vector-valued automorphic forms and vector bundles › ... › 3 ›...
TRANSCRIPT
Vector-valued automorphic forms andvector bundles
by
Hicham Saber
Thesis submitted to the
Faculty of Graduate and Postdoctoral Studies
In partial fulfillment of the requirements
For the Ph.D. degree in
Mathematics
Department of Mathematics and Statistics
Faculty of Science
University of Ottawa
c© Hicham Saber, Ottawa, Canada, 2015
Abstract
In this thesis we prove the existence of vector-valued automorphic forms for
an arbitrary Fuchsian group and an arbitrary finite dimensional complex
representation of this group. For small enough values of the weight as well as
for large enough values, we provide explicit formulas for the spaces of these
vector-valued automorphic forms (holomorphic and cuspidal).
To achieve these results, we realize vector-valued automorphic forms as
global sections of a certain family of holomorphic vector bundles on a certain
Riemann surface associated to the Fuchsian group. The dimension formulas
are then provided by the Riemann-Roch theorem.
In the cases of 1 and 2-dimensional representations, we give some ap-
plications to the theories of generalized automorphic forms and equivariant
functions.
ii
Acknowledgements
I would like to thank everyone who has helped me in any way, particularly my
supervisor Dr. Abdellah Sebbar. I would also like to thank the Department
of Mathematics and Statistics for its financial support.
iii
Contents
Introduction 1
1 Preliminaries 8
1.1 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.3 The Riemann-Roch theorem. . . . . . . . . . . . . . . . 17
1.2 Fuchsian groups and their Riemann surfaces. . . . . . . . . . . 19
1.2.1 Classification of Mobius transformations . . . . . . . . 20
1.2.2 Fuchsian groups . . . . . . . . . . . . . . . . . . . . . . 22
1.2.3 The Riemann surface of a Fuchsian group. . . . . . . . 25
1.2.4 Congruence subgroups of PSL(2,Z) . . . . . . . . . . . 27
1.3 Automorphic forms. . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3.1 Definition of automorphic forms. . . . . . . . . . . . . 29
1.3.2 Examples of modular forms . . . . . . . . . . . . . . . 32
2 Vector-valued automorphic forms and vector bundles 34
2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 The family EΓ,R,k of vector bundles . . . . . . . . . . . . . . . 37
iv
2.2.1 The unbranched case. . . . . . . . . . . . . . . . . . . 37
2.2.2 The elliptic case. . . . . . . . . . . . . . . . . . . . . . 40
2.2.3 The cuspidal case. . . . . . . . . . . . . . . . . . . . . 43
2.2.4 Construction of the 1−cocycle. . . . . . . . . . . . . . 46
2.3 Behavior at the cusps . . . . . . . . . . . . . . . . . . . . . . . 50
2.4 The divisor of a vector-valued automorphic form . . . . . . . 51
2.5 Mk(Γ, R) and Sk(Γ, R) as global sections of vector bundles . . 57
3 The dimension Formula 69
3.1 The degree of EΓ,R,k . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 The holomorphic degree of a vector bundle. . . . . . . . . . . 75
3.3 The dimension of Mk(Γ, R) and Sk(Γ, R) . . . . . . . . . . . 81
3.4 Finite image representations . . . . . . . . . . . . . . . . . . . 87
4 Applications 90
4.1 Generalized automorphic forms . . . . . . . . . . . . . . . . 90
4.2 ρ−equivariant functions . . . . . . . . . . . . . . . . . . . . . 95
4.2.1 Differential equations . . . . . . . . . . . . . . . . . . . 96
4.2.2 The correspondence . . . . . . . . . . . . . . . . . . . . 99
4.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 103
v
Introduction
This thesis deals with the topic of vector-valued automorphic forms. These
are considered as a natural extension of classical automorphic forms and are
defined as follows. Let SL(2,R) be the group of real 2 × 2 matrices with
determinant one, and PSL(2,R) = SL(2,R)/±I. The group SL(2,R) acts
on the Poincare half-plane
H = z ∈ C : Im z > 0
by Mobius transformations, i.e.,
γz =az + b
cz + d, z ∈ H , γ =
(a b
c d
)∈ SL(2,R).
If Γ is a discrete subgroup of SL(2,R), and R is an n-dimensional complex
representation of Γ, that is a homomorphism
R : Γ −→ GL(n,C) ,
then an unrestricted vector-valued automorphic form for Γ of multiplier R
and integral weight k is a meromorphic map F : H −→ Cn satisfying
F (γ · z) = Jkγ (z)R(γ) F (z) , z ∈ H , γ ∈ Γ ,
where Jγ(z) = cz+d if γ =
(∗ ∗c d
). When F is meromorphic (resp. holomor-
phic) at the cusps of Γ, then we say that F is a vector-valued automorphic
form (resp. holomorphic vector-valued automorphic form) for Γ of multiplier
1
R and weight k. This definition can be extended to real weights k using the
notion of multiplier system, see [23]. In this thesis we content ourselves to
the case of integral weights.
The theory of vector-valued automorphic forms has been around for a long
time, first, as a generalization of the classical theory of scalar automorphic
forms, then as natural objects appearing in mathematics and physics. For
instance, Selberg suggested vector-valued forms as a tool to study modular
forms for finite index subgroups of the modular group [49] and they also
appear as Jacobi forms in the work of M. Eichler and D. Zagier [10], and in
[57]. In the last decade, there has been a growing interest in the study of
vector-valued forms, and several important results have been obtained by M.
Knopp, G. Mason and other authors [4, 17, 30, 31, 35, 36, 37].
In order to prove the existence of vector-valued automorphic forms or to
compute the dimensions of their vector spaces, people used a wide range of
tools coming from geometry, spectral theory, differential equations and clas-
sical complex analysis. For instance, J. Fischer [14], D. A. Hejhal [22, 23],
and W. Roelcke [47, 48] used the Selberg trace formula to compute the di-
mensions of the spaces of holomorphic vector-valued automorphic forms, but
they imposed the tight condition of the unitarity of the representation R, or
more generally the multiplier system. This condition was also imposed by
R. Borcherds [6, 7] for the metaplectic group of a general Fuchsian group of
the first kind. In the special case of the modular group SL(2,Z), G. Mason
[37] employed linear differential equations to construct vector-valued auto-
morphic forms of negative weight. In the paper [17], T. Gannon used the
Riemann-Hilbert problem to prove the existence of vector-valued automor-
phic forms for SL(2,Z), with a possible generalization to the case of Γ being a
Fuchsian group of genus zero as announced there. Classical methods such as
Poincare series have been used by M. Knopp, and G. Mason [30] to solve the
existence problem, but always restricted to the special case of Γ = SL(2,Z).
A different approach based on the Riemann-Roch theorem has been used by
C. Meneses [39], but only when the weight is two and the representation is
2
unitary. E. Freitag [15] also used the Riemann-Roch theorem in the case
when Γ has a finite index subgroup Γ0 such that the image of the elements
of Γ0 by the multiplier system are simultaneously diagonalizable. In other
words, up to a change of basis, the components of a vector-valued automor-
phic form for Γ are simply classical automorphic forms for Γ0. A sketch of a
proof has been given in the case of unitary multiplier system.
These different restrictions on the representation R or the group Γ have
the disadvantage of excluding many important cases which are highly relevant
to the theory of automorphic forms. For example, weight two vector-valued
automorphic forms for the symmetric power representations [34], which are
not unitary, play an essential role in the theory of periods of modular forms,
see Shimura’s work in [56]. Also, in [42] it is shown that the theory of quasi-
modular forms [38] can be covered by vector-valued modular forms with
respect to the symmetric tensor representations. In the case of 2-dimensional
representations, one misses the theory of equivariant functions developed
by A. Sebbar and others, see [11, 12, 51, 54]. Even in the 1-dimensional
case, these restrictions are strict enough to exclude the emerging theory of
generalized modular forms, see [28, 25, 26, 29, 33, 44, 45].
What we propose in this thesis is to prove the existence of vector-valued
automorphic forms in full generality, i.e., without any restriction on the group
Γ or the representation. Also, to show that given a discrete subgroup Γ of
SL(2,R), an n-dimensional complex representation R of Γ, and an integer
k ∈ Z, the dimensions dΓ,R,k, sΓ,R,k of the spaces Mk(Γ, R), Sk(Γ, R) of
holomorphic, cusp vector-valued automorphic forms for Γ of multiplier R
and weight k are infinite when Γ is Fuchsian group of the second kind, and
they are finite when Γ is Fuchsian group of the First kind. In this last case,
we provide two rational numbers k+Γ,R, and k−Γ,R depending on Γ and R such
that:
(A) For k < k−Γ , we have dΓ,R,k = 0, and sΓ,R,k = 0.
(B) For k > k+Γ , we have explicit formulas for dΓ,R,k and sΓ,R,k expressed in
3
terms of k, n, and some invariants of Γ and R.
Our method consists of associating to the triplet (Γ, R, k) a holomorphic
vector bundle EΓ,R,k on a certain, well known, Riemann surface XΓ. Then
we prove that the global sections of EΓ,R,k, when lifted to H, correspond to
vector-valued automorphic forms for Γ of multiplier R and weight k. In case
XΓ is noncompact, we show that dΓ,R,k and sΓ,R,k are both infinite. When XΓ
is compact, then we use the Riemann-Roch theorem to give explicit formulas
for dΓ,R,k and sΓ,R,k as explained in (A) and (B). The existence of vector-
valued automorphic forms will be proved by showing that dΓ,R,k and sΓ,R,k,
when they are finite, are asymptotically equivalent to k, that is
dΓ,R,k
k−→ nmΓ
2, as k →∞,
andsΓ,R,k
k−→ nmΓ
2, as k →∞,
where mΓ is some positive constant depending on Γ, see Remark 3.1.5.
It should be pointed that in a joint work with A. Sebbar [50], we proved
the existence of vector-valued automorphic forms in the case of subgroups
of PSL(2,R) rather than subgroups of SL(2,R). This was achieved as fol-
lows: we used the solution to the Riemann-Hilbert problem for non-compact
Riemann surfaces [13] to attach to each pair (Γ, R) of a discrete subgroup Γ
of PSL(2,R) and an n-dimensional complex representation R of Γ, a holo-
morphic vector bundle EΓ,R on XΓ in such a way that the global sections
of EΓ,R correspond to vector-valued automorphic forms for Γ of multiplier R
and weight 0. Then by the Kodaira vanishing theorem for compact Riemann
surfaces [19], in case XΓ is compact, or by the fact that XΓ is a Stein variety
[21] in case it is not compact, we proved the existence of n linearly indepen-
dent vector-valued automorphic forms for Γ of multiplier R and a certain
integral weight k.
The 2-dimensional case is of special interest since it has a connection
to ρ-equivariant functions. More precisely, given a discrete subgroup Γ of
4
SL(2,R) and a representation ρ : Γ −→ GL(2,C), a meromorphic function h
on H is called a ρ−equivariant function with respect to Γ if
h(γz) = ρ(γ)h(z) for all γ ∈ Γ , z ∈ H , (0.0.1)
where the action on both sides is by linear fractional transformations. When
ρ is the defining representation of Γ, that is ρ(γ) = γ for all γ in Γ, h is
simply an equivariant function, see (4.2.1).
ρ-equivariant functions and 2−dimensional vector-valued automorphic
forms are connected in the following way: if F = (f1, f2)t is an unrestricted
vector-valued automorphic form for Γ of multiplier ρ and weight k, then
one can check that if f2 is nonzero, then the function hF = f1/f2 is a ρ-
equivariant function for Γ. The main result of our joint work [52] with A.
Sebbar is that the converse is also true: for any ρ−equivariant function there
exists an unrestricted vector-valued automorphic form F = (f1, f2)t for Γ of
multiplier R and some weight k ∈ Z such that:
h = f1/f2.
In this thesis we enhance this result by proving that for any 2-dimensional
representation ρ of Γ, there exists a ρ−equivariant function of the form
hF = f1/f2,
where F = (f1, f2)t is a holomorphic vector-valued automorphic form for Γ
of multiplier R and a certain nonnegative weight k.
The 1-dimensional case corresponds to the theory of generalized auto-
morphic forms. Due to their similarity with classical automorphic forms and
their richness, the generalized automorphic forms are receiving an increasing
interest, specially after the work of M. Knopp and G. Mason in [28], see
[25, 26, 29, 33, 44, 45]. A meromorphic function f on H is called a general-
ized automorphic form of weight k for a discrete subgroup Γ of SL(2,Z) if
for all γ ∈ Γ and z ∈ H, we have
f(γ · z) = µ(γ) Jkγ (z) f(z)
5
where µ : Γ −→ C is a character. In addition, f should be meromorphic
at the cusps. The fundamental difference between generalized automorphic
forms and classical automorphic forms is that the character µ need not be
unitary. The existence theorem of vector-valued automorphic forms in di-
mension 1 guarantees the existence of generalized automorphic forms for an
arbitrary character of an arbitrary discrete group Γ. Our main contribution
to this theory will be the computation of the dimensions of the spaces of
holomorphic and cusp generalized automorphic forms in the sense of (A) and
(B).
This thesis is organized as follows: in chapter one we shall review some
classical results from the theories of holomorphic vector bundles on Riemann
surfaces, Fuchsian groups, and automorphic forms.
In chapter two we realize vector-valued automorphic forms as global sec-
tions of holomorphic vector bundles. More precisely, given a Fuchsian sub-
group Γ of SL(2,R), a finite-dimensional representation R of Γ, and an in-
teger k such that the pair (R, k) is simple, see Definition 2.1.2, we construct
a holomorphic vector bundle EΓ,R,k over the Riemann surface XΓ = Γ\H∗
associated to Γ. The spaces Mk(Γ, R), Sk(Γ, R) of holomorphic, cusp vector-
valued automorphic forms for Γ of multiplier R and weight k will be respec-
tively isomorphic to H0(XΓ,O(EΓ,R,−k)) and H0(XΓ,O(−DS,R,−k+EΓ,R,−k)),
where DS,R,−k is a certain holomorphic line bundle depending on the cusp-
idal points of XΓ. As a result, we deduce that the dimensions dΓ,R,k, sΓ,R,k
of Mk(Γ, R), Sk(Γ, R) are finite if and only if Γ is Fuchsian group of the first
kind, see Theorem 2.5.9.
In chapter three we introduce the notion of the holomorphic degree of a
holomorphic vector bundle, which consists of associating to each holomorphic
vector bundle E , defined on a compact Riemann surface, an integer d(E) in
a such a way that:
d(E) < 0 =⇒ H0(X,O(E)) is trivial.
We will give some basic properties of the holomorphic degree and some useful
6
bounds of d(E). Applying the Riemann-Roch theorem to the holomorphic
vector bundles EΓ,R,−k and −DS,R,−k+EΓ,R,−k, we prove the results mentioned
in (A) and (B).
The last chapter provides applications of our results in the special cases
of 1 and 2-dimensional representations. In particular, we compute the di-
mensions of the spaces of holomorphic and cusp generalized automorphic
forms. Also, we show the existence of ρ−equivariant functions, and prove
their parametrization by 2-dimensional unrestricted vector-valued automor-
phic forms of multiplier ρ. This parametrization relies on the existence of
global solutions to a certain second degree differential equation, as well as on
the fact that the Schwarz derivative of a ρ−equivariant function is a weight 4
unrestricted automorphic form for Γ. We end the chapter by constructing ex-
amples of ρ−equivariant functions when ρ is the monodromy representation
of second degree ordinary differential equations.
7
Chapter 1
Preliminaries
The aim of this chapter is to give a review of some basic facts from the theories
of holomorphic vector bundles on Riemann surfaces, Fuchsian groups, and
automorphic forms.
1.1 Vector Bundles
The content of this section is entirely based on the references:
[2]: Chapter 1: §1.5, §1.8.
[13]: Chapter 1: §29, §30, and Chapter 2: §16.
[19] Chapter 0: §0.5, Chapter 1: §1.1, and Chapter 1: §2.1 .
[20] Chapter C.
[?] Chapter 6: §6.5.
X will be a Riemann surface, that is, a one-dimensional complex manifold,
and n a positive integer.
8
1.1.1 Basic properties
Definition 1.1.1. Let X be a Riemann surface, E be a complex manifold,
and p : E −→ X be a holomorphic map. Suppose that each fiber Ex :=
p−1(x) has the structure of an n−dimensional vector space over C. Then
p : E −→ X, or simply E , is called a holomorphic vector bundle of rank n on
X if every point x ∈ X has an open neighborhood U such that :
1. There exists a biholomorphic map
φU : EU := p−1(U) −→ U × Cn
taking Ex to x × Cn for every x ∈ U .
2. For every x ∈ U , the map φU is a linear isomorphism from Ex to
x × Cn ∼= Cn.
The map φU : EU −→ U ×Cn is called a trivialization of E over U . A vector
bundle of rank 1 is called a line bundle.
Definition 1.1.2. Let E and F be two holomorphic vector bundles on X
of rank n and m respectively. A holomorphic map Ψ : E −→ F is called a
homomorphism (resp. isomorphism) if for all x ∈ X
1. Ψ(Ex) ⊆ Fx
2. Ψx := Ψ|Ex : Ex −→ Fx is a linear homomorphism (resp. isomorphism).
E and F are called isomorphic if there exists an isomorphism between them.
In particular, E is called a trivial holomorphic vector bundles on X if it is
isomorphic to the trivial holomorphic vector bundel X × Cn.
Note that for any pair of trivializations φU and φV we have a holomorphic
map
gUV : U ∩ V −→ GL(n,C),
9
such that
φUV := φU φ−1V : (U ∩ V )× Cn −→ (U ∩ V )× Cn
is given by
φUV (x, t) = (x, gUV (x)t) for every x ∈ (U ∩ V )× Cn.
