of [2], proved by Hick-
erson in [23], we can show that
B(F0; q) = F0(q) + φ0(−q)− 1 =(q5; q5)∞G(q2)H(q)
H(q2)− qK(q5)H(q2),
B(F1; q) = F1(q)− q−1φ1(−q) = 3K(q5)G(q2)− H(q)2(q5; q5)∞G(q)
.
Then, since B(F0; q) = B(φ0;−q) and B(F1; q) = −q−1B(φ1;−q), we have
shown the modularity of each of these bilateral series.
9.3 Proof of theorems 9.1.1 and 9.1.2
9.3.1 Proof of Theorem 9.1.1
Proof. We start by considering f0. Let ζ be a primitive root of unity of order
2k. From (9.2.1), we have that
limq→ζ
(f0(q)−B(f0; q)
)= −2 lim
q→ζ
∑n≥0
q(n+1)(n+2)/2(−q; q)n
= −2 limr→1−
∑n≥0
(rζ)(n+1)(n+2)/2(−rζ; rζ)n
= −2k−1∑n=0
ζ(n+1)(n+2)/2(−ζ; ζ)n.
We can interchange the limit and the summation since, as shown in Section
6 of [51], the above infinite sum is absolutely convergent for all 0 < r ≤ 1.
Then, the last equality follows from the fact that, since ζ has order 2k,
(−ζ; ζ)n = (1 + ζ)(1 + ζ2) · · · (1 + ζk−1)(1 + ζk) · · · (1 + ζn) = 0
for all n ≥ k. The modularity of B(f0; q) is shown in Lemma 9.2.1. This
proves the result for f0. The remaining statements of the theorem are proved
in a similar fashion.
187
9.3.2 Proof of Theorem 9.1.2
Proof. We start by considering ψ0. Let ζ be a primitive root of unity of order
2k − 1. From (9.2.1), we have that
limq→ζ
(ψ0(q)−B(ψ0; q)
)= −1
2limq→ζ
(∞∑n=0
qn2
(−q; q)n
)= −1
2limr→1−
(∞∑n=0
(rζ)n2
(−rζ; rζ)n
).
(9.3.1)
We will write this limit as a finite sum. Due to the shape of the series, the
proof in this case is more subtle. Therefore, we provide more details.
For any complex number u = cos θ + i sin θ of modulus 1, let us consider
|1r
+ u| as a function of r. For 0 < r ≤ 1, the function |1r
+ u| has a minimum
at r = 1 sinced
dr
∣∣∣∣1r + u
∣∣∣∣ =−(1 + r cos θ)
r3
√1r2 + 2
rcos θ + 1
≤ 0.
Using this, we can construct a dominating series, which shows that the right-
most sum in (9.3.1) is absolutely convergent for all 0 < r ≤ 1. Therefore, we
can interchange the limit with the infinite sum, yielding
limq→ζ
(ψ0(q)−B(ψ0; q)
)= −1
2limr→1−
∞∑n=0
(rζ)n2
(−rζ; rζ)n
= −1
2
∞∑n=0
ζn2
(−ζ; ζ)n
= −1
2
(1 +
∞∑N=0
2k−1∑n=1
ζ(N(2k−1)+n)2
(1 + ζ) · · · (1 + ζN(2k−1)+n)
)
= −1
2
(1 +
∞∑N=0
1
2N
2k−1∑n=1
ζn2
(1 + ζ) · · · (1 + ζn)
)
= −1
2−
2k−1∑n=1
ζn2
(−ζ; ζ)n.
The modularity of B(ψ0; q) is shown in Lemma 9.2.1. The remaining state-
ments in the theorem are proved in a similar fashion.
188
9.4 Results for χ0 and χ1
To complete our study of Ramanujan’s 5th order mock theta functions, we
now consider χ0 and χ1. As previously mentioned, the bilateral series B(χ0; q)
and B(χ1; q) are not modular forms. In fact, B(χ0; q) and B(χ1; q) are not
even defined on any half-plane. However, we can modify our definition of the
bilateral series for these two mock theta functions to attain similar results.
To do so, we use the following alternative forms for χ0 and χ1 from the
mock theta conjectures:
χ0(q) = 2F0(q)− φ0(−q),
χ1(q) = 2F1(q) + q−1φ1(−q).
We then define the bilateral series associated to χ0 and χ1 as follows:
B(χ0; q) := 2B(F0; q)−B(φ0;−q) = B(F0; q),
B(χ1; q) := 2B(F1; q) + q−1B(φ1;−q) = B(F1; q),
where the second equality in each definition follows from (9.2.2). We can then
use Lemma 9.2.1 to immediately show that B(χ0; q) and B(χ1; q) are modular
forms of weight 1/2 with level Γ1(10). Therefore, we can use this definition
of the bilateral series for χ0 and χ1 to obtain analogous radial limits for these
mock theta functions.
Theorem 9.4.1. Let ζ be a primitive root of unity, k ∈ N, and suppose q → ζ
radially within the unit disk:
(a) If ζ has order 2k − 1, then we have that
limq→ζ
(χ0(q)− 2B(χ0; q)
)= −3
k−1∑n=0
(−ζ)n2
(ζ; ζ2)n + 2,
limq→ζ
(χ1(q)− 2B(χ1; q)
)= 3ζ−1
k−1∑n=0
(−ζ)(n+1)2
(ζ; ζ2)n.
189
(b) If ζ has order 2k, then we have that
limq→ζ
(χ0(q) + B(χ0; q)
)= 6
k∑n=1
ζ2n2
(ζ; ζ2)n+ 2,
limq→ζ
(χ1(q) + B(χ1; q)
)= 6
k∑n=1
ζ2n(n−1)
(ζ; ζ2)n.
Moreover, B(χ0; q) and B(χ1; q) are modular forms of weight 1/2 with level
Γ1(10).
Note that the denominators indicated in part (b) of Theorem 9.4.1 do not
vanish since ζ has order 2k.
Proof. To show part (a), we note that
χ0(q)− 2B(χ0; q) = −3φ0(−q) + 2 and χ1(q)− 2B(χ1; q) = 3φ1(−q).
Then the result follows directly from Theorem 9.1.1. To show part (b), we
note that
χ0(q) + B(χ0; q) = 3F0(−q) + 2 and χ1(q) + B(χ1; q) = 3F1(−q).
Then, using a similar argument to that in the proof of Theorem 9.1.2, we
obtain the desired result.
9.5 Mock theta functions of other orders
Having fully explored all of Ramanujan’s 5th order mock theta functions using
the method of bilateral series, we now turn our attention to applying this
method to mock theta functions of other orders. We discuss in this section
the cases where bilateral series can be exploited to give results akin to those
listed above.
We start by exploring 3rd order mock theta functions. Then we move on
to those of even order.
190
9.5.1 3rd order mock theta functions
The method of looking at the associated bilateral series can be applied to the
following three mock theta functions:
φ(q) :=∞∑n=0
qn2
(−q2; q2)nψ(q) :=
∞∑n=1
qn2
(q; q2)nν(q) :=
∞∑n=0
qn(n+1)
(−q; q2)n+1
.
We form the bilateral series as before. Using (9.1.2) and equation (6.1)
from [12], ∑n≥0
(aq; q)ntn =
1
1− t∑n≥0
(−at)nq(n2+n)/2
(tq; q)n, (9.5.1)
it is not hard to show that
B(φ; q) = φ(q) + 2ψ(q) = 2B(ψ; q) and B(ν; q) = ν(q) + ν(−q).
We first establish the modularity of these bilateral series.
Lemma 9.5.1. B(φ; q), B(ψ; q), and B(ν; q) are modular forms of weight 1/2
and level Γ0(4).
Proof. We make use of Ramanujan’s 1ψ1 formula,
1ψ1(α, β; q; z) :=∑n∈Z
(α; q)n(β; q)n
zn =(β/α; q)∞(αz; q)∞(q/αz; q)∞(q; q)∞(q/α; q)∞(β/αz; q)∞(β; q)∞(z; q)∞
for |β/α| < |z| < 1 (see, for example, [19]). Letting n 7→ −n in B(φ; q), we
see that
B(φ; q) =∑n∈Z
qn2(−1; q2)n
q−2nqn2+n=∑n∈Z
qn(−1; q2)n = 1ψ1(−1, 0; q2; q).