The map gUV is called a transition function for E relative to the trivializa-
tions φU and φV . Also, one has the relation
gUV gVW = gUW on U ∩ V ∩W. (1.1.1)
Definition 1.1.3. Let U be an open subset of X. Let O(U) (resp. M(U))
denotes the group of holomorphic (resp. meromorphic) functions on U , and
GL(n,O(U)) (resp. GL(n,M(U)) ) the group of all n × n invertible ma-
trices with coefficients in O(U) (resp. M(U) ). Together with the natural
restrictions when V ⊆ U , they define sheaves of groups O, M, GL(n,O),
and GL(n,M) on X ( GL(n,O) and GL(n,M) are not abelian for n ≥ 2).
If U = (Ui)I is an open cover of X, we define Z1(U ,GL(n,O)) to be
the set of all 1−cocycles with values in GL(n,O) with respect to U , i.e., all
families (gUV )U with
gUV ∈ GL(n,O(U ∩ V ))
and verifying (1.1.1).
Remark 1.1.1. Note that for n ≥ 2 the set Z1(U ,GL(n,O)) is not a group
with respect to component-wise multiplication.
As seen above, any holomorphic vector bundle gives rise to one a 1−cocycle
with respect to an open cover U of X by means of the transition functions.
Conversely, we have:
Theorem 1.1.2. Let U = (Ui)I be an open cover of X, and (gUV )U be an
element of Z1(U ,GL(n,O)). Then up to isomorphism, there exits a unique
holomorphic vector bundle E on X of rank n having transition functions
(gUV )U .
10
Also, we have
Theorem 1.1.3. Let U = (Ui)I be an open cover of X, and (gUV )U , (hUV )U
be elements of Z1(U ,GL(n,O)). Suppose that there exist elements
fU ∈ GL(n,O(U)), U ∈ U , such that
hUV = fU gUV f−1V on U ∩ V ,
then (gUV )U and (hUV )U represent the same holomorphic vector bundle.
In principale, all operations on vector spaces induce operations on vector
bundles. For example, if E and F are holomorphic vector bundles over X of
rank n and m having transition functions (gUV )U and (hUV )U one can define:
1. The dual bundle E∗, given by the transition functions
g∗UV (x) = (g−1UV )T (x).
2. The direct sum E ⊕ F , given by the transition functions
gUV (x)⊕ hUV (x) ∈ GL(Cn ⊕ Cm).
3. The tensor product E ⊗ F , given by the transition functions
gUV (x)⊗ hUV (x) ∈ GL(Cn ⊗ Cm).
4. The exterior product∧k E , given by the transition functions
k∧gUV (x) ∈ GL(∧kCn).
In particular,∧n E is a line bundle given by the transition functions
det(gUV )(x) ∈ GL(1,C) = C∗,
also called the determinant bundle of E .
11
We now come to the notion of a section over a holomorphic vector bundle.
Definition 1.1.4. Let p : E −→ X be a holomorphic vector bundle of rank
n, and U be an open subset of X. A meromorphic section σ of E over U is a
meromorphic map σ : U −→ E such that p σ = idU , i.e., σ(x) ∈ Ex for all
x ∈ U\Pσ, where Pσ is the set of poles of σ in U . If U = X, then σ is called
a global meromorphic section of E .
If σ has no poles, it is called a holomorphic section.
If φU is a trivialization of E over U , then we can associate to each holo-
morphic section σ of E over U a unique meromorphic map fU : U −→ Cn
such that
φU(σ(x)) = (x, fU(x)), for all x ∈ U\Pσ.
If φV is a trivialization of E over V , then we have
fU = gUV fV on (U ∩ V )\Pσ,
where gUV is the transition function of E relative to φU and φV . Thus we
have
Proposition 1.1.4. In terms of trivializations EU : U −→ U × Cn, a mero-
morphic section σ of E over⋃U∈U U , corresponds to a collection (fU)U of
meromorphic maps fU : U −→ Cn such that
fU = gUV fV on (U ∩ V )\Pσ,
where the gUV , U, V ∈ U , are transition functions of E relative to (φU)U .
1.1.2 Divisors
In this subsection, we define the notion of a divisor of a section of a holo-
morphic vector bundle and give its basic properties.
12
Let U be an open subset of C, and F : U −→ Cn be a meromorphic map.
Then for any w in H, there exists an open disc Dw centered at w, and a
unique integer nW such that
F (z) = (z − w)nwGw(z), z ∈ Dw,
where Gw is a holomorphic map on Dw satisfying Gw(w) 6= 0. The integer
nw is called the order of F at w and is denoted by ordw(F ).
Remark 1.1.5. If Ψ: Dw −→ GL(n,C) is a holomorphic map, Then
ordw(F ) = ordw(ΨF ).
Suppose that U is an open cover of X. Let E be a holomorphic vector
bundle over X having transition functions (gUV )U , σ be a meromorphic sec-
tion of E over X, and (fU)U be its corresponding collection of meromorphic
maps. Let P ∈ U and take an open neighborhood UP of P in U such that
under a coordinate chart ϕ for X, we have ϕ(UP ) = D and ϕ(P ) = 0, where
D is the unit disc. The map FU,P = fU ϕ−1 is then meromorphic on D, and
we define the order νP (σ) of σ at P by
νP (σ) = ord0(FU,P ). (1.1.2)
According to [58], the integer νP (σ) is invariant under coordinate changes.
Hence, it is enough to show that νP (σ) is independent of U . Indeed, if P lies
in U ∩ V , then we have
fU = gUV fV on (U ∩ V )\Pσ.
Since gUV ∈ GL(n,O(U ∩ V )), by the above remark, we have
ordP (fU) = ordP (fV ).
The divisor of σ is defined by the formal sum
div(σ) =∑P∈X
νP (σ)P. (1.1.3)
13
Similarly, the divisor of a meromorphic function f on X is given by
div(f) =∑P∈X
νP (f)P, (1.1.4)
where νP (f) is the order of f at P . More generally
Definition 1.1.5. A divisor D on X is a locally finite formal sum
D =∑P∈X
dP P, dP ∈ Z,
where locally finite means that for any point P in X, there exists a neigh-
borhood U of P such that dP 6= 0 for only finitely many P in U . If U is an
open subset of X, we set
D|U =∑P∈U
dP P.
Div(X) will be the abelian group of all divisors on X. For D,D′ ∈Div(X),
set D ≥ D′ if dP ≥ d′P for all P ∈ X. A divisor D is called effective if
D ≥ 0.
The support Supp(D) of D will be the closure in X of the set
P ∈ X | dP 6= 0.
Remark 1.1.6. When X is compact, a locally finite sum is simply a finite
sum. Hence Supp(D) is a finite set.
Now, we focus on the relationship between divisors and line bundles. Let
D be a divisor on X. By the local finiteness assumption on D, one can find
an open cover U of X and functions fU ∈M(U), U ∈ U , such that
div(fU) = D on U.
Since fU and fV have the same divisor D|U∩V on U ∩ V , we see that
gUV := fU/fV ∈ O∗(U ∩ V ) = GL(1,O(U ∩ V )).
14
It is clear that (gUV )U is an element of Z1(U ,GL(n,O)). Hence it defines a
line bundleD onX, and by construction, σD = (fU)U is a global meromorphic
section of D such that
div(σD) = D.
If the set of line bundles constructed in the above way is denoted by
AD, then it can be shown that any two line bundles in AD are naturally
isomorphic. We use this fact to identify all elements of AD to one line bundle
that will be called the associated line bundle to the divisor D, and will be
denoted by |D|.
Suppose that L is a holomorphic line bundle given by transition functions
(gUV )U . If L has a nonzero global meromorphic section σ = (fU)U , then by
Proposition 1.1.4 we have
gUV = fU/fV ,
that is L = |div(σ)|. From this we deduce that a line bundle is associated
to a divisor (resp. an effective divisor) if and only if it has a nonzero global
meromorphic (resp. holomorphic) section.
Theorem 1.1.7. Any holomorphic line bundle L on X has a nontrivial
global meromorphic section. In other words it is of the form L = |D| for
some divisor D on X.
Recall that when X is compact, the support of any divisor on X is a finite
set.
Definition 1.1.6. The degree of a divisor D =∑
P∈X dP P, on a compact
Riemann surface X is defined by
deg(D) =∑P∈X
dP .
We have the following
Proposition 1.1.8. Suppose that X is compact. If f is a meromorphic
function on X, then
deg(div(f)) = 0.
15
The degree of a holomorphic line bundle over a compact Riemann surface
X is defined as follows. Let U be an open cover of X, L a holomorphic line
bundle over X having transition functions (gUV )U , σ a meromorphic section
of E over X, and (fU)U its corresponding collection of meromorphic maps.
Such σ always exists according to Theorem 1.1.7. If σ1 = (hU)U is an other
meromorphic section of L, then from the relations
fU = gUV fV , hU = gUV hV on U ∩ V
we see that
hU/fU = hV /fV , on U ∩ V.
Hence f := σ1/σ is a global meromorphic function on X, and we have
div(σ1) = div(σ) + div(f).
Therefore
deg(div(σ1)) = deg(div(σ)) + deg(div(f)) = deg(div(σ)),
since deg(div(f)) = 0 according to Theorem 1.1.8. Thus, the divisors of
global meromorphic sections have the same degree.
Definition 1.1.7. Let X be a compact Riemann surface, and let E be a
holomorphic vector bundle over X, and L be a holomorphic line bundle over
X.
1. The degree of L is defined by
deg(L) = deg(div(σ)),
where σ is any global meromorphic section of L.
2. The degree of E is defined by
deg(E) = deg(det(E)),
where det(E) is the determinant bundle of E .
16
The degree has the following useful property.
Proposition 1.1.9. Let X be a compact Riemann surface, and let E and Fbe a holomorphic vector bundles over X, and L be a holomorphic line bundle
over X. We have
deg(E ⊕ F) = deg(E) + deg(F),
and
deg(L ⊗ E) = n deg(L) + deg(E),
where n is the rank of E.
1.1.3 The Riemann-Roch theorem.
The goal of this subsection is to state the Riemann-Roch theorem for holo-
morphic vector bundles over compact Riemann surfaces.
Let E be a holomorphic vector bundle over X of rank n, and D a divisor
on X, σ a meromorphic section of E over X, and (fU)U its corresponding
collection of meromorphic maps. For any open subset U of X set O(E)(U)
(resp. M(E)(U) ) to be the O(U)−module of holomorphic (resp. mero-
morphic) sections of E over U . Together with the natural restrictions when
V ⊆ U , they define sheaves O(E) and M(E) of O−modules on X. The
vector space H0(X,O(E)) (resp. H0(X,M(E))) of global sections of O(E))
(resp. M(E))) is exactly the vector space of global holomorphic (resp. mero-
morphic) sections on X.
Remark 1.1.10. The dimension of the C−vector space H0(X,O(E)) will be
denoted by h0(E) when it is finite.
Example 1.1.11. If (U, zU), U ∈ U , is a covering of X by coordinate neigh-
borhoods, then on U ∩ V the function
gUV = dzU/dzV
17
lies in GL(1,O(U ∩ V )) = O∗(U ∩ V ), and it is clear that
(gUV )U ∈ Z1(U ,GL(1,O)).
The attached holomorphic line bundle is called the canonical line bundle of
X, and will be denoted by KX .
The sheaf O(KX) is isomorphic to the sheaf ΩX of holomorphic 1−forms
on X, the latter is defined by taking Ω(U) to be the O(U)−module of holo-
morphic 1−forms on U together with the natural restrictions. Thus the
sections of KX over U can be identified with holomorphic 1−forms on U . We
will adopt this identification for the rest of this thesis.
Similarly, if for an open subset U of X we set
OD(E)(U) = σ ∈M(E)(U) | div(σ) ≥ −D|U,
then OD(E)(U) is an O(U)−module, and together with the natural restric-
tions, we get a sheaf OD(E) of O−modules on X. By definition, we have
H0(X,OD(E)) = σ ∈ H0(X,M(E)) | div(σ) ≥ −D.
Moreover, if |D| is the associated line bundle to the divisor D, and σ is a
section over |D| such that div(σ) = D, then the multiplication by σ gives an
isomorphism of sheaves
OD(E) −→ O(|D| ⊗ E) : f 7→ σf.
This induces the isomorphism
H0(X,OD(E)) −→ H0(X,O(|D| ⊗ E)) : f 7→ σf, (1.1.5)
which we use to identify H0(X,OD(E)) with H0(X,O(|D| ⊗ E)). We have
the following fundamental theorem:
Theorem 1.1.12. Suppose that X is compact, and let E be a holomorphic
vector bundle over X of rank n. Then H0(X,O(E)) is a finite dimensional
C−vector space.
18
Definition 1.1.8. The genus g of a compact Riemann surface X is defined
to be the dimension of the space of holomorphic 1−forms on X, that is
g = h0(KX).
Finally, the Riemann-Roch theorem for holomorphic vector bundles on a
compact Riemann surfaces reads as follows:
Theorem 1.1.13. Suppose that X is compact and having genus g. If E is a
holomorphic vector bundle over X of rank n, then
h0(E)− h0(KX ⊗ E∗) = deg(E)− n(g − 1).
Remark 1.1.14. To ease the notations, in the sequel, the tensor product
L ⊗ E of a vector bundle E and a line bundle L over X will be denoted by
L + E .
1.2 Fuchsian groups and their Riemann sur-
faces.
All the content of this section is taken from the references:
[5]: Chapter 4.
[24]: Chapter 2.
[46] Chapter 1: §1.4 .
[55] Chapter 1.
We give the classification of Mobius transformations, and some standard
results on Fuchsian groups and their Riemann surfaces.
19
1.2.1 Classification of Mobius transformations
Let C = C ∪ ∞ and a, b, c, d be complex numbers with ad − bc 6= 0, then
the map g : C→ C defined by
g(z) =az + b
cz + d
is called a Mobius transformation. The set M of all Mobius transformations
equipped with the composition of maps is a group. Moreover, the map given
by A 7→ gA, where
gA(z) =az + b
cz + d, A =
(a b
c d
),
is a group homomorphism φ : GL(2,C)→M, and
kerφ =
(a 0
0 a
), a 6= 0
.
In general, we shall be more concerned with the restriction of φ to SL(2,C).
The kernel of this restriction is
kerφ ∩ SL(2,C) = −I, I.
Hence M is isomorphic to PSL(2,C) = SL(2,C)/−I, I.
Let A =
(a b
c d
)∈ GL(2,C) and tr(A) = a+ d, then the function
tr2(A)
det(A)
is invariant under the transformation A → λA, λ 6= 0, and so it induces a
function on M, namely
Tr(g) =tr2(A)
det(A),
where A is any matrix which projects on g. Notice that Tr(g) is invariant
under conjugation.
20
We now proceed to classify Mobius transformations. First, we introduce
some normalized Mobius transformations. For each k 6= 0 in C, we define
mk by
mk(z) = kz if k 6= 1,
and
m1(z) = z + 1.
Notice that for all k 6= 0, we have
Tr(mk) = k +1
k+ 2.
If g 6= I is any Mobius transformation, we denote by α ∈ C its fixed point
if it has a unique one, and by α and β in C, α 6= β, if it has two. Now let h
be any Mobius transformation such that
h(α) =∞, h(β) = 0, h(g(β)) = 1 if g(β) 6= β,
then
hgh−1(∞) =∞, hgh−1(0) =
0 if g(β) = β
1 if g(β) 6= β .
If g fixes α and β, then hgh−1 fixes 0 and ∞ and thus hgh−1 = mk for some
k 6= 1. If g fixes α only, then hgh−1 fixes ∞ only and hgh−1(0) = 1 and thus
hgh−1 = m1. Therefore, any nonidentity Mobius transformation is conjugate
to one of the standard form mk.
Definition 1.2.1. Let g 6= I be any Mobius transformation. We say that
1. g is parabolic if and only if g is conjugate to m1(equivalently g has a
unique fixed point in C);
2. g is loxodromic if and only if g is conjugate to mk for some k satisfying
|k| 6= 1 (g has exactly two fixed points in C);
3. g is elliptic if and only if g is conjugate to mk for some k satisfying
|k| = 1 (g has exactly two fixed points in C).
21
It is convenient to subdivide the loxodromic class by reference to invariant
discs rather than fixed points.
Definition 1.2.2. Let g be a loxodromic transformation. We say that g is
hyperbolic if g(D) = D for some open disc or half-plane D in C. Otherwise
g is said to be strictly loxodromic.
Now, using the fact that Tr(g) is invariant under conjugation, we can give
a complete classification of all Mobius transformations. Indeed, we have the
following results.
Theorem 1.2.1. Let f and g be two Mobius transformations, neither of
which is the identity. Then Tr(f) = Tr(g) if and only if f and g are conju-
gate.
Theorem 1.2.2. Let g 6= I be any Mobius transformation. Then
1. g is parabolic if and only if Tr(g) = 4;
2. g is elliptic if and only if Tr(g) ∈ [0, 4);
3. g is hyperbolic if and only if Tr(g) ∈ (4,+∞);
4. g is strictly loxodromic if and only if Tr(g) 6∈ [0,+∞).
1.2.2 Fuchsian groups
Let H = z ∈ C, Im(z) > 0 be the Poincare upper half-plane, SL(2,R) be
the group of real 2 × 2 matrices with determinant one, R := R ∪ ∞, and
H := H ∪ R. If A =
(a b
c d
)∈ SL(2,R) and gA(z) =
az + b
cz + dis the associated
Mobius transformation, then gA maps H into H. Indeed, for z in H
Im(gA(z)) =Im(z)
|cz + d|2. (1.2.1)
Also, it is clear that gA maps R into R.
22
Let MR denote the group of real Mobius transformations. As we have
seen, the map φ : SL(2,R) → MR given by A −→ gA is a surjective group
homomorphism, and kerφ = ±1 (1 here stands for I2, this will be used for
the rest of the thesis). Thus MR can be identified with
PSL(2,R) = SL(2,R)/±1.
Recall that an automorphisms of H is a biholomorphic bijection from Hto H.
Theorem 1.2.3. The group of automorphisms of H is MR = PSL(2,R).
The group PSL(2,C) inherits the topology of SL(2,C), which is a topo-
logical space with respect to the standard norm of C4. A subgroup Γ of
PSL(2,C) is called discrete if the subspace topology on Γ is the discrete
topology. Thus Γ is discrete if and only if: For any sequence Tn in Γ,
Tn → T ∈ GL(2,C) implies Tn = T for all sufficiently large n. Giving
PSL(2,R) ⊆ PSL(2,C) the subspace topology of PSL(2,C), we have
Definition 1.2.3. A discrete subgroup of PSL(2,R) is called a Fuchsian
group.
Example 1.2.4. The modular group PSL(2,Z) = SL(2,Z)/±1 and its
subgroups are Fuchsian groups.
For γ in PSL(2,R), let Fix(γ) be the set of fixed points of γ in H. By
Theorem 1.2.2, γ cannot be strictly loxodromic, and one can show that
1. If γ is elliptic, then Fix(γ) = zγ, for some zγ in H.
2. If γ is parabolic, then Fix(γ) = sγ, for some sγ in R.
3. If γ is hyperbolic, then Fix(γ) = aγ, bγ, for some aγ, bγ in R.
Definition 1.2.4. Let Γ be a Fuchsian subgroup of PSL(2,R), and z be a
point of H. Then:
23
1. The point z is called cuspidal or a cusp (resp. elliptic, hyperbolic) if it
is the fixed point of a parabolic (resp. elliptic, hyperbolic) element of
Γ.
2. The set of all elliptic (resp. cuspidal) points of Γ will be denoted by
EΓ (resp. CΓ).
3. The stabilizer Γz of z in Γ is the subgroup defined by
Γz = γ ∈ Γ | γ.z = z.
We have
Proposition 1.2.5. Let Γ be a Fuchsian subgroup of PSL(2,R), and z be a
point of H. Then
1. For any γ ∈ PSL(2,R), we have
(γΓγ−1)γ.z = γ(Γz)γ−1.
In particular, if γ ∈ Γ, then Γγ.z = γ(Γz)γ−1.
2. Γz is cyclic.
3. If z ∈ H, then Γz is finite and Γz = ±1 if z is not elliptic.
Definition 1.2.5. Let z be a point of H. The cardinality of Γz will be called
the order of z, and will be denoted by nz. In particular, nz = 1 if z is not
elliptic.
Remark 1.2.6. Since the conjugate of an elliptic (resp. parabolic) element
of PSL(2,C) is also elliptic (resp. parabolic), we see that Γ acts on EΓ and
CΓ.
24
1.2.3 The Riemann surface of a Fuchsian group.
In this subsection, Γ will be a Fuchsian subgroup of PSL(2,R), and
H∗ := H ∪ CΓ.
Since Γ acts on CΓ, it then acts on H∗, and so we can form the quotient
X = XΓ := Γ\H∗. Our aim here is to give X a structure of a Riemann
surface.
First we define a topology on H∗ as follows. H is given its usual topology.
For a cusp s 6= ∞, we take as a fundamental system of neighborhoods all
sets of the form:
s ∪ the interior of a circle in H tangent to the real axis at s
If∞ is cusp, then a fundamental system of neighborhoods of∞ are the sets:
∞ ∪ z ∈ H | =(z) > c,
for all positive numbers c. Endowed with this topology, H∗ becomes a Haus-
dorff space, and Γ acts on H∗ by homeomorphisms.
We give X a structure of a Riemann surface in the following way. Let φ
denotes the natural projection map of H∗ onto X = Γ\H∗, p be an arbitrary
point of H∗, and P = φ(p). Then
Lemma 1.2.7. There exists an open neighborhood U of p in H∗ such that
Γp = γ ∈ Γ | (γ.U) ∩ U 6= ∅.
Hence we have a natural injection of Γp\U −→ X, and Γp\U is an open
neighborhood of P in X. If P is neither an elliptic point nor a cusp, then ΓP
contains only the identity, so that the map U −→ Γp\U is a homeomorphism.
We take the pair (Γp\U, φ−1) as a member of the local charts defining the
complex structure on X.
25
In case p is elliptic, setting
α(z) =z − pz − p
, z ∈ H,
then α(H) is the unit disc D, α(p) = 0, and αΓpα−1 is the group
〈σ′p : w 7−→ ζpw〉,
where ζp is a primitive np−th root of unity, np is the order of p. Then we
define a map ψ : Γp\U −→ C by
ψ(φ(z)) = α(z)np .
We see that ψ is a homeomorphism onto an open subset of C. Thus we
include (Γp\U, ψ) in our local charts.
Finally, if p is a cusp of Γ, and α ∈ SL(2,R) is such that α · s =∞, then
we have
αΓpα−1 = 〈th : z 7−→ z + h〉,
h being a positive number. We define a homeomorphism ψ of Γp\U onto an
open subset of C by
ψ(φ(z)) = exp(2πiα(z)/h),
and include (Γp\U, ψ) in the local charts.
The atlas formed by these local charts gives X the structure of a Riemann
surface.
Definition 1.2.6. Keeping the above notations, we have :
1. A point P in X will be called elliptic, if it corresponds to an elliptic
point of Γ. The function t = wnp will be called the local parameter at
P (or at p ).
2. A point P in X will be called a cusp, if it corresponds to a cusp of Γ.
The function qh := exp(2πiz/h) will be called the local parameter at
P (or at p ).
26
An important class of Fuchsian groups consists of Fuchsian groups of the
first kind; they are defined as follows:
Definition 1.2.7. Γ is called a Fuchsian group of the first kind, if XΓ = Γ\H∗
is compact. A subgroup Γ1 of SL(2,R) is a Fuchsian group of the first kind,
if its associated subgroup Γ1 := ±1\Γ1.±1 of PSL(2,R) is of the first
kind.
As we have seen above, Γ acts on the sets EΓ (resp. CΓ) of elliptic (resp.
cuspidal) points of Γ. Set
EΓ := Γ\EΓ, SΓ := Γ\CΓ. (1.2.2)
Proposition 1.2.8. If Γ is of the first kind, then EΓ and SΓ are both finite
sets. Their cardinality will be respectively denoted by eΓ and cΓ.
Also, we have following useful result:
Proposition 1.2.9. Suppose that Γ is of the first kind, then any finite index
subgroup of Γ is also a Fuchsian group of the first kind.
Suppose that p ∈ H is an elliptic fixed point of Γ. By Proposition 1.2.5,
the orders of γ.p and p are equal for all γ ∈ Γ. Therefore, the order of an
elliptic point can be lifted to X.
Definition 1.2.8. If P ∈ X is not a cuspidal point, then we define the order
of P to be the order of any point in H corresponding to P .
1.2.4 Congruence subgroups of PSL(2,Z)
The modular group PSL(2,Z) and its finite index subgroups form a very rich
category of Fuchsian groups of the first kind. Here we list some well known
examples.
27
Let
T =
(1 1
0 1
), S =
(0 −1
1 0
), P = ST =
(0 −1
1 1
).
Then S is elliptic of order 2 with fixed point i, and P is elliptic of order 3 and
fixes ρ = e2πi/3. Moreover PSL(2,Z) is generated by S and T , or equivalently,
by S and P .
An important class of subgroups of PSL(2,Z) are the so-called congruence
subgroups. A subgroup Γ of PSL(2,Z) is called a congruence subgroup if it
contains some principal congruence subgroup Γ(n) for some positive integer
n, where Γ(n) is defined by
Γ(n) = γ ∈ SL(2,Z), γ ≡ ±1 mod n/±1.
The smallest such n is called the level of Γ.
One easily shows that Γ(n) is a normal subgroup of PSL(2,Z) of finite
index and that the conjugates in PSL(2,Z) of a congruence subgroup are also
congruence subgroups.
Example 1.2.10. Important examples of congruence subgroups of PSL(2,Z)
are:
Γ1(n) = γ ∈ SL(2,Z), γ ≡ ±
(1 ∗0 1
)mod n/±1,
Γ0(n) =
(a b
c d
)∈ SL(2,Z), c ≡ 0 mod n
/±1,
Γ0(n) =
(a b
c d
)∈ SL(2,Z), b ≡ 0 mod n
/±1.
It is clear that Γ(n) ⊆ Γ1(n) ⊆ Γ0(n) and that Γ0(n), Γ1(n), Γ0(n) are of
level n. Note that Γ0(n) is conjugate to Γ0(n) by(n 00 1
)
28
1.3 Automorphic forms.
In this section we give the definition of automorphic forms and provide some
well known examples.
1.3.1 Definition of automorphic forms.
In this subsection we follow The second chapter of [55].
Let k ∈ Z, γ ∈ GL+(2,R), Γ be a Fuchsian, and f be a meromorphic
function on H, we set
f |kγ = (det γ)k2 J−kγ f γ , (1.3.1)
where Jγ(z) = cz + d if γ =
(∗ ∗c d
). We have for all α, γ ∈ Γ,
Jαγ(z) = Jα(γz)Jγ(z), z ∈ H. (1.3.2)
Notice that for k odd, Jk−γ = −Jkγ , hence f |k(−γ) = −f |kγ. If k is even, then
f |k(−γ) = f |kγ.
Definition 1.3.1. Let Γ be a Fuchsian subgroup of SL(2,R), and k be an in-
teger. A meromorphic function f on H is called an unrestricted automorphic
form of weight k for Γ, if it satisfies
(i) f |kγ = f for all γ ∈ Γ.
If in addition
(ii) f is meromorphic at every cusp of Γ,
then f is called an automorphic form of weight k for Γ. When k = 0, f is
called an automorphic function.
29
By a cusp of Γ we mean a cusp of its associated subgroup Γ := ±1\Γ.±1of PSL(2,R). The last condition in the definition means the following. If Γ
has no cusp, then this condition is empty. In case Γ has a cusp s, then the
stabilizer of s in Γ is equal to Γs for some subgroup Γs ⊆ Γ. Moreover, if
α =
(a b
c d
)∈ SL(2,R) is such that α · s =∞, then we have
αΓsα−1±1 =
⟨±(
1 h
0 1
)⟩,
with h being a positive real number referred to as the cusp width at s. From
(i) we deduce that fs := f |k(α−1) is invariant under the group αΓα−1.
Case I: k is even. Here fs is invariant under the translation
th : z 7−→ z + h,
hence fs = Gs(qh) for some meromorphic function Gs in the punctured disc
: 0 < |qh| < r, where r is a positive number, and qh := exp(2πiz/h). We say
that f is meromorphic at the cusp s if Gs is meromorphic at qh = 0, this
means that f , when it is nonzero, has a Laurent expansion∑n≥ns
anqnh , 0 < |qh| < r,
where ns is the order ords(Gs) of Gs at 0. We define the order of f at s by
ords(f) = ord0(Gs).
Case II: k is odd. If Γ contains −1, then condition (i) implies that
f = −f . Hence there is no nontrivial automorphic form of weight k for Γ.
Therefore we may assume that −1 6∈ Γ. Then αΓsα−1 is generated by
(1 h0 1
),
or by −(
1 h0 1
). We say that the cusp s is regular or irregular, accordingly. If s
is regular, then the meromorphy at s is treated in the same way as in Case
I. If s is irregular, then fs(z + h) = −fs. Letting
gs(z) = exp(−πiz/h)fs, z ∈ H,
then gs is invariant under the translation th : z 7−→ z + h, hence gs = Gs(qh)
for some meromorphic function Gs in the punctured disc : 0 < |qh| < r,
30
where r is a positive number, and qh := exp(2πiz/h). We say that f is
meromorphic at the cusp s if Gs is meromorphic at qh = 0. We define the
order of f at s by ords(f) = ord0(Gs) + 1/2.
f is said to be holomorphic (resp. cuspidal) at s if ords(f) ≥ 0 (resp.
ords(f) > 0). An unrestricted automorphic form which is meromorphic at
any point of H∗, is called a meromorphic automorphic form. An automor-
phic form which is holomorphic at any point of H∗, is called a holomorphic
automorphic form. A holomorphic automorphic form which is cuspidal at
any cusp is called a cusp automorphic form, or simply a cusp form. The vec-
tor space of automorphic forms of weight k for Γ will be denoted by Ak(Γ).
The vector spaces of unrestricted, holomorphic, cusp automorphic forms of
weight k for Γ will be respectively denoted by Gk(Γ), Mk(Γ), and Sk(Γ).
If Γ is Fuchsian group of the first kind, then Mk(Γ) and Sk(Γ) have finite
dimensions. Indeed, we have
Theorem 1.3.1. Let Γ be a Fuchsian group of the first kind, and k be an
even integer. Let g be the genus of XΓ = Γ\H∗, e and c be the number of
elliptic and cuspidal points of XΓ respectively. If n1, . . . , ne are the orders of
the elliptic points, then
dimMk(Γ) =
(k − 1)(g − 1) + (k/2)c+∑e
i=1[k(ni − 1/2ni)] (k > 2)
g + c− 1 (k = 2, c > 2)
g (k = 2, c = 0)
1 (k = 0)
0 (k < 0),
31
and
dimSk(Γ) =
(k − 1)(g − 1) + (k/2− 1)c+∑e
i=1[k(ni − 1/2ni)] (k > 2)
g (k = 2)
1 (k = 0, m = 0)
1 (k = 0, m > 0)
0 (k < 0),
where [.] denotes the integral part of a real number.
1.3.2 Examples of modular forms
The main reference for this subsection is the first chapter of [38]. When
working with the modular group PSL(2,Z) and its subgroups, the word au-
tomorphic is replaced by modular.
We start our examples by the well known Eisenstein series. They are
defined for every even integer k ≥ 2 and z ∈ H by
Gk(z) =∑n
∑m
′ 1
(nz +m)k,
where the symbol∑ ′ means that the summation is over the pairs (m,n)
different from (0, 0). Notice that this series is not absolutely convergent for
k = 2. If we normalize the Eisenstein series by letting
Ek = 2ζ(k)Gk,
where ζ is the Riemann Zeta function, then we get
Ek(z) = 1− 2k
Bk
∞∑n=1
σk−1(n)qn, q = e2πiz.
Here Bk is the k-th Bernoulli number and σk−1(n) =∑
d|n dk−1. The most
familiar Eisenstein series are:
E2(z) = 1− 24∞∑n=1
σ1(n)qn,
32
E4(z) = 1 + 240∞∑n=1
σ3(n)qn,
E6(z) = 1− 504∞∑n=1
σ5(n)qn.
For k ≥ 4, the series Ek are holomorphic modular forms of weight k. The
Eisenstein series E2 is holomorphic on H and at the cusps, but it is not a
modular form as it does not satisfy the modularity condition. The Eisenstein
series E2 is an example of a quasimodular form and plays an important role
in the theory of vector valued modular forms, see [37]. Moreover, E2 satisfies
E2(z) =1
2πi
∆′(z)
∆(z), (1.3.3)
where ∆ is the weight 12 cusp form for PSL(2,Z) given by
∆(z) = q∏n≥1
(1− qn)24.
The Eisenstein series satisfy the Ramanujan relations
6
πiE ′2 = E2
2 − E4 , (1.3.4)
3
2πiE ′4 = E4E2 − E6 , (1.3.5)
1
πiE ′6 = E6E2 − E2
4 . (1.3.6)
The Dedekind j-function given by
j(z) =E3
4 − E26
∆
is a modular function and it generates the function field of modular functions
for PSL(2,Z).
33
Chapter 2
Vector-valued automorphic
forms and vector bundles
The aim of this chapter is to realize vector-valued automorphic forms as
global sections of holomorphic vector bundles. More precisely, given a Fuch-
sian subgroup Γ of SL(2,R), a finite-dimensional representation R of Γ,
and an integer k such that the pair (R, k) is simple, see Definition 2.1.2,
we construct a holomorphic vector bundle EΓ,R,k over the Riemann surface
XΓ = Γ\H∗ associated to Γ. The global meromorphic sections of EΓ,R,k will
correspond to vector-valued automorphic forms of weight k and multiplier
R. More interestingly, the spaces Mk(Γ, R), Sk(Γ, R) of holomorphic, cusp
vector-valued automorphic forms of weight k and multiplier R, will be respec-
tively isomorphic to H0(XΓ,O(EΓ,R,−k)) and H0(XΓ,O(−DS,R,−k+EΓ,R,−k)),
where DS,R,−k is a certain holomorphic line bundle depending on the cuspidal
points of XΓ. As an immediate consequence, we deduce that Mk(Γ, R) and
Sk(Γ, R) are finite-dimensional C−vector spaces if and only if Γ is Fuchsian
group of the first kind, see Theorem 2.5.9.
34
2.1 Notations
For the remainder of this thesis, n is a positive integer, Γ is a Fuchsian
subgroup of SL(2,R), R is a representation of Γ in GL(n,C). We will denote
the matrix R(γ), γ ∈ Γ, by Rγ. R∗ will be the adjoint representation to R,
that is
R∗γ = (Rγ−1 )T ,
where (.)T is the transpose operator. Γ will be the image of Γ in PSL(2,R),
that is Γ = ±1\Γ.±1. E will be the set of elliptic fixed points of Γ and
C the set of its cusps in R ∪ ∞. We define
H∗ := H ∪ C , S := Γ\C, E := Γ\E,
X = X(Γ) := Γ\H∗ , X ′ = X ′(Γ) := Γ\(H− E) ,
Y ′ = Y ′(Γ) := H− E.
We have an unbranched covering map
π : Y ′ −→ X ′,
and since Γ acts properly and discontinuously on Y ′, then the group of cov-
ering transformations is
Deck (Y ′/X ′) = Γ.
Moreover, this is a Galois covering in the sense that for all y1 and y2 in Y ′
with π(y1) = π(y2), there exists σ ∈ Γ such σ(y1) = y2.
Also, when Γ does not contain −1, we can identify Γ with Γ via the
canonical map, and so Γ will be viewed as a group of matrices as well as a
group of Mobıus transformations.
If k is an integer, then we define the slash operator |k on meromorphic
maps F : H −→ Cm, m being an integer, by
F |kγ = J−kγ F γ , γ ∈ GL+(2,R) , (2.1.1)
where Jγ(z) = cz + d if γ =
(∗ ∗c d
).
35
Definition 2.1.1. Let k be an integer. An unrestricted vector-valued au-
tomorphic form for Γ of multiplier R and weight k is a meromorphic map
F : H −→ Cn satisfying
F |kγ = Rγ F.
The C-vector space of these maps will be denoted by Gk(Γ, R).
If R = R1 ⊕ R2 is the direct sum of two representations R1 and R2,
then it is clear that Gk(Γ, R) = Gk(Γ, R1)⊕Gk(Γ, R1), and when Γ contains
−1, then after [3], the representation R will be called even (resp. odd) if
R(−1) = In (resp. R(−1) = −In). Since −1 commutes with Γ, then modulo
a conjugation in GL(n,C), any representation R may be decomposed into a
direct sum R = R+ ⊕ R− of even and odd representations. It follows from
Definition 2.1.1 that for an even (resp. odd) representation R there are no
nontrivial unrestricted vector-valued automorphic forms of odd (resp. even)
weight. Therefore
Gk(Γ, R) =
Gk(Γ, R+) if k is even,
Gk(Γ, R−) if k is odd.
This shows that it suffices to treat the case when both R and k are even, or
when they are both odd.
To avoid trivial cases, we adopt the following definition:
Definition 2.1.2. The pair (R, k), k being an integer, is called simple if
1. Γ does not contain −1, or
2. Γ contains −1, R and k have the same parity.
Remark 2.1.1. In the sequel, we only consider the integers k such that
(R, k) is a simple pair. In particular, R is either even or odd when −1 ∈ Γ.
36
Definition 2.1.3. The integer ε = ε(R) defined by
ε =
0 if R is even, or −1 6∈ Γ
1 if R is odd
is called the parity index of R.
Notice that ε is the smallest nonnegative integer k such that (R, k) is a
simple pair.
Remark 2.1.2. 1. In this work, almost all the constructed objects will
depend on at least three parameters, namely : Γ, R, and the weight k.
When the context is clear, we will only keep the relevant indices.
2. If the context is clear, an object defined on Y ′ which is invariant under
the covering transformations, will be considered as an object defined
on X ′.
2.2 The family EΓ,R,k of vector bundles
The goal of this section is to associate to each triplet (Γ, R, k), with (R, k)
being a simple pair, a holomorphic vector bundle EΓ,R,k on X. This will
be achieved by constructing a 1-cocycle in Z1(U ,GL(n,OX)), where U is
an open cover of X. Then EΓ,R,k will be its associated holomorphic vector
bundle.
2.2.1 The unbranched case.
Let (R, k) be a simple pair, we construct a matrix-valued automorphic form
of weight k and multiplier R on the unbranched covering Y ′ of X ′. More
precisely, we have
37
Theorem 2.2.1. There exists a holomorphic map
Ψk = ΨR,k : Y ′ −→ GL(n,C)
such that for all σ ∈ Γ, we have
Ψk|k σ = Ψk Rσ−1 .
Proof. Since π : Y ′ −→ X ′ is an unbranched Galois covering, there exists an
open covering U = (Ui)i∈I of X ′ and homeomorphisms
φi = (π, ηi) : π−1(Ui) −→ Ui × Γ
which are compatible with the action of Γ in the sense that φi(y) = (x, σ)
implies φ(τy) = (x, τσ). In other words, the mapping ηi : π−1(Ui) −→ Γ
satisfies ηi(τy) = τηi(y) for all y ∈ π−1(Ui) and τ ∈ Γ.
On Yi = π−1(Ui) we define Ψk,i : Yi −→ GL(n,C) by
Ψk,i(y) = J−kηi(y)−1(y)Rηi(y)−1 , y ∈ Yi,
which is well defined since the expression
J−kσ Rσ
is well defined on Γ for a simple pair (R, k). Moreover, Ψk,i is holomorphic
since it is locally constant, and so Ψk,i ∈ GL(n,O(Yi)). Now, if y ∈ Yi and
σ ∈ Γ, then
Ψk,i(σy) = J−kηi(σy)−1(σy)Rηi(σy)−1
= J−kηi(y)−1σ−1(σy)Rηi(y)−1σ−1
= J−kσ−1(σy)J−kηi(y)−1(y)Rηi(y)−1Rσ−1
= Jkσ (y)Ψk,i(y)Rσ−1 .
38
Hence, for σ ∈ Γ,
Ψk,i|k σ = Ψk,iRσ−1 on Yi.
Therefore, Ψk,i has the required property on Yi. If we set
Fk,ij = FR,k,ij = Ψk,iΨ−1k,j ∈ GL(n,O(Yi ∩ Yj)) ,
then for all y ∈ Yi ∩ Yj we have Fk,ijσ = Fk,ij, that is, the Fk,ij is invariant
under covering transformations and hence may be considered as an element
of GL(n,O(Ui ∩ Uj)). Thus, we have a cocycle
(Fk,ij) ∈ Z1(U ,GL(n,O)) .
As X ′ is a noncompact Riemann surface, there exist elements
Fk,i = FR,k,i ∈ GL(n,O(Ui))
such that
Fk,ij = Fk,iF−1k,j on Ui ∩ Uj , (2.2.1)
see [13]. We now look at the Fk,i as elements of GL(n,O(Yi)) that are
invariant under covering transformations and set
Ψi = F−1k,i Ψk,i ∈ GL(n,O(Yi)) .
Then, for every σ ∈ Γ, we have
Ψk,i|k σ = (Fk,iσ)−1Ψk,i|kσ = F−1i Ψk,iRσ−1 = ΨiRσ−1 .
Moreover, on Yi ∩ Yj we have
(Ψk,i)−1Ψk,j = Ψ−1
k,iFk,iF−1k,j Ψk,j = Ψ−1
k,iFk,ijΨk,j = Ψ−1k,iΨk,iΨ
−1k,jΨk,j = 1.
Thus, the Ψk,i’s define a global function Ψk ∈ GL(n,O(Y ′)) with
Ψk|k σ = ΨkRσ−1 , σ ∈ Γ .
39
We see that the ΨR,k constructed above depends on the choice of the
elements FR,k,i ∈ GL(n,O(Ui)), i ∈ I, let us call it Ψk,F . Suppose that
GR,k,i ∈ GL(n,O(Ui)), i ∈ I, is another choice, then by (2.2.1), we have
Fk,iF−1k,j = Fk,ij = Gk,iG
−1k,j on Ui ∩ Uj .
Hence, the element AF,G ∈ GL(n,O(X ′)) given by
AF,G|Ui = G−1k,iFk,i on Ui
is well-defined and we
Ψk,G = AF,GΨk,F . (2.2.2)
Now, we take an arbitrary choice of the elements FR,k,i ∈ GL(n,O(Ui)),
i ∈ I, and we fix it for the rest of the thesis. Its corresponding map will be
our ΨR, k.
Remark 2.2.2. From the proof of Theorem 2.2.1, we see that
F ∗R,k,i(F∗R,k,j)
−1 = FR∗,−k,i F−1R∗,−k,j on Ui ∩ Uj ,
and hence, as in the above discussion, we have
(ΨR,k)∗ = ∆R,k ΨR∗,−k,
for some ∆R,k ∈ GL(n,O(X ′)).
2.2.2 The elliptic case.
Now, let e be an elliptic fixed point of Γ, then the stabilizer of e in Γ is equal
to Γe for some subgroup Γe ⊆ Γ. Moreover, Γe = 〈σe〉 with σe = γe, for some
γe in Γe. If e denotes the complex conjugate of e, then we have a character
µe on Γe defined by
µe(γ) = Jγ(e), γ ∈ Γe, (2.2.3)
40
such that µe(−1) = −1, when −1 ∈ Γ. Also, the holomorphic function on Hgiven by
fe(z) =1
z − e, z ∈ H,
verifies
fe|1γ = µe(γ)fe, γ ∈ Γe.
Lemma 2.2.3. There exists a meromorphic map Ψk,e = ΨR,k,e in H, having
a pole only at e, such that for all γ ∈ Γe, we have
Ψk,e|kγ = Ψk,eRγ−1 .
Proof. If
α(z) =z − ez − e
, z ∈ H,
then α(H) is the unit disc D, α(e) = 0, and αΓeα−1 is the group
〈σ′e : w 7−→ ζew〉
where ζe is a primitive ne−th root of unity, ne being the order of e.
Let R = Rk,e := µ−ke R, then R is a well defined representation of Γe. Since
Rneγe = 1, then Rγe is diagonalizable and for some Ak,e = AR,k,e ∈ GL(n,C)
we have
Ak,eRσeA−1k,e = diag (λ1, . . . , λn), λnej = 1 .
Write λj = ζmk,je , 0 ≤ mk,j ≤ ne − 1, and for w ∈ D∗ set
Φk,e(w) = ΦR,k,e(w) := diag (w−mk,1 , . . . , w−mk,n) .
We have
Φk,e(σ′ew) = Φk,e(ζew)
= diag (ζ−mk,1e w−mk,1 , . . . , ζ
−mk,ne w−mk,n)
= diag (λ−11 w−mk,1 , . . . , λ−1
n w−mk,1)
= Φk,e(w)diag (λ1, . . . , λn)
= Φk,e(w)Ak,eRσ−1eA−1k,e .
41
Therefore,
Φk,e(σ′e)Ak,e = Φk,eAk,eRσ−1 ,
If we set
Φk,e = (Φk,eα)Ak,e ,
then for all σ ∈ Γe,
Φk,e(σ) = Φk,eRσ−1 .
Now, let
Ψk,e = fke Φk,e,
then for all γ ∈ Γe,
Ψk,e|kγ = µke(γ)fke Φk,eR(γ)−1 = µke(γ)Ψk,eR(γ)−1 .
But for γ ∈ Γe we have
R(γ)−1 = µ−ke (γ−1)Rγ−1 .
Hence for all γ ∈ Γe, we have
Ψk,e|kγ = Ψk,eRγ−1
as desired.
Notice that the construction of ΨR,k,e depends on the choice of the AR,k,e.
We fix a choice for the matrix AR,k,e once for all, and its corresponding map
will be our ΨR,k,e .
Remark 2.2.4. The relationship between (ΨR,k,e) and ΨR∗,−k,e is given by
the following: Since
AR,k,e(µ−ke (γe)Rγe
)A−1R,k,e = diag (ζ
mR,k,1e , . . . , ζ
mR,k,ne ),
where (mR,k,j)j is a sequence of integers in [0, ne), then
A∗R,k,e(µke(γe)R
∗γe
)(A∗R,k,e)
−1 = diag (ζne−mR,k,1e , . . . , ζ
ne−mR,k,ne ).
42
But
diag (ζne−mR,k,1e , . . . , ζ
ne−mR,k,ne ) = AR∗,−k,e
(µke(γe)R
∗γe
)A−1R∗,−k,e.
This implies that the matrix
BR,k,e = A∗R,k,eA−1R∗,−k,e
commutes with
diag (ζne−mR,k,1e , . . . , ζ
ne−mR,k,ne ),
hence with ΦR∗,−k,e, and we have
(ΨR,k,e)∗ = (α(z))ne BR,k,e ΨR∗,−k,e.
Definition 2.2.1. With the above notations, we define the k−order of R at
e to be
νk,e(R) = −(1/ne)n∑j=1
mR,k,j,
where the integers 0 ≤ mR,k,j ≤ ne − 1 are such that
AR,k,e(µ−ke (γe)Rγe )A−1
R,k,e = diag (ζmR,k,1e , . . . , ζ
mR,k,ne ).
Since it only depends on the class of e modulo Γ, we define the k−order of
R at an elliptic point P ∈ X to be
νk,P (R) = νk,e(R),
with e ∈ H is an elliptic point corresponding to P .
2.2.3 The cuspidal case.
Similarly, If s is a cusp of Γ, then the stabilizer of s in Γ is equal to Γs
for some subgroup Γs ⊆ Γ. Also, Γs = 〈γs〉 with γs in Γs, and we have a
character µs on Γs defined by
µs(γ) = Jγ(s), γ ∈ Γs, (2.2.4)
43
such that µs(−1) = −1, when −1 ∈ Γ. Notice that for γ =
(aγ bγcγ dγ
)∈ Γs,
a simple computation shows that
s = (aγ − dγ)/2cγ.
This implies that
µs(γ) = Trace(γ)/2, γ ∈ Γs, (2.2.5)
which is also valid for s =∞. Since the trace of γ is ±2, we see that
µs(γ) = ±1 γ ∈ Γs.
Moreover, the holomorphic function on H defined by
fs(z) =
1z−s , z ∈ H, if s 6=∞
1 if s =∞ ,
verifies
fs|1γ = µs(γ)fs, γ ∈ Γs.
If α =
(a b
c d
)∈ SL(2,R) is such that α · s = ∞, then, using the notations
of [55], we have
αΓsα−1 · ±1 = 〈th : z 7−→ z + h〉,
with h being a positive real number referred to as the cusp width of s. Also,
from s = α−1 · ∞ = −d/c, we have fs = cα /Jα for some nonzero constant
cα ∈ C.
Lemma 2.2.5. [50] There exists a holomorphic map Ψk,s = ΨR,k,s in H,
such that for all γ ∈ Γs, we have
Ψk,s|kγ = Ψk,sRγ−1 .
44
Proof. Let R = Rk,s := µ−ks R, then R is a well-defined representation of Γs.
Take Bk,s = BR,k,s ∈M(n,C) such that
Rγs = exp(2πiBk,s).
If we set
Φk,s(z) = exp(−2πi
z
hBk,s
), z ∈ H ,
we have
Φk,s(thz) = exp
(−2πi
z + h
hBk,s
)= exp
(−2πi
z
hBk,s − 2πiBk,s
)= R(γs)−1Φk,s(z)
= Φk,s(z)R(γs)−1 .
Hence if
Ψk,s = fks Φk,sα = ckα Φk,s|kα,
then Ψk,s has the required property.
Remark 2.2.6. Note that Ψk,s depends on the choice of Bk,s. In the sequel
we take Bk,s such that
0 ≤ <(λ) < 1
for every eigenvalue λ of Bk,s, which is always possible, see [18]. With this
choice, one can check that [18, 16]
BR∗,−k,s = In − (BR,k,s)t.
Therefore
(ΦR,k,s)∗ = qh(α)ΦR∗,−k,s,
where qh(α(z)) = exp (2πiα(z)/h), z ∈ C.
45
Definition 2.2.2. Let Bk,s = BR,k,s ∈M(n,C) such that
µ−ks (γs)Rγs = exp(2πiBk,s),
and for all eigenvalues λ of Bk,s
0 ≤ <(λ) < 1 .
Then the number
νk,R(s) = −Trace(Bk,s)
is called the k−order of R at s. Since it only depends on the class of s modulo
Γ, we define the k−order of R at a cuspidal point P ∈ X to be
νk,P (R) = νk,s(R),
where s ∈ H∗ is a cusp corresponding to P .
Definition 2.2.3. If for a point P ∈ X \ (E ∪ S ) we set νk,P (R) = 0, then
the k−divisor of R is defined by
DR,k =∑P∈X
νk,P (R)P.
2.2.4 Construction of the 1−cocycle.
We now come to the main construction in this section. Write
E ∪S = aii≥1
for the discrete closed set of classes of cusps and elliptic fixed points in X.
In particular, they correspond to inequivalent points ai in H∗. For each ai
we choose a neighborhood in X in the following way:
If ai is a cusp, let Ui be a neighborhood of ai in X given by
Ui = (Γai\Dai) ∪ ai,
46
where Dai is a horocycle in H tangent at ai [55] (if ai =∞, this horocycle is
a half-plane). Here Γai is infinite cyclic.
If ai is an elliptic fixed point, then Ui is given by
Ui = Γai\Dai ,
where Dai is an open disc in H centered at ai. Here Γai is a finite cyclic
group.
Further, these neighborhoods can be taken such that for ai 6= aj, we have
Ui ∩ Uj = ∅. Also, for each ai choose a chart z for X such that z(Ui) = Dand z(ai) = 0 where D is the unit disc.
Set U0 = X ′, V0 = H − E = π−1(X ′) = Y ′, and Vi = π−1(Ui − ai),i ≥ 1. Then U = (Ui)i≥0 is an open covering of X, and we have
Vi =⊔
γ∈Γ/Γai
Zγ,
where γ is a representative of γ in Γ/Γai , and Zγ is a connected component
of Vi. If Ψk,0 = ΨR,k is the map constructed in Theorem 2.2.1, then we have
the following.
Proposition 2.2.7. [50] If ai is a cusp, then there exists a holomorphic map
Ψk,i : Vi −→ GL(n,C)
such that Ψk,i Ψ−1k,0 is invariant under Deck (Zγ/Ui − ai) for all connected
components Zγ of Vi.
Proof. We have
Vi = ΓDai =⊔
γ∈Γ/Γai
γDai =⊔
γ∈Γ/Γai
Dγai .
Here we have a disjoint union of connected components Zγ = γDai of Vi and
γ is a chosen representative of γ in Γ/Γai . Then
Zγ −→ Ui − ai
47
is a universal covering with the covering transformations given by the fun-
damental group
π1(Ui − ai) = Γγai .
By Theorem 2.2.1 and Lemma 2.2.5, the maps Ψk,γai and Ψk,0 on Zγ have
the same automorphic behaviour with respect to Γγai , and so Ψk,γai Ψ−1k,0 is
invariant under Γγai , hence under Γγai . The map Ψk,i defined on Vi by
Ψk,i|Zγ = Ψk,γai
is then a holomorphic map on Vi such that Ψk,i Ψ−1k,0 is invariant under
Deck (Zγ/Ui − ai) for all connected components Zγ of Vi.
Proposition 2.2.8. [50] If ai is an elliptic fixed point, then there exists a
holomorphic map
Ψk,i : Vi −→ GL(n,C)
such that Ψk,i Ψ−1k,0 is invariant under Deck (Zγ/Ui − ai) for all connected
components Zγ of Vi. Moreover, Ψk,i is meromorphic at each point of π−1ai.
Proof. If ai is the class of an elliptic fixed point ai in H, then
Ui − ai = Γai\D∗ai ,
where D∗ai = Dai − ai is the punctured disc. Here, the stabilizer Γai of
cyclic of finite order ki. Furthermore, we have
Vi =⊔
γ∈Γ/Γai
γD∗ai =⊔
γ∈Γ/Γai
Zγ ,
where the Zγ = γD∗ai are the connected components of Vi. In the meantime,
the group of covering transformations of the unbranched covering Zγ −→ D∗aiis given by the fundamental group
π1(Ui − ai) = Γγai .
48
Indeed, it is isomorphic to the covering
D∗ −→ D∗
z 7−→ zki
for which the covering transformations are given by the group 〈σi : z 7−→ ζiz〉where ζi is a primitive ki−th root of unity, ki is the order of ai.
By Theorem 2.2.1 and Lemma 2.2.3, the maps Ψk,γai and Ψk,0 on Zγ
have the same automorphic behavior with respect to Γγai , and so Ψk,γai Ψ−1k,0
is invariant under Γγai , hence under Γγai . The map Ψk,i defined on Vi by
Ψk,i|Zγ = Ψk,γai
is then a holomorphic map on Zγ = γD∗ai , and meromorphic at γai ∈ γDai .
Hence, Ψk,i is holomorphic in Vi and meromorphic at each point of π−1ai.Also, Ψk,i Ψ
−1k,0 is invariant under Deck (Zγ/Ui − ai) for all connected com-
ponents of Zγ of Vi.
Now define the 1−cocycle
Fk,ij = Ψk,iΨ−1k,j ∈ GL(n,O(Vi ∩ Vj)) if i 6= j and Fk,ii = id . (2.2.6)
Theorem 2.2.9. [50] The 1−cocycle (Fk,ij) defines a 1−cocycle (Fk,ij) in
Z1(U ,GL(n,O)) such that Fk,ij = π∗Fk,ij.
Proof. We need to prove that the Fk,ij’s defined in (2.2.6) can be considered
as elements of GL(n,O(Ui ∩ Uj)). By construction, for i 6= 0 and j 6= 0 we
have
Ui ∩ Uj = Vi ∩ Vj = ∅ , U0 ∩ Ui = Ui − ai , V0 ∩ Vi = Vi .
Thus we only need to prove that Ψk,0Ψ−1k,i ∈ GL(n,O(Vi)) defines an el-
ement of GL(n,O(Ui − ai)), that is to show that Ψk,0Ψ−1k,i is invariant
under the action of Deck (Zγ/Ui − ai) which is a direct consequence of
49
Proposition 2.2.7 and Proposition 2.2.8. Therefore, there exists an element
(Fk,ij) ∈ Z1(U ,GL(n,O)) such that Fk,ij = π∗Fk,ij.
According to Theorem 1.1.2, a 1−cocycle in Z1(U ,GL(n,O)) gives rise
to a holomorphic vector bundle on X.
Definition 2.2.4. We define EΓ,R,k to be the holomorphic vector bundle
p : EΓ,R,k −→ X of rank n whose transition functions are the (Fk,ij). If the
context is clear EΓ,R,k will be denoted by Ek.
If R is the trivial character χ0 sending all elements of Γ to 1, then EΓ,χ0,k
is a line bundle which will be denoted by Lχ0,k = Lk. Since χ0 is an even
representation, then Lk is defined only for even k’s when −1 ∈ Γ, and for
any k when −1 6∈ Γ.
2.3 Behavior at the cusps
In this section we shall make explicit the behavior of unrestricted vector-
valued automorphic forms at cusps.
Let s be a cusp of Γ. As we have seen in §2.2.3, the stabilizer of s in
Γ is equal to Γs for some subgroup Γs ⊆ Γ. Also, Γs = 〈γs〉 with γs in Γs.
Moreover, if α ∈ SL(2,R) is such that α · s =∞, then we have
αΓsα−1 = 〈th : z 7−→ z + h〉,
h being the cusp width of s.
Let F be an unrestricted vector-valued automorphic form for Γ of mul-
tiplier R and weight k, Fs := Ψ−k,sF , and Fs := Fsα−1. By Lemma 2.2.5,
Fs is invariant under Γs, hence under Γs. Therefore Fs is invariant under
αΓsα−1, and so
Fsth = Fs .
50
Hence Fs(z) = Gs(qh) for some meromorphic map
Gs : qh; 0 < |qh| < r −→ Cn,
where r is a positive number, and qh := exp(2πiz/h). We say that F is
meromorphic at the cusp s if Gs is meromorphic at qh = 0, this means that
Gs, when it is nonzero, has a Laurent expansion∑n≥ns
anqnh , 0 < |qh| < r,
where ns is the order ords(Gs) of Gs at 0. We define the order of F at s by
ords(F ) := ord0(Gs).
F is said to be holomorphic at s if ords(F ) ≥ 0. An unrestricted vector-
valued automorphic form which is meromorphic (resp. holomorphic) at every
point of H∗, is called a vector-valued automorphic form (resp. holomorphic
vector-valued automorphic form). The vector space of vector-valued automor-
phic forms for Γ of multiplier R and weight k will be denoted by Ak(Γ, R),
and its subspace of holomorphic vector-valued automorphic forms will be
denoted by Mk(Γ, R).
2.4 The divisor of a vector-valued automor-
phic form
The goal of this section is to associate to each vector-valued automorphic
form F a divisor div(F ) defined on the Riemann surface X.
Suppose that F is a vector-valued automorphic form for Γ of multiplier
R and weight k. Then by Remark 1.1.5, we have ordγw(F ) = ordw(F ) for
any w ∈ H∗, γ ∈ Γ. This means that the order of F can be lifted to X.
If P ∈ X corresponds to a point w ∈ H, then set νP (F ) = ordw(F )/nw,
where nw is the order of the point w, see Definition 1.2.5. If P ∈ X corre-
sponds to a cusp s, then set νP (F ) = ords(F ). We define the divisor of F on
51
X to be the formal sum
div(F ) =∑P∈X
νP (F )P. (2.4.1)
From this we see that F is holomorphic in H∗ exactly when div(F ) ≥ 0.
Proposition 2.4.1. we have
Mk(Γ, R) =F ∈ Ak(Γ, R) | div(F ) ≥ 0
.
As usual, one can define a cusp vector-valued automorphic form of mul-
tiplier R and weight k, to be an element F of Mk(Γ, R) vanishing at all the
cusps of Γ. More precisely, suppose that s is cusp, then using notations of
Lemma 2.2.5, and Definition 2.2.2, we have
µks(γs)Rγs = exp(2πiBR,−k,s),
such that the eigenvalues λ−k,s,j of BR,k,s satisfy the condition
0 ≤ <(λ−k,s,j) < 1, 0 ≤ j ≤ n.
In terms of the local parameter qh at s, we have
F |kα−1 = ckα exp
(2πi
z
hBR,−k,s
)Gs(qh). (2.4.2)
By the Jordan decomposition theorem, we can write BR,−k,s in the form
BR,−k,s = DR,−k,s +NR,−k,s,
where DR,−k,s is diagonalizable matrix, NR,−k,s is nilpotent matrix, and
DR,−k,sNR,−k,s = NR,−k,sDR,−k,s.
Therefore
exp(
2πiz
hBR,−k,s
)= exp
(2πi
z
hNR,−k,s
)exp
(2πi
z
hDR,−k,s
).
52
Let nR be the smallest integer such that NnRR,−k,s = 0. From the definition of
the exponential of a matrix, we see that the entries of the matrix
exp(
2πiz
hNR,−k,s
)are polynomials in z of degree less than nR.
Since DR,−k,s is diagonalizable matrix, we have
DR,−k,s = AR,−k,s diag (λ−k,s,1, . . . , λ−k,s,nR)A−1R,−k,e,
for some A = AR,k,e ∈ GL(n,C). By (2.4.2), we have
A−1F |kα−1 =
ckα
(A−1 exp
(2πi
z
hNR,−k,s
)A)
exp(
2πiz
hdiag(λ−k,s,1, . . . , λ−k,s,nR)
)A−1Gs(qh).
Set
Fs = A−1 F |kα−1, Hs = A−1Gs(qh), Q(z) = ckαA
−1 exp(2πiz
hNR,−k,s)A.
Then it is clear that ord0(Hs) = ord0(G), and that the entries of Q(z) are
polynomials in z. Also, we have
exp(
2πiz
hdiag(λ−k,s,1, . . . , λ−k,s,nR)
)=
diag(
exp(2πiλ−k,s,1z
h), . . . , exp(2πiλ−k,s,nR
z
h))
:=
diag(qλ−k,s,1h , . . . , q
λ−k,s,nRh
)With these notations, we get
Fs(z) = Q(z) diag(qλ−k,s,1h , . . . , q
λ−k,s,nRh
)Hs(qh).
We say that F vanishes at s if the right hand side of this equation vanishes
when z → i∞ (or qh → 0).
53
We want to describe this condition in terms of the order of F at s, i.e., in
terms of ords(F ) = ord0(Gs) = ord0(Hs). Let
δR,−k,s = min<(λ−k,s,j), 0 ≤ j ≤ n.
Suppose that δR,−k,s > 0. Since the entries of Q(z) are polynomials in z, it
suffices to ask that Hs is holomorphic in qh (ord0(Hs) ≥ 0) in order to get
the vanishing of F at s. If δR,−k,s = 0, we need that ords(F ) = ord0(Hs) ≥ 1.
This leads to the following definition.
Definition 2.4.1. Let BR,k,s = Bk,s ∈M(n,C) such that
µ−ks (γs)Rγs = exp(2πiBk,s),
and
0 ≤ <(λk,s) < 1 ,
for all eigenvalues λk,s of Bk,s. Let
δR,k,s = min <(λk,s), λk,s is an eigenvalue of Bk,s .
Then
1. The number
ρk,s(R) =
0 if δR,k,s > 0
1 if δR,k,s = 0
is called the k−cuspidal order of R at s. Since it only depends on the
class of s modulo Γ, we define the k−cuspidal order of R at a cuspidal
point P ∈ X to be
ρk,P (R) = ρk,s(R),
where s ∈ H∗ is a cusp corresponding to P .
2. The k−cuspidal Divisor of R is defined by
DS,R,k =∑P∈S
ρk,P (R)P.
The associated line bundle |DS,R,k| to the divisor DS,R,k will be denoted
by DS,R,k.
54
We see that an element F in Ak(Γ, R) vanishes at a cuspidal point s of Γ
if and only if νP (F ) ≥ ρ−k,s(R), in this case we say that F is cuspidal at s.
An element F in Mk(Γ, R) is called a cusp vector-valued automorphic form of
multiplier R and weight k, if it is cuspidal at all cusps of Γ. The vector space
of cusp vector-valued automorphic forms for Γ of multiplier R and weight
k will be denoted by Sk(Γ, R). We summarize the above discussion in the
following result
Proposition 2.4.2. In terms of divisors, Sk(Γ, R) can be written as
Sk(Γ, R) = F ∈Mk(Γ, R) | div(F ) ≥ DS,R,−k .
Now, suppose that χ is a character on Γ and that f ∈ Al(Γ, χ). If
F ∈ Ak(Γ, R), then f F ∈ A(k+l)(Γ, χR). In the following, we shall express
the divisor of fF in terms of div(f) and div(F ). Set R = χR. If P ∈ X
corresponds to a point of H, then it is clear that
νP (fF ) = νP (f) + νP (F ).
If P ∈ X corresponds to a cusp s, the above formula is no longer valid.
Indeed, using notations of Lemma 2.2.5, and Definition 2.2.2, we have
µks(γs)Rγs = exp(2πiBR,−k,s),
with eigenvalues λk,s of BR,k,s verifying 0 ≤ <(λk,s) < 1. Similarly
µls(γs)χγs = exp(2πibχ,−l,s),
with 0 ≤ <(bχ,l,s) < 1. This implies that
µ(k+l)s (γs)Rγs = exp(2πi(bχ,−l,sIn +BR,−k,s).
Since the eigenvalues of bχ,l,sIn +BR,k,s are the eigenvalues of BR,k,s plus the
scalar bχ,l,s, we see that
BR,−(k+l),s = (bχ,−l,sIn +BR,−k,s) + diag (−m1, . . . ,−mn),
55
where mj ∈ 0, 1, 1 ≤ j ≤ n.
Using notations of Lemma 2.2.5, then in terms of the local parameter qh
at s, we have
c−kα exp(−2πiz
hBR,−k,s)F |kα−1 = Gs(qh),
c−lα exp(−2πiz
hbχ,−l,s) f |l(α−1) = gs(qh),
c−(k+l)α exp(−2πi
z
hBR,−(k+l),s) (fF )|
k+l(α−1) = Gs(qh),
and by definition, we have
νP (F ) = ord0(Gs), νP (f) = ord0(gs), νP (fF ) = ord0(Gs).
From this we see that
c−(k+l)α exp
(−2πi
z
h(bχ,−l,sIn +BR,−k,s)
)(fF )|
k+l(α−1) = gs(qh)Gs(qh),
and so
exp(−2πi
z
h(bχ,−l,sIn +BR,−k,s −BR,−(k+l),s)
)Gs(qh) = gs(qh)Gs(qh).
Hence
diag (qm1h , . . . , qmnh )Gs(qh) = gs(qh)Gs(qh).
Let gs,j (resp. gs,j), 1 ≤ j ≤ n, be the components of Gs (resp. Gs). Then
for some 1 ≤ j0, j1 ≤ n, we have
mj0 + ord0(Gs) = ord0(gs) + ord0(gs,j0),
mj1 + ord0(gs,j1) = ord0(gs) + ord0(Gs).
From the definition of the order, we deduce that
ord0(gs) + ord0(Gs) ≤ mj0 + ord0(Gs)
mj1 + ord0(Gs) ≤ ord0(gs) + ord0(Gs),
56
which yield
mj1+ ≤ ( ord0(gs) + ord0(Gs)− ord0(Gs) ) ≤ mj0 ,
Hence
0 ≤ νP (f) + νP (F )− νP (fF ) ≤ 1.
Thus, we have proved
Theorem 2.4.3. Let χ be a character, and f be an element of Al(Γ, χ). If
F ∈ Ak(Γ, R), then f F ∈ A(k+l)(Γ, χR), and we have
div(fF ) = div(f) + div(F )−Df,F ,
where
Df,F =∑Q∈S
aQ,f,F Q
is a divisor having support in S with coefficients aQ,f,F ∈ 0, 1.
2.5 Mk(Γ, R) and Sk(Γ, R) as global sections of
vector bundles
We now come to the main construction of this chapter. We establish the
correspondence between vector-valued automorphic forms of weight k and
multiplier k for Γ, and global sections of the the holomorphic vector bundles
Ek = EΓ,R,k on the Riemann surface X.
In this section we keep the notations of §2.2.4 used to construct the vector
bundle Ek. Recall that the transition functions of Ek are the elements of the
1−cocycle (Fk,ij) ∈ Z1(U ,GL(n,O)), and that Fk,ij = π∗Fk,ij, with
Fk,ij = Ψk,iΨ−1k,j ∈ GL(n,O(Vi ∩ Vj)) if i 6= j and Fk,ii = id ,
where Ψk,i is the map constructed in Proposition 2.2.7 and Proposition 2.2.8.
Now, suppose that G is a collection (Gi)Vi of meromorphic functions Gi on
57
Vi satisfying
Gi = Fk,ijGj on Vi ∩ Vj. (2.5.1)
If G is invariant under Γ, that is
Giγ = Gi, for all γ ∈ Γ, (2.5.2)
then there exists a collection G = (Gi)Ui of meromorphic functions Gi on Ui
such that
Gi = π∗Gi on Vi.
Note that, by construction, Vi is Γ-invariant. Since Fk,ij = π∗Fk,ij, then, by
(2.5.1), we have
Gi = Fk,ijGj on Ui ∩ Uj.
Hence by Proposition 1.1.4, we deduce that G = (Gi)Ui is a meromorphic
section of Ek.
Conversely, if G = (Gi)Ui is a meromorphic section of Ek, then
G = π∗G = (π∗Gi)Vi
clearly satisfies (2.5.1) and (2.5.2). Thus we have proved the following
Proposition 2.5.1. There is a correspondence between the global meromor-
phic sections of Ek, and the collections (Gi)Vi of meromorphic functions Gi
on Vi satisfying
Gi = Fk,ijGj on Vi ∩ Vj,
and
Giγ = Gi, for all γ ∈ Γ.
Theorem 2.5.2. We have a linear isomorphism between A−k(Γ, R) and
H0(X,M(Ek)) given by
F 7→ ΘF = (Ψk,iF )Vi ,
58
with inverse
Θ 7→ FΘ = (Ψ−1k,iπ
∗Θi)Vi ,
where Θ = (Θi)Ui .
Proof. By Proposition 2.5.1, the only nontrivial part is to prove that FΘ is
globally defined on H. For this, it suffices to show that
Ψ−1k,iπ
∗Θi = Ψ−1k,jπ
∗Θj on Vi ∩ Vj.
Since π∗Fk,ij = Ψk,iΨ−1k,j on Vi ∩ Vj, we have
Ψ−1k,iπ
∗Θi = Ψ−1k,iπ
∗(Fk,ijΘj)
= Ψ−1k,iπ
∗(Fk,ij)π∗(Θj)
= Ψ−1k,iFk,ijπ
∗Θj
= Ψ−1k,iΨk,iΨ
−1k,jπ
∗Θj
= Ψ−1k,jπ
∗Θj .
The next step is to find the relationship between the divisor of an element
F of A−k(Γ, R) and its corresponding element ΘF of H0(X,M(Ek)), see
(1.1.3) and (2.4.1). First, suppose that P ∈ X ′ corresponds to a point
z0 ∈ V0, then in a local chart ΘF is just Ψ0F . Since Ψ0 lies in GL(n,O(V0)),
then by Remark 1.1.5, we have νp(ΘF ) = νp(F ).
In case P corresponds to an elliptic point e of order ne, then using the no-
tations in the proof of Lemma 2.2.3 and letting α(z) = w, the local parameter
at P is t = wne . Letting ΘF = G(t), we have in terms of w
G(wne) = Ψe(α−1w)Ak,e F (α−1w) =
fke (α−1w) diag (w−m1 , . . . , w−mn)Ak,e F (α−1w),
which implies that
ne νP (ΘF ) = ord0(Fe), (2.5.3)
59
where Fe is defined near w = 0 by
Fe(w) = diag (w−m1 , . . . , w−mn)Ak,eF (α−1w). (2.5.4)
If Fe = (Ak,eF )α−1, then it is clear that ord0(Fe) = orde(F ). Moreover,
if ni denotes the order of the ith component of Fe, 1 ≤ i ≤ n, then we have
ord0(Fe) = ni0 −mi0 ,
for some index i0, 1 ≤ i0 ≤ n . Since orde(F ) = ord0(Fe) = nj for some j,
then we have
ord0(Fe) ≤ nj −mj = orde(F )−mj.
In other words,
mj ≤ orde(F )− ord0(Fe).
Let me = maxmi, 1 ≤ i ≤ n,
F ∗(w) := diag (wme−m1 , . . . , wme−mn)(w−ord0(Fe)Fe(w)),
and
Fe(w) := w−ord0(Fe)Fe(w).
Then F ∗ is holomorphic in w, and by (2.5.4), we have
Fe(w) = w(ord0(Fe)−ord0(Fe)−me) F ∗(w).
If ord0(Fe) − ord0(Fe) − me > 0, then Fe(0) = 0 which contradicts the
definition of the order of Fe at 0, and so ord0(Fe)− ord0(Fe) ≤ me. Hence
mj ≤ orde(F )− ord0(Fe) ≤ me. (2.5.5)
We are led to the following definition.
Definition 2.5.1. The elliptic error τe(F ) of F at the elliptic point e ∈ H is
defined by
τe(F ) =orde(F )− ord0(Fe)
ne.
60
Notice that by (2.5.5), we have
0 ≤ τe(F ) ≤ 1− 1/ne.
Since τe(F ) depends only on the class of e modulo Γ, then:
1. If P ∈ X corresponds to an elliptic point e ∈ H, then we define
τP (F ) = τe(F ).
2. If Θ ∈ H0(X,M(Ek)), then we define the elliptic error of Θ at an
elliptic point P of X by
τ(Θ) = τ(FΘ),
where FΘ is the element of A−k(Γ, R) constructed in Theorem 2.5.2.
3. We set
τ(F ) = τ(ΘF ) =∑P∈E
τP (F )P =∑P∈E
τP (ΘF )P.
Using the fact that ne νP (F ) = orde(F ) and (2.5.3), we have
νP (ΘF ) = νP (F )− τP (F ). (2.5.6)
Finally, suppose that P corresponds to a cusp s. Since F has weight
−k, then from §2.3, we see that the behavior of F at s is by definition the
behavior of ΘF at P . Therefore
νP (ΘF ) = νP (F ). (2.5.7)
Combining (2.5.6) and (2.5.7), we get the following.
Theorem 2.5.3. Let F be an element of A−k(Γ, R), and ΘF be its corre-
sponding element in H0(X,M(Ek)). We have
div(ΘF ) = div(F )− τ(F ).
61
At this point we need to give an explicit formula for the line bundle
Lk = EΓ,χ0,k for even integers k, where χ0 is the trivial character sending all
elements of Γ to 1, see Definition 2.2.4.
It is well-known that to each automorphic form f of even weight k is
associated a (k/2)−form η = f(dz)k/2 on X, see [55].
Proposition 2.5.4. [55] Let f be a nonzero automorphic form for Γ of even
weight k, and η = f(dz)k/2 be the associated (k/2)−form on X. Then
div(f) = div(η) + (k/2)
(∑P∈E
(1− 1/ne)P +∑Q∈S
Q
).
Definition 2.5.2. As in [55], we define the integral part of a rational divisor
D =∑P∈X
aP P , ap ∈ Q,
by
[D] =∑P∈X
[aP ]P,
where [x] is the integral part of x ∈ R.
We have the following useful lemma.
Lemma 2.5.5. [55] Let A =∑
P∈X aP P , ap ∈ Q, be a rational divisor on
X. Then for an integral divisor D on X, i.e., having coefficients in Z, we
have
D ≥ −A ⇐⇒ D ≥ −[A].
Remark 2.5.6. Notice that for Θ in H0(X,M(Ek)) and FΘ the correspond-
ing element of A−k(Γ, R), see Theorem 2.5.3, we have
div(Θ) = [div(FΘ).]
Recall that the line bundle associated to a divisor D on X is denoted by
|D|, see §1.1.2.
62
Proposition 2.5.7. Let KX be the canonical bundle of X,
DS =∑P∈S
P, D′E =∑P∈E
(1− 1/nP )P,
and DS = |DS|. Then for k even, we have
L−k = (k/2)KX + (k/2)DS + | [(k/2)D′E] |.
In particular
L−2 = KX +DS.
Proof. Let Θ be an element of H0(X,M(L−k)), and FΘ be the corresponding
element in Ak(Γ, R). By Proposition 2.5.7 and Remark 2.5.6, we have
div(Θ) = [div(FΘ)] = [ div(ηFΘ) + (k/2)
(∑P∈E
(1− 1/ne)P +∑Q∈S
Q
)],
that is
div(Θ) = div(ηFΘ) + (k/2)DS + [(k/2)D′E].
Hence
|div(Θ)| = |div(ηFΘ)| + (k/2)DS + | [(k/2)D′E] |.
Since L−k = |div(Θ)| and (k/2)KX = |div(ηFΘ)|, we have
L−k = (k/2)KX + (k/2)DS + | [(k/2)D′E] |,
as desired. When k = 2, the divisor (k/2)[D′E] = [D′E] is equal to the zero
divisor, and hence L−2 = KX +DS.
We now come to the main theorem of this chapter. Recall that the
(−k)−cuspidal divisor of R is given by DS,R,−k =∑
P∈S ρ−k,P (R)P , see
Definition 2.4.1.
63
Theorem 2.5.8. If DS,R,−k = |DS,R,−k| is the line bundle associated to
DS,R,−k, then we have
Mk(Γ, R) ∼= H0(X,O(E−k)),
and
Sk(Γ, R) ∼= H0(X,O(−DS,R,−k + E−k)) ∼=Θ ∈ H0(X,O(E−k)) | div(Θ) ≥ DS,R,−k
.
Proof. Let Θ be an element of H0(X,M(E−k)), and FΘ be its corresponding
element in Ak(Γ, R). By Theorem 2.5.3, we have
div(Θ) = div(FΘ)− τ(Θ).
According to Definition 2.5.1, the coefficients of τ(Θ) are in [0, 1), hence
[τ(Θ)] = 0. By Lemma 2.5.5, we have
div(FΘ) ≥ 0 ⇐⇒ div(Θ) ≥ −τ(Θ) ⇐⇒ div(Θ) ≥ −[τ(Θ)] = 0.
Similarly, since DS,R,−k is an integral divisor, we have
div(FΘ) ≥ DS,R,−k ⇐⇒ div(Θ)−DS,R,−k ≥ −τ(Θ) ⇐⇒
div(Θ)−DS,R,−k ≥ −[τ(Θ)] = 0 ⇐⇒ div(Θ) ≥ DS,r,−k .
Now, the result is a straightforward consequence of (1.1.5), Theorem 2.5.2,
Proposition 2.4.1, and Proposition 2.4.2.
As a first consequence we have
Theorem 2.5.9. Let Γ be a Fuchsian group. Then for any simple pair (R, k),
k ∈ Z, the dimensions of Mk(Γ, R) and Sk(Γ, R) are finite if and only if Γ is
of the first kind.
64
Proof. If Γ is of the first kind, then X is a compact Riemann surface.
From Theorem 1.1.12 and Theorem 2.5.8 we deduce that the dimensions
of Mk(Γ, R) and Sk(Γ, R) are finite.
If Γ is not of the first kind, then X is a noncompact Riemann surface.
This implies that any holomorphic vector bundle V on a X is trivial, see
[13]. Therefore H0(X,O(V)) is isomorphic to O(V)n, where n is the rank
of V . But the dimension of O(V) is infinite since any effective divisor is
the divisor of a holomorphic function on X, see [13]. Hence H0(X,O(V) is
an infinite-dimensional vector space. By the Theorem 2.5.8, Mk(Γ, R) and
Sk(Γ, R) can be realized as global holomorphic sections of some holomorphic
vector bundles on X, hence they have infinite dimensions.
From now on Γ will be a Fuchsian group of the first kind, and so X will
be a compact Riemann surface. We set
1. gX for the genus of X.
2. cX = |S| for the number of cuspidal points of X, which is finite since
Γ is of the first kind.
3. eX = |E| for the number of elliptic points of X, which is finite since Γ
is of the first kind.
4. dΓ,R,k = dk and sΓ,R,k = sk for the dimension of Mk(Γ, R) and Sk(Γ, R)
respectively.
Recall that the parity index of R is denoted by ε = ε(R), see Defini-
tion 2.1.3.
Theorem 2.5.10. Let DE =∑
P∈E P , and DE = |DE| be its associated
holomorphic line bundle. Then for any line bundle L on X, we have
h0(−DS + L+ L−(k−ε) + E−ε) ≤ h0(L+ E−k) ≤ h0(DE + L+ L−(k−ε) + E−ε).
65
Proof. Let L be a divisor on X such that L = |L|. From the definition of the
parity index, we see that the space Ak−ε(Γ) of automorphic forms of weight
k − ε for Γ is nontrivial. We fix a choice of a nonzero element f ∈ Ak−ε(Γ),
and we let α be its corresponding element of H0(X,M(L−(k−ε))). Then by
(1.1.5), H0(X,O(L+ E−k) is isomorphic toΘ ∈ H0(X,M(E−k)) | div(Θ) ≥ −L
,
which is, by Theorem 2.5.3 and Theorem 2.5.2, isomorphic to
F ∈ Ak(Γ, R) | div(F )− τ(F ) ≥ −L.
Since the multiplication by f gives an isomorphism
Aε(Γ, R) −→ Ak(Γ, R) : G 7→ f G,
we have
F ∈ Ak(Γ, R) | div(F )− τ(F ) ≥ −L ∼=
G ∈ Aε(Γ, R) | div(fG)− τ(fG) ≥ −L.
On the other hand, using Theorem 2.4.3, we have
div(fG) = div(f) + div(G)−Df,G,
where
Df,G =∑P∈S
aP,f,G P
is a divisor having support in S with coefficients aP,f,G ∈ 0, 1. HenceG ∈ Aε(Γ, R) | div(fG)− τ(fG) ≥ −L
=
G ∈ Aε(Γ, R) | div(f) + div(G)− τ(fG)−Df,G ≥ −L,
which is, by Theorem 2.5.3 and Theorem 2.5.2, isomorphic toΘ ∈ H0(X,M(E−ε)) | div(α)+τ(f)+div(Θ)+τ(GΘ)−τ(fGΘ)−Df,GΘ
≥ −L
66
which is in turn equal toΘ ∈ H0(X,M(E−ε)) | div(α)+div(Θ)+L−Df,GΘ
≥ −(τ(f)+τ(Θ)−τ(fGΘ)).
By Lemma 2.5.5, the latter space is equal toΘ ∈ H0(X,M(E−ε)) | div(α)+div(Θ)+L−Df,GΘ
≥ −[τ(f)+τ(GΘ)−τ(fGΘ)]
which is equal toΘ ∈ H0(X,M(E−ε)) | div(α)+div(Θ)+L ≥ Df,GΘ
−[τ(f)+τ(GΘ)−τ(fGΘ)]
=: Vk,L .
According to Definition 2.5.1, the divisors τ(f), τ(GΘ), and τ(fGΘ) have
coefficients in [0, 1). Hence, the divisor τ(f) + τ(GΘ) − τ(fGΘ) has co-
efficients in (−1, 2 ). If Θ1 is the element of H0(X,M(E−k)) correspond-
ing to the element fGΘ of Ak(R,Γ), then from the relation div(fGΘ) =
div(f) + div(GΘ)−Df,GΘ, and Theorem 2.5.3 applied to GΘ, we have
div(Θ1) + τ(fGΘ) = div(α) + τ(f) + div(Θ) + τ(GΘ)−Df,GΘ,
which is equivalent to
div(Θ1)− div(α)− div(Θ) +Df,GΘ= τ(f) + τ(GΘ)− τ(fGΘ).
Therefore the coefficients of τ(f) + τ(GΘ)− τ(fGΘ) are all integers, and so
lie in 0, 1. Hence
0 ≤ [τ(α) + τ(Θ)− τ(fGΘ)] ≤∑P∈E
P = DE.
By Theorem 2.4.3, the divisor Df,GΘhas support in S with coefficients
in 0, 1, and so Df,GΘ≤ DS. Therefore, we have
Θ ∈ H0(X,M(E−ε)) | div(α) + div(Θ) + L ≥ DS
⊆ Vk,L,
67
and
Vk,L ⊆
Θ ∈ H0(X,M(E−ε)) | div(α) + div(Θ) + L ≥ −DE
.
But (1.1.5) implies thatΘ ∈ H0(X,M(E−ε)) | div(α) + div(Θ) + L ≥ DS
∼=H0(X,O(−DS + L+ L−(k−ε) + E−ε)),
and Θ ∈ H0(X,M(E−ε)) | div(α) + div(Θ) + L ≥ −DE
∼=H0(X,O(DE + L+ L−(k−ε) + E−ε)).
By Theorem 2.5.8, we conclude that
h0(−DS + L+ L−(k−ε) + E−ε) ≤ h0(L+ E−k) ≤ h0(DE + L+ L−(k−ε) + E−ε).
Taking L = −DS,R,−k or the trivial line bundle, then by Theorem 2.5.8,
we get the following.
Corollary 2.5.11. We have
h0(−DS + L−(k−ε) + E−ε) ≤ dk ≤ h0(DE + L−(k−ε) + E−ε).
h0(−DS−DS,R,−k+L−(k−ε) +E−ε) ≤ sk ≤ h0(−DS,R,−k+DE+L−(k−ε) +E−ε).
The above two results will be of fundamental use in the next chapter
when computing the dimensions of Mk(Γ, R) and Sk(Γ, R).
68
Chapter 3
The dimension Formula
3.1 The degree of EΓ,R,k
Our main tool to compute the dimensions of Mk(Γ, R) and Sk(Γ, R) is the
Riemann-Roch theorem applied to the holomorphic vector bundles E−k =
EΓ,R,−k and −DS,R,−k + E−k, where DS,R,−k is the holomorphic line bundle
associated to the (−k)−cuspidal divisor of R, see Definition 2.4.1. Therefore
we need to compute the degree of E−k.
Let Θ1, . . . ,Θn be global meromorphic sections of the holomorphic vector
bundle E−k such that
Θ = Θ1 ∧ . . . ∧Θn (3.1.1)
is a nonzero section of the holomorphic line bundle∧n E−k (the determinant
bundle of E−k ). These always exist according to [50]. By Theorem 2.5.2,
each Θi corresponds to an element Fi of Ak(Γ, R). Hence F = F1 ∧ . . . ∧ Fnis an element of Ank(Γ, det(R)).
Recall that for an integer l, the space of automorphic forms of weight l
and trivial character for Γ is denoted by Al(Γ). Set χ = det(R), and let g
be a fixed element of A2nk(Γ). Then F1 = F 2/g lies in A0(Γ, χ2), and so
f = F ′1/F1 is a weight 2 automorphic form for Γ. Write η = fdz for its
69
corresponding 1−form on X. To find the degree of E−k, we follow the same
strategy used in [43]. First, we express the residues of η in terms of the orders
of Θ and g, then use the fact that the sum of residues of η is 0.
Lemma 3.1.1. Let f be a weight 2 automorphic form, and η be its associated
1−form on X. Suppose that P ∈ X corresponds to x ∈ H∗, then
1. If x ∈ H, then ResP (η) = (1/nP )Resx(f), where nP is the order of P .
2. If x is a cusp s, and ∑n≥ns
an(f) qnh
is the expansion of f at s, qh being the local parameter at s, and h the
cusp width at s, then we have
(2πi/h)ResP (η) = a0(f).
Proof. Suppose that x ∈ H. If
α(z) =z − xz − x
, z ∈ H,
then α(H) is the unit disc D, α(e) = 0. With α(z) = w, the local parameter
at P is t = wnx , where nx = nP is the order of P . Set β := α−1. By
definition, Resx(f) is the residue of the 1−form f(z)dz on H. Writing
β∗(f(z)dz) = h(w)dw
and using the fact that the residue of a differential form is invariant under
changes of local parameter, we deduce that
Resx(f) = Res0(h). (3.1.2)
if η = h1(t)dt, then according to [55], we have
h(w) = nxwnx−1 h1(wnx). (3.1.3)
70
Suppose that the t−expansion of h1 near t = 0 is given by
h1(t) =∑n≥n0
cn tn,
where n0 = ord0(h1), then
h1(wnx) =∑n≥n0
cnwnnx .
Using (3.1.3), we find
Res0(h) = nx Res0(h1) = nx ResP (η),
and from (3.1.2), we conclude that
ResP (η) = (1/nP )Resx(f).
As for 2., if x is a cusp s ∈ H∗ with cusp width h, then if we take
α =
(a b
c d
)∈ SL(2,R) such that α · s = ∞, the local parameter at P is
given by qh = exp(2πiz/h). Suppose that η = F (qh)dqh, then according to
[55], we have
(2π/h)F (qh) = q−1h
∑n≥ns
an(f) qnh . (3.1.4)
Thus
(2π/h) Res0(F ) = a0(f).
Since Res0(F ) = ResP (η), we conclude that
(2πi/h)ResP (η) = a0(f).
For non-cuspidal points of X, we have the following
Lemma 3.1.2. If P ∈ X ′, then
ResP (η) =
2 νP (Θ) − 2 ν−k,P (R) − νP (g) if P is an elliptic point
2 νP (Θ) − νP (g) otherwise.
71
Proof. Let x ∈ H be above P . Then
ResP (η) = Resx(f) = νx(F1) = 2νx(F )− νx(g) = 2νP (F )− νP (g),
and so it suffices to express νP (F ) in terms of νP (Θ). Suppose that x is an
elliptic fixed point e of Γ. Using the notations in the proof of Lemma 2.2.3
and letting α(z) = w, the local parameter at P is t = wne , where ne = nP is
the order of P . If Θi = Gi(t), then by the correspondence in Theorem 2.5.2,
we have in terms of w
Gi(wne) = Ψ−k,e(α
−1w)A−k,e Fi(α−1w)
= f−ke (α−1w) diag (w−m−k,1 , . . . , w−m−k,n)A−k,e Fi(α−1w).
If Θ = G(t), then G = G1 ∧ . . . ∧Gn and
G(wne) = f−nke (α−1w)w−(∑nj=1m−k,j) det(A−k,e)F (α−1w).
Hence
νP (Θ) = νP (F )− (1/ne)n∑j=1
m−k,j = νP (F ) + ν−k,P (R).
In case x is not an elliptic point, then in a local chart Θi is given by
Ψ−k,0Fi, and so Θ = det(Ψ−k,0)F . Since Ψ−k,0 ∈ GL(n,O(Y ′)), we have
νp(Θ) = νp(F ).
As for the cusps, we have:
Lemma 3.1.3. Suppose that P ∈ X corresponds to a cusp s ∈ H∗, then
ResP (η) = 2 νP (Θ) − 2 ν−k,P (R) − νP (g).
Proof. Let α =
(a b
c d
)∈ SL(2,R) such that α · s = ∞, then the local
parameter at P is given by qh = exp(2πiz/h), h being the cusp width at
72
s. Suppose that Θi = Gi(qh) and Θ = G(t), then using notations from the
proof of Lemma 2.2.5, we have
Gi(qh) = f−ks (α−1z) exp(−2πi
z
hB−k,s
)Fi(α
−1z).
This, together with the fact that for a square matrix A,
det( exp(A) ) = exp( trace(A) ),
gives
G(qh) = f−nks (α−1z) exp(−2πi
z
htrace(B−k,s)
)F (α−1z) = .
G(qh) = f−nks (α−1z) qν−k,P (R)
h F (α−1z),
As −trace(B−k,s) is ν−k,P (R).
Since s = α−1 · ∞ = −d/c, we have fs(α−1z) = cα Jα−1(z) for some
nonzero constant cα ∈ C. Hence
G(qh) = cα qν−k,P (R)
h F |nk
(α−1).
Applying the logarithmic derivative on both sides we get
2πi qhh
(∂qhG)(qh)
G(qh)=
2πi ν−k,P (R)
h+
(F |nk
(α−1) )′
F |nk
(α−1),
and since f = 2F ′/F − g′/g, we get
f |2(α−1) = 2(F |
nk(α−1) )
′
F |nk
(α−1)− ( g|
2nk(α−1) )
′
g|2nk
(α−1).
Therefore
f |2(α−1) =4πi qhh
(∂qhG)(qh)
G(qh)− 4πi ν−k,P (R)
h− ( g|
nk(α−1) )
′
g|nk
(α−1).
Comparing the constant terms of the qh−expansions and using Lemma 3.1.1,
we have
(2πi/h)ResP (η) = (4πi/h)νP (Θ) − (4πi/h)ν−k,P (R) − (2πi/h)νP (g)
and hence the formula.
73
Combining the preceding two lemmas with the fact that∑p∈X
ResP (η) = 0,
as X is compact, we get
div(Θ) = DR,−k + (1/2) div(g),
where DR,−k is the (−k)−divisor of R, see Definition 2.2.3. If ηg is the
(nk)−form associated to g, then by Proposition 2.5.4 we have
div(Θ) = DR,−k + (1/2) div(ηg) + (nk/2)(D′E +DS).
Hence
deg(E−k) = deg( div(Θ) ) =
deg(DR,−k) + nk(gX − 1) + (nk/2)( deg(D′E) + deg(DS) ),
where gX is the genus of the compact Riemann surface X. We summarize
the above in the following:
Theorem 3.1.4. Let cX = |S| be the number of cuspidal points of X, gX be
the genus of X. Then the degree of E−k is given by
deg(E−k) =nk
2
(2gX − 2 + cX +
∑P∈E
(1− 1/nP )
)+ deg(DR,−k).
Remark 3.1.5. The number mX defined by
mX = 2gX − 2 + cX +∑P∈E
(1− 1/nP )
is positive, see [55], and 2πmX is called the hyperbolic area of X. Also, it
can be shown that
mX ≥ 1/42,
see [55].
74
According to Definition 2.2.3 the k−divisor of R is given by
DR,k =∑P∈X
νk,P (R)P.
Set
DR,k =∑P∈X
<(νk,P (R))P. (3.1.5)
The above theorem implies that the degree of DR,k is a rational number.
Hence
deg(DR,k) = deg(DR,k). (3.1.6)
But from Definition 2.2.1, and Definition 2.2.2, we see that
−n < <(νk,P (R)) ≤ 0,
for all P ∈ E ∪S, and since the support of DR,k is by definition in E ∪S,
we conclude that
−n (cX + eX) < deg(DR,k) ≤ 0.
Now, using (3.1.6), we find
Proposition 3.1.6. Let DR,k be the k−divisor of R, we have
−n (cX + eX) < deg(DR,k) ≤ 0.
3.2 The holomorphic degree of a vector bun-
dle.
Recall that the dimensions of Mk(Γ, R) and Sk(Γ, R) are respectively denoted
by
75
dk = dΓ,R,k and sk = sΓ,R,k. Applying the Riemann-Roch theorem to the
holomorphic vector bundles E−k and −DS,R,−k + E−k yields
h0(E−k)− h0(KX + E∗−k) = deg(E−k)− n(gX − 1),
h0(−DS,R,−k + E−k)− h0(KX +DS,R,−k + E∗−k) =
deg(−DS,R,−k + E−k)− n(gX − 1),
where E∗−k is the dual holomorphic vector bundle of E−k. Combining this
with Theorem 2.5.8 gives:
dk = deg(E−k)− n(gX − 1) + h0(KX + E∗−k),
sk = deg(−DS,R,−k + E−k)− n(g − 1) + h0(KX +DS,R,−k + E∗−k).
Therefore we need to compute h0(KX + E∗−k) and h0(KX +DS,R,−k + E∗−k).
The key point in computing dk and sk in the case of automorphic forms
(i.e., R is the trivial character χ0) relies on the vanishing of these two di-
mensions, which is in fact equivalent to the non-existence of holomorphic au-
tomorphic forms of negative weight. In higher dimensions, this is no longer
valid since for some representations vector-valued automorphic forms of neg-
ative weight do exist, see (4.2.7). To handle this problem we shall introduce
the notion of the holomorphic degree of a vector bundle, which associates to
each holomorphic vector bundle E , defined over a compact Riemann surface,
an integer d(E) in a such way that:
d(E) < 0 =⇒ H0(X,O(E)) is trivial.
We will give some basic properties of the holomorphic degree, and some useful
bounds of d(E) that will serve in the next section.
For this section X will be an arbitrary compact Riemann surface of genus
g.
76
Proposition 3.2.1. Let E be a vector bundle over a compact Riemann sur-
face X. There exists a constant CE ∈ Z such that
deg(div(Θ)) ≤ CE for all Θ ∈ H0(X,M(E)).
Proof. If no such constant exists, then we will have a sequence
(Θn)n∈N ∈ H0(X,M(E))
such that deg(div(Θn)) > n. Let Dn = div(Θn), and Dn = |Dn| be its
associated holomorphic line bundle. By the Riemann-Roch theorem, we have
h0(Dn) = deg(Dn) + h0(KX −Dn)− (g − 1).
If we take n > deg(KX) = 2(g − 1), then h0(KX −Dn) = 0, and so
h0(Dn) = deg(Dn)− (g − 1).
Hence
h0(Dn) > n− (g − 1). (3.2.1)
Let L(Dn) = f ∈ M(X) | div(f) + Dn ≥ 0. Then the injective homomor-
phism
L(Dn) −→ H0(X,M(E)) : f 7→ fΘn
has its image in H0(X,O(E)). Indeed, if f ∈ L(Dn), then
div(fΘn) = div(f) + div(Θn) = div(f) +Dn ≥ 0.
Hence L(Dn) imbeds in H0(X,O(E)). Now, using the standard fact that
L(Dn) is isomorphic to H0(X,O(Dn)), we deduce that H0(X,O(Dn)) imbeds
in H0(X,O(E)). Therefore h0(E) ≥ h0(Dn). If n > 2(g − 1), then according
to (3.2.1), we have
h0(E) > n− (g − 1),
Thus h0(E) =∞, which contradicts Theorem 1.1.12.
As a consequence, maxdeg(div(Θ)); Θ ∈ H0(X,M(E)) is finite.
77
Definition 3.2.1. Let E be a holomorphic vector bundle over a compact
Riemann surface X, then the holomorphic degree d(E) of E is defined by
d(E) = maxdeg(div(Θ)) : Θ ∈ H0(X,M(E)).
Here are some basic properties of the holomorphic degree:
Proposition 3.2.2. Let X be a compact Riemann surface, E a holomorphic
vector bundle over X and L a holomorphic line bundle over X. Then we
have
1. d(L) = deg(L).
2. d(L+ E) = d(L) + d(E).
3. If d(E) < 0, then h0(E) = 0.
Proof. By definition of d(L), there exits an element α ∈ H0(X,M(L)) such
that
d(L) = deg(div(α)) = deg(L)
as L is a line bundle, which proves 1.
To prove 2., take α ∈ H0(X,M(L)) and Θ ∈ H0(X,M(E)) such that
deg(div(α)) = d(L), deg(div(Θ)) = d(E),
then αΘ ∈ H0(X,M(L+ E)) and we have
deg(div(αΘ)) = deg(div(α)) + deg(div(Θ)) = d(L) + d(E).
Let Θ1 ∈ H0(X,M(L+ E)). Then Θ1/α ∈ H0(X,M(E)) and we have
deg(div(Θ1)) = deg(div(α)) + deg(div(Θ1/α)) ≤
deg(div(α)) + deg(div(Θ)) = d(L) + d(E).
78
Therefore
d(L+ E) ≤ d(L) + d(E) = deg(div(αΘ)) ≤ d(L+ E).
Hence d(L+ E) = d(L) + d(E).
As for 3., suppose that d(E) < 0. If there exists a nonzero element
Θ ∈ H0(X,O(E)), then we will have
0 ≤ deg(div(Θ) ≤ d(E),
which is impossible. Hence h0(E) = 0.
We have the following upper bound of d(E):
Theorem 3.2.3. Let E be a holomorphic vector bundle over X of holomor-
phic degree d(E). We have
d(E) ≤ h0(E) + g − 1.
Proof. Take an element Θ of H0(X,M(E)) such that deg(div(Θ)) = d(E),
and set D = div(Θ) and D = |D|. As in the proof of Proposition 3.2.1, we
have an embedding of H0(X,O(D)) in H0(X,O(E)). Therefore
h0(D) ≤ h0(E).
By the Riemann-Roch theorem, we have
h0(D) = deg(D) + h0(KX −D)− (g − 1),
which implies that
deg(D)− (g − 1) ≤ h0(D).
Since deg(D) = deg(D) = d(E), we have
d(E)− (g − 1) ≤ h0(D) ≤ h0(E),
hence
d(E) ≤ h0(E) + g − 1.
79
Now we will give a lower bound for d(E). Applying the Riemann-Roch
theorem to the holomorphic vector bundle E , we get
h0(E) = h0(KX + E∗) + deg(E)− n(g − 1),
where n is the rank of E . Thus
deg(E)− n(g − 1) ≤ h0(E). (3.2.2)
Let P be any point of X, and L = |P | be the line bundle associated to the
divisor P . If E1 is defined by
E1 = −(d(E) + 1)L+ E ,
then we have
d(E1) = −(d(E) + 1)d(L) + d(E) = −(d(E) + 1) + d(E) = −1.
Hence, by Proposition 3.2.2, we have h0(E1) = 0. By (3.2.2) applied to E1,
we have
deg(E1)− n(g − 1) ≤ h0(E1) = 0.
Using Proposition 1.1.9, we get
−n(d(E) + 1) deg(L) + deg(E)− n(g − 1) ≤ 0,
and since deg(L) = 1, we conclude that
(1/n) deg(E)− g ≤ d(E).
Thus we have proved:
Theorem 3.2.4. Let E be a rank n holomorphic vector bundle over X of
holomorphic degree d(E). We have
deg(E)
n− g ≤ d(E).
80
3.3 The dimension of Mk(Γ, R) and Sk(Γ, R)
We now come to one of the main results of this thesis. The goal is to give
formulas for the dimensions of Mk(Γ, R) and Sk(Γ, R). We will show that
there are some constants k+R,X and k−R,X such that:
1. For k < k−R,X , Mk(Γ, R) will be trivial.
2. For k > k+R,X , we will have an explicit formula for the dimensions of
Mk(Γ, R) and Sk(Γ, R) expressed in terms of some invariants of R and
X.
Recall that
DE =∑P∈E
P, DS =∑P∈S
P,
DR,−k is the (−k)−divisor of R, and that DS,R,−k is the (−k)−cuspidal
divisor of R, see Definition 2.2.3 and Definition 2.4.1. Their associated line
bundles are respectively denoted by DE, DS, DR,−k, and DS,R,−k. Also, recall
that
mX = 2gX − 2 + cX +∑P∈E
(1− 1/nP ),
and that the parity index of R is denoted by ε = ε(R) according to Defini-
tion 2.1.3.
Applying the Riemann-Roch theorem to the holomorphic vector bundles
E−k and −DS,R,−k + E−k, we get:
h0(E−k)− h0(KX + E∗−k) = deg(E−k)− n(gX − 1),
h0(−DS,R,−k+E−k)−h0(KX+DS,R,−k+E∗−k) = deg(−DS,R,−k+E−k)−n(gX−1).
Combining this with Theorem 2.5.8 gives:
dk = deg(E−k)− n(gX − 1) + h0(KX + E∗−k),
sk = deg(−DS,R,−k + E−k)− n(gX − 1) + h0(KX +DS,R,−k + E∗−k),
81
where E∗−k is the dual holomorphic vector bundle of E−k. We have the fol-
lowing formula for E∗−k.
Lemma 3.3.1. If R∗ is the adjoint representation of R, then
E∗R,−k = DE +DS + ER∗,k .
Proof. This is a direct consequence of the Remarks (2.2.2, 2.2.4, 2.2.6), and
Theorem 1.1.3. The only nontrivial part is the contribution of the line bun-
dles DE = |DE| and DS = |DS|. Using notations of Remark 2.2.4 and
Remark 2.2.6, this can be justified by the fact that the local parameter at an
elliptic (resp. cuspidal) point P ∈ X corresponding to e ∈ H (resp. s ∈ H∗ )
is t = wne (resp. exp(2πiz/h) ), and that the section defining DE (resp. DS)
is given near P by the function t (resp. qh ), see [19].
As a consequence we have:
Proposition 3.3.2. We have
dk = deg(E−k)− n(gX − 1) + h0(KX +DE +DS + ER∗,k),
and
sk = deg(−DS,R,−k +E−k)−n(gX−1)+h0(KX +DE +DS +DS,R,−k +ER∗,k).
We will show that for k large enough, the numbers h0(KX +DE +DS +
ER∗,k) and h0(KX +DE +DS +DS,R,−k + ER∗,k) vanish, thus providing exact
formulas for the dimensions dk and sk. We only treat the first case, the
second one follows in a similar way. Using Theorem 2.5.10, we have
h0(KX +DE +DS + ER∗,k) ≤ h0(KX + 2DE +DS + L(k+ε) + ER∗,−ε).
Hence, by Proposition 3.2.2, it suffices to show that
d(KX + 2DE +DS + L(k+ε) + ER∗,−ε) < 0,
82
which is equivalent to
d(KX) + 2 d(DE) + d(DS) + d(L(k+ε)) + d(ER∗,−ε) < 0,
that is
(2gX − 2 + 2eX + cX) + d(L(k+ε)) + d(ER∗,−ε) < 0. (3.3.1)
By Theorem 3.1.4, the degree of L(k+ε) is given by
d(L(k+ε)) = − (k + ε)mX
2+ deg(Dχ0,k+ε),
where χ0 is the trivial character sending all elements of Γ to 1, see Defini-
tion 2.2.4. By definition, the support of the divisor
Dχ0,k+ε =∑P∈X
νk+ε,P (χ0)P
lies in E ∪S, with coefficients in (−1, 0]. Therefore
deg(Dχ0,k+ε) ≤ 0.
Hence, to get (3.3.1), it suffices to have
d(ER∗,−ε) + (2gX − 2 + cX + 2 eX) − (k + ε)mX
2< 0 ,
or equivalently,
2
mX
(d(ER∗,−ε) + 2gX − 2 + cX + 2 eX)− ε < k.
Now, using Theorem 3.1.4 and Proposition 3.3.2, we get
Theorem 3.3.3. Let Γ be a Fuchsian group of the first kind, and X be the
associated compact Riemann surface. If
k+X,R =
2
mX
( d(ER∗,−ε) + 2gX − 2 + cX + 2 eX )− ε,
then
83
1. For k > k+R,X , the dimension of Mk(Γ, R) is given by
dk =n kmX
2− n (gX − 1) + deg(DR,−k).
2. For k > k+X,R + (2/mX) deg(DS,R,−k) , the dimension of Sk(Γ, R) is
given by
sk = dk − n deg(DS,R,−k).
If we relax the lower bound of k, we get the following:
Corollary 3.3.4. For
k ≥ 84(h0(ER∗,−ε) + gX + cX + 2 eX
),
we have
1. The dimension of Mk(Γ, R) is given by
dk =n kmX
2− n (gX − 1) + deg(DR,−k).
2. The dimension of Sk(Γ, R) is given by
sk = dk − n deg(DS,R,−k).
Proof. By Remark 3.1.5, we have
( 2/mX ) ≤ (2) 42 = 84.
Also, by Theorem 3.2.3, we have
d(ER∗,−ε) ≤ h0(ER∗,−ε) + (gX − 1).
By definition, DS,R,−k ≤ DS, and so deg(DS,R,−k) ≤ deg(DS) = cX . Com-
bining these inequalities with the fact that (2gX − 2 + cX)/mX ≤ 1, and
ε ∈ 0, 1, we have
k+X,R ≤ k+
X,R + (2/mX) deg(DS,R,−k) ≤ k+X,R + (2/mX)cX =
84
(2/mX) (d(ER∗,−ε) + 2gX − 2 + 2cX + 2 eX)− ε ≤
(2/mX)(h0(ER∗,−ε) + (gX − 1) + cX + 2 eX
)+ 2/(nmX)(2gX−2 + cX) ≤
84(h0(ER∗,−ε) + (gX − 1) + cX + 2 eX
)+ (2/n) ≤
84(h0(ER∗,−ε) + gX + cX + 2 eX
).
We conclude using Theorem 3.3.3.
Now, recall from Proposition 3.1.6 that
−n (cX + eX) < deg(DR,k) ≤ 0,
so that | deg(DR,k)| < n (cX + eX). Since 0 ≤ deg(DS,R,−k) ≤ cX , then by
Corollary 4.1.3, we have
Corollary 3.3.5. Let Γ be a Fuchsian group of the first kind. The dimensions
dk of Mk(Γ, R), and sk of Sk(Γ, R) are asymptotically equivalent to k. More
precisely, we havedkk−→ nmX
2, as k →∞,
andskk−→ nmX
2, as k →∞.
Combining this result with Theorem 2.5.9, we have
Theorem 3.3.6. Holomorphic, cusp vector-valued automorphic forms exist
for any Fuchsian group Γ and any n−dimensional representation R of Γ.
Using the upper bounds of dk and sk from Corollary 2.5.11, we have
Theorem 3.3.7. Let Γ be a Fuchsian group of the first kind, X be the asso-
ciated compact Riemann surface. If
k−X,R =−2
mX
( d(ER,−ε) + eX )− ε,
then
85
1. For k < k−R,X , the dimension dk of Mk(Γ, R) is zero.
2. For k < k−X,R + (2/mX) deg(DS,R,−k) , the dimension sk of Sk(Γ, R)
is zero.
Proof. By Corollary 2.5.11, we have
dk ≤ h0(DE + L−(k−ε) + E−ε),
sk ≤ h0(−DS,R,−k +DE + L−(k−ε) + E−ε).
Hence it suffices to show that
d(DE + L−(k−ε) + E−ε) < 0, (3.3.2)
d(−DS,R,−k +DE + L−(k−ε) + E−ε) < 0. (3.3.3)
We only treat the first case. As in the proof of the Theorem 3.3.3, we
have
d(DE + L−(k−ε) + ER,−ε) =
d(DE) + d(L−(k−ε)) + d(ER,−ε) ≤
eX +(k + ε)mX
2+ d(E−ε).
To get (3.3.2), it is enough to have
eX +(k + ε)mX
2+ d(E−ε) < 0,
that is
k <−2
mX
( d(ER,−ε) + eX )− ε = k−X,R .
As in Corollary 4.1.3, we have the following simpler upper bound for k:
Corollary 3.3.8. If
k < − 84(h0(ER,−ε) + (gX − 1) + eX
)− 1,
then Mk(Γ, R) and Sk(Γ, R) are trivial.
86
Remark 3.3.9. Notice that for values of k in the interval [k−X,R , k+X,R] we
have:
dk ≥ deg(E−k)− n(gX − 1),
and
sk ≥ deg(−DS,R,−k + E−k)− n(gX − 1).
This a direct consequence of Proposition 3.3.2.
3.4 Finite image representations
Due to its richness and simplicity, the case of finite image representations
has been treated by many authors, see [3, 15]. Our aim in this section is to
provide bounds for the weight k simpler than k+X,R and k−X,R . For example, we
will see that k−X,R can be replaced by 0. For the sake of clarity and to avoid
technical details, we restrict ourselves to the case of even representations,
that is ε = ε(R) = 0.
Let
ΓR := ker(R), ΓR := ±1\ΓR.±1, XR := ΓR\H∗.
Since the image of R is finite, ΓR has finite index in Γ. Therefore, ΓR is a
Fuchsian group of the first kind, and XR is a compact Riemann surface, see
Proposition 1.2.9. Suppose that F ∈ Ak(Γ, R), by definition we have
F |kγ = Rγ F, γ ∈ Γ ,
and so for γ ∈ ΓR we get
F |kγ = F.
Hence all components of F are weight k automorphic forms for ΓR, that
is F ∈ (Ak(ΓR) )n. As ΓR has finite index in Γ, Γ and ΓR have the same
cusps, and the cuspidal behavior of F is the same with respect to Γ and ΓR.
Therefore we have the following two embedding
Mk(Γ, R) −→ (Mk(ΓR) )n : F −→ F, (3.4.1)
87
Sk(Γ, R) −→ (Sk(ΓR) )n : F −→ F. (3.4.2)
Since Mk(ΓR) is trivial for k < 0, we deduce that Mk(Γ, R) and Sk(Γ, R)
are both trivial for k < 0.
The next step is to simplify the bound
k+X,R =
2
mX
(d(ER∗,−ε) + 2gX − 2 + cX + 2 eX)− ε
given by Theorem 3.3.3 (here ε = 0). Suppose that d(ER∗,−ε) − (gX−1) > 0,
and take Θ1 ∈ H0(X,M(ER∗,0)) such that deg(div(Θ1)) = d(ER∗,0). Set
D = div(Θ1) and D = |D| its associated holomorphic line bundle. By the
Riemann-Roch theorem, we have
h0(D) = deg(D) + h0(KX −D)− (gX − 1) ≥ d(ER∗,−ε) − (gX − 1) > 0.
Hence H0(X,O(D)) is nontrivial. Let δ be an element of H0(X,M(D))
such that div(δ) = D, and take σ ∈ H0(X,O(D)). Then f = σ/δ is a
meromorphic function on X satisfying
div(f) + D = div(σ) ≥ 0.
Therefore,
div(f Θ1) = div(f) + D ≥ 0,
and hence Θ = f Θ1 ∈ H0(X,O(ER∗,0)). Let F be the element of M0(Γ, R∗)
corresponding to Θ. Since the elements of M0(ΓR) correspond to holomorphic
functions on the compact Riemann surface XR, we have M0(ΓR∗) ⊆ Cn.
By the embedding of M0(Γ, R∗) in M0(Γ∗R), see (3.4.1), we deduce that Θ
is a nonzero vector in Cn. Therefore, div(F ) is the trivial divisor. Since
[div(F )] = div(Θ) and deg( div(f) ) = 0, we have
0 = deg( div(Θ) ) = deg( div(Θ1) ) = deg(D) = d(ER∗,0).
We conclude that if d(ER∗,−ε) − (gX − 1) > 0, then d(ER∗,0) = 0, and so
gX = 0. Hence, in all cases
d(ER∗,0) ≤ gX .
88
Using the fact that mX ≥ 2gX − 2 + cX , we see that
k+X,R ≤ 2 +
2
mX
(gX + 2 eX) .
Now, by Theorem 3.3.3 we conclude that
Theorem 3.4.1. Let Γ be a Fuchsian group of the first kind, X be the associ-
ated compact Riemann surface, and R be an even finite image representation.
If
fX,R = 2 +2
mX
(gX + 2 eX) ,
then
1. For k > fR,X , the dimension of Mk(Γ, R) is given by
dk =n kmX
2− n (gX − 1) + deg(DR,−k).
2. For k > fX,R + (2/mX) deg(DS,R,−k) , the dimension of Sk(Γ, R) is
given by
sk = dk − n deg(DS,R,−k).
3. For k < 0, Mk(Γ, R) and Sk(Γ, R) are both trivial.
As we have seen in the discussion preceding Theorem 3.4.1, M0(Γ, R) is a
subset of Cn. Combining this with the definition of vector valued automor-
phic forms of weight 0 and multiplier R, and the fact that the elements of
S0(Γ, R) ⊆M0(Γ, R) ⊆ Cn vanish at the cusps, we have
Corollary 3.4.2. Let Γ be a Fuchsian group of the first kind, X be the associ-
ated compact Riemann surface, and R be an even finite image representation.
Then S0(Γ, R) is trivial, and we have
M0(Γ, R) = v ∈ Cn |Rγ(v) = v for all γ ∈ Γ.
89
Chapter 4
Applications
4.1 Generalized automorphic forms
Since the work of M. Knopp and G. Mason in [28], the theory of gen-
eralized automorphic forms has been receiving an increasing interest, see
[25, 26, 29, 32, 33, 44, 45]. We propose here to compute the dimensions of
the vector spaces of generalized automorphic forms. In this section R will be
a 1−dimensional representation (a character) and will be denoted by χ. The
holomorphic vector bundle EΓ,χ,k is then a line bundle, and will be denoted
by IΓ,χ,k = Ik.
The generalized automorphic forms look like classical modular forms with
a multiplier system except for the fact that the character χ need not be
unitary. They can be defined as 1−dimensional vector valued automorphic
forms. More precisely,
Definition 4.1.1. Let (χ, k), k ∈ Z, be a simple pair. Then
1. An element of Ak(Γ, χ) is called a generalized automorphic form for Γ
of multiplier χ and weight k.
2. An element of Gk(Γ, χ) (resp. Mk(Γ, χ), Sk(Γ, χ) ) is called an unre-
90
stricted (resp. holomorphic, cusp) generalized automorphic form for Γ
of multiplier χ and weight k.
Recall that the dimensions of Mk(Γ, χ) and Sk(Γ, χ) are denoted by
dΓ,χ,k = dk and sΓ,χ,k = sk respectively. Also, according to Definition 2.4.1,
DS,χ,−k denotes the holomorphic line bundle associated to the k−cuspidal
divisor of χ
DS,χ,k =∑P∈S
ρk,P (χ)P.
To compute dk and sk, we follow the same steps as in §3.3. Indeed, applying
the Riemann-Roch theorem to I−k and −DS,χ,−k + I−k yields
h0(I−k)− h0(KX + I∗−k) = deg(I−k)− (gX − 1),
h0(−DS,χ,−k+I−k)−h0(KX+DS,χ,−k+I∗−k) = deg(−DS,χ,−k+I−k)−(gX−1),
where I∗−k is the dual holomorphic line bundle of E−k. Combining this with
Theorem 2.5.8, and the fact that I∗−k = −I−k gives:
dk = deg(I−k)− (g − 1) + h0(KX − I−k),
sk = deg(−DS,R,−k + I−k)− (g − 1) + h0(KX +DS,R,−k − I−k).
We will show that for k large enough, the numbers h0(KX − I−k) and
h0(KX +DS,χ,−k−I−k) vanish. Thus we find an exact formula for the dimen-
sions dk and sk. As usual, we only treat the first case. By Proposition 3.2.2,
h0(KX − I−k) is zero if
d(KX − I−k) = d(KX)− d(I−k) < 0,
that is
2(gX − 1)− d(I−k) < 0.
By Theorem 3.1.4, the degree of I−k is given by
d(I−k) = (k/2)mX + deg(Dχ,−k).
91
Hence, d(KX − I−k) < 0 if and only if
2(gX − 1) − (k/2)mX − deg(Dχ,−k) < 0.
By definition, the support of the divisor
Dχ,−k =∑P∈X
ν−k,P (χ)P
lies in E ∪S, with coefficients in (−1, 0], and so
− deg(Dχ,−k) ≤ cX + eX .
Therefore, to get d(KX − I−k) < 0 if suffices to have
2(gX − 1) − (k/2)mX + cX + eX < 0,
in other words,
(2/mX) ( 2(gX − 1) + cX + eX) < k.
But
(2/mX) ( 2(gX − 1) + cX + eX) = mX +∑P∈E
1
nP.
Thus we conclude that h0(KX − I−k) = 0 if
2 +2
mX
∑P∈E
1
nP< k.
Using Theorem 3.1.4 and Proposition 3.3.2, we have
Theorem 4.1.1. Let Γ be a Fuchsian group of the first kind, X be its compact
Riemann surface, and let
mX = 2gX − 2 + cX +∑P∈E
(1− 1/nP ),
l+X,χ = 2 +2
mX
∑P∈E
1
nP.
Then:
92
1. For k > l+X,χ , the dimension of Mk(Γ, χ) is given by
dk = (k/2)mX − (gX − 1) + deg(Dχ,−k).
2. For k > l+X,χ + (2/mX) deg(DS,χ,−k) , the dimension of Sk(Γ, χ) is
sk = dk − deg(DS,χ,−k).
We have the following simplification for the lower bound of k.
Corollary 4.1.2. Suppose that gX ≥ 1. Then
1. For k > 4, the dimension of Mk(Γ, χ) is
dk = (k/2)mX − (gX − 1) + deg(Dχ,−k).
2. For k > 6, the dimension of Sk(Γ, χ) is
sk = dk − deg(DS,χ,−k).
Proof. First we show that mX is greater than cX and∑
P∈E(1/nP ). Indeed,
if gX ≥ 1, then 2(gX − 1) ≥ 0, and hence all the terms in the expression of
mX are nonnegative. This implies that mX ≥ cX . If Γ has an elliptic point
P , then by definition nP ≥ 2. Therefore∑P∈E
1
nP≤∑P∈E
(1− 1/nP ) ≤ mX .
If Γ is torsion-free, this inequality is obviously satisfied, since∑
P∈E(1/nP )
is interpreted as 0. Now, we have
l+χ,X = 2 + (2/mX)∑P∈E
(1/nP ) ≤ 2 + 2 = 4.
Also, since DS,R,−k ≤ DS, we have deg(DS,R,−k) ≤ deg(DS) = cX , and so
l+χ,X + (2/mX) deg(DS,χ,−k) ≤ 4 + (2/mX) cX ≤ 4 + 2 = 6.
The formulas follow from Theorem 4.1.1.
93
When gX is zero, we have the following.
Corollary 4.1.3. Suppose that gX = 0. If cX ≥ 2, then
1. For k > 4, the dimension of Mk(Γ, χ) is
dk = (k/2)mX + deg(Dχ,−k) + 1.
2. For k > 4 + max6, 4/mX, the dimension of Sk(Γ, χ) is
sk = dk − deg(DS,χ,−k).
Moreover, we have max6, 4/mX ≤ 168.
Proof. Suppose that gX = 0. If cX ≥ 2, then all the terms in the expression
of mX are nonnegative. As in the preceding proof, we conclude that l+χ,X ≤ 4.
Since deg(DS,R,−k) ≤ cX , we have
l+χ,X + (2/mX) deg(DS,χ,−k) ≤ 4 + (2/mX) cX .
Suppose that cX > 2. Since mX ≥ (cX − 2), we have
(2/mX) cX ≤ 2cX/(cX − 2) = 2 + 4/(cX − 2) ≤ 6.
Hence
l+χ,X + (2/mX) deg(DS,χ,−k) ≤ 4 + max6, 4/mX.
The result follows from Theorem 4.1.1.
By Remark 3.1.5, we have
( 4/mX ) ≤ (4) 42 = 168.
Hence max6, 4/mX ≤ 168.
Remark 4.1.4. In the remaining case, that is gX = 0 and cX < 2, one can
show that Γ is generated by elliptic elements, and so the image of χ is fi-
nite. Therefore, our generalized automorphic forms are classical automorphic
forms.
94
Rewriting Theorem 3.3.7 in the 1-dimensional case gives the following:??
Theorem 4.1.5. Let Γ be a Fuchsian group of the first kind, X be its compact
Riemann surface, and
mX = 2gX − 2 + cX +∑P∈E
(1− 1/nP ).
Then:
1. For k < 0, the dimension dk of Mk(Γ, χ) is zero.
2. For k < (2/mX) deg(DS,χ,−k) , the dimension sk of Sk(Γ, χ) is zero.
Corollary 4.1.6. If k < 0, then Mk(Γ, χ) and Sk(Γ, χ) are trivial.
4.2 ρ−equivariant functions
Throughout this section, Γ will be a Fuchsian subgroup of SL(2,R), and
ρ : Γ −→ GL(2,C) will be a 2-dimensional complex representation of Γ.
Definition 4.2.1. A meromorphic function h on H is called a ρ−equivariant
function with respect to Γ if
h(γ · z) = ρ(γ) · h(z) for all z ∈ H , γ ∈ Γ ,
where the action on both sides is by Mobius transformations. The set of
ρ−equivariant functions for Γ will be denoted by Eρ(Γ).
In the case ρ is the defining representation of Γ, that is ρ(γ) = γ for all
γ ∈ Γ, then elements of Eρ(Γ) are simply called equivariant functions. These
were studied extensively in [11, 12, 51, 54] and have various connections to
modular forms, quasi-modular forms, and elliptic functions. In particular,
one shows that the set of equivariant functions for a Fuchsian group Γ without
the trivial one h0(z) = z has a vector space structure isomorphic to the space
95
of weight 2 unrestricted automorphic forms for Γ. Nontrivial examples are
constructed from automorphic forms. Indeed, if f is a weight k automorphic
form for Γ, then
hf (z) = z + kf(z)
f ′(z)
is equivariant for Γ. These are referred to as the rational equivariant functions
[12].
The first main result is that ρ−equivariant functions are parameterized by
2-dimensional unrestricted vector-valued automorphic form of multiplier ρ.
More precisely, if F = (f1, f2)t is an unrestricted vector-valued automorphic
form of multiplier ρ and an arbitrary weight such that f2 is nonzero, then
hF (z) = f1(z)/f2(z) is a ρ-equivariant function. We will show that, in fact,
every ρ−equivariant function arises in this way. To achieve this parametriza-
tion, we use the fact that the Schwarz derivative of a ρ−equivariant function
is a weight 4 unrestricted