Similarly, we have
B(ν; q) = 1ψ1(−q, 0; q2; q).
Therefore, by Ramanujan’s 1ψ1 formula, we can realize these bilateral series
as products of q-Pochhammer symbols. Then, using the definition of η, one
can show
B(φ; q) =q1/24η(2τ)7
η(τ)3η(4τ)3and B(ν; q) =
2η(4τ)3
q1/3η(2τ)2.
191
Written in this form and given the fact that B(ψ; q) = 12B(φ; q), modularity
follows with the desired weight and level easily computed.
We now provide radial limits for these mock theta functions using their
associated bilateral series.
Theorem 9.5.2. Let ζ be a primitive root of unity, k ∈ N, and suppose q → ζ
radially within the unit disk:
(a) If ζ has order 4k, then we have that
limq→ζ
(φ(q)−B(φ; q)
)= −2ζ
k−1∑n=0
ζn(−ζ2; ζ2)n.
(b) If ζ has order 2k − 1, then we have that
limq→ζ
(ψ(q)−B(ψ; q)
)= −
k−1∑n=0
(−1)n(ζ; ζ2)n.
(c) If ζ has order 4k − 2, then we have that
limq→ζ
(ν(q)−B(ν; q)
)= −
k−1∑n=0
ζn(−ζ; ζ2)n.
Moreover, for M ∈ φ, ψ, ν, B(M ; q) is a modular form of weight 1/2 and
level Γ0(4).
Proof. Note that by using equation (9.5.1) we have the following relations:
2ψ(q) = 2q∑n≥0
qn(−q2; q2)n,
1
2φ(q) =
∑n≥0
(−1)n(q; q2)n,
ν(−q) =∑n≥0
qn(−q; q2)n.
Now with these equations, the proof of the theorem is similar to that of Theo-
rem 9.1.1. The modularity of the bilateral series follows from Lemma 9.5.1.
192
The mock theta function f(q) is treated in [13, 14]. In that case, the
bilateral series B(f ; q) is the product of a modular form and a mock modular
form. In the cases of the 3rd order mock theta functions ω, ρ, and χ, as well
as for the 7th order mock theta functions, it is not immediately clear how the
bilateral series could be utilized.
9.5.2 Even order mock theta functions
We now turn our attention to the associated bilateral series for even order
mock theta functions. Using similar methods to Section 9.2, we can use the
linear relations between functions of the same order to prove that the bilateral
series are modular forms for eight of the 6th order and six of the 8th order
mock theta functions. (See [20] for a full description of these linear relations.)
Through arguments similar to those in the proofs of Theorems 9.1.1 and 9.1.2,
we then are able to provide the associated closed forms.
For the sake of brevity, we summarize the relevant information for each
of these cases in the tables below. We list only the essential information in
order to recreate the corresponding statement associating the closed form to
the radial limits. For example, given the mock theta function M(q), we list
the associated bilateral series, the order of the roots of unity for which the
radial limit will be taken, and the closed formula for this radial limit. Given
this information we can form the following statement:
Theorem 9.5.3. Let ζ be an appropriate root of unity with k ∈ N, and suppose
q → ζ radially within the unit disk. We have that
limq→ζ
(M(q)−B(M ; q)
)= C(M ; ζ),
where M(q), C(M ; ζ), and ζ are as in Tables 5.1 and 5.2. Moreover, B(M ; q)
is a modular form of weight 1/2.
193
Remark. The level and character of B(M ; q) can be explicitly calculated using
methods as in the proofs of Theorems 9.1.1 and 9.1.2.
Note that Table 5.1 is specifically for mock theta functions of order 6,
despite the reuse of notation with mock theta functions of other orders. For
full definitions of these q-series, see Section 5 of [20]. Further, as noted in
the remarks after Theorems 9.1.2, the denominators in the closed formulas
C(M ; ζ) in the tables never vanish under the hypotheses on the given root of
unity ζ.
By writing the bilateral series as linear combinations of mock theta func-
tions, we can then use the linear relations in [20] to show that these series are
in fact modular. For example, for the 6th order mock theta function λ(q), we
have that B(λ; q) = λ(q) + 2ρ(q). Then, from relations (5.8) in [20], we see
that
B(λ; q) = (q; q2)2∞(q; q6)∞(q5; q6)∞(q6; q6)∞
+2(−q; q2)2∞(−q; q6)∞(−q5; q6)∞(q6; q6)∞ .
Then, using methods similar to those in Section 9.2 (or more generally in [41]
for example), we can easily recognize this sum of infinite products as a modular
form of weight 1/2. The proof of all remaining B(M ; q) in the tables is similar.
194
TABLE 5.1: Mock theta functions of order 6
M(q) B(M ; q) Order of ζ C(M ; ζ)
λ λ(q) + 2ρ(q) 2k −2k−1∑n=0
ζ12n(n+1)(−ζ; ζ)n(ζ; ζ2)n+1
µ µ(q) + 2σ(q) 2k −2k−1∑n=0
ζ12
(n+1)(n+2)(−ζ; ζ)n(ζ; ζ2)n+1
φ φ(q) + 2ν(q) 2k −2k−1∑n=0
ζn+1(−ζ; ζ)2n+1
(ζ; ζ2)n+1
ψ ψ(q) + 2ξ(q) 2k −2k−1∑n=0
ζn+1(−ζ; ζ)2n
(ζ; ζ2)n+1
ρ ρ(q) +1
2λ(q) 2k − 1 −1
2
k−1∑n=0
(−1)nζn(ζ; ζ2)n(−ζ; ζ)n
σ σ(q) +1
2µ(q) 2k − 1 −1
2
k−1∑n=0
(−1)n(ζ; ζ2)n(−ζ; ζ)n
ν ν(q) +1
2φ(q) 2k − 1 −1
2
k−1∑n=0
(−1)nζn2(ζ; ζ2)n
(−ζ; ζ)2n
ξ ξ(q) +1
2ψ(q) 2k − 1 −1
2
k−1∑n=0
(−1)nζ(n+1)2(ζ; ζ2)n
(−ζ; ζ)2n+1
195
TABLE 5.2: Mock theta functions of order 8
M(q) B(M ; q) Order of ζ C(M ; ζ)
S0 S0(q) + 2T0(q) 4k −2k−1∑n=0
ζ(n+1)(n+2)(−ζ2; ζ2)n(−ζ; ζ2)n+1
S1 S1(q) + 2T1(q) 4k −2k−1∑n=0
ζn(n+1)(−ζ2; ζ2)n(−ζ; ζ2)n+1
T0 T0(q) +1
2S0(q) 4k − 2 −1
2
k−1∑n=0
ζn2(−ζ; ζ2)n
(−ζ2; ζ2)n
T1 T1(q) +1
2S1(q) 4k − 2 −1
2
k−1∑n=0
ζn(n+2)(−ζ; ζ2)n(−ζ2; ζ2)n
V0 V0(q) + V0(−q) + 1 2k − 1 −2k−1∑n=0
−ζn2(ζ; ζ2)n
(−ζ; ζ2)n
V1 V1(q)− V1(−q) 2k − 1k−1∑n=0
−ζ(n+1)2(ζ; ζ2)n
(−ζ; ζ2)n+1
This appears to be an exhaustive list of the mock theta functions for which
this method of using bilateral series can be applied to obtain closed formulas for
the radial limits. For those mock theta functions listed in [20] but not included
here, the method seems to fail because the bilateral series B(M ; q) associated
to each of these mock theta functions do not appear to be modular forms. In
a small number of cases, we can apply the method of Section 9.4 to obtain a
modular form. For example, for the 8th order mock theta function U0, we can
define B(U0; q) using the equation U0 = S0(q2) + qS1(q2). However, despite
the modularity of these bilateral series, they do not immediately reveal closed
formulas for the radial limits. Therefore, the authors believe an alternate
method is needed for the mock theta functions not addressed here.
196
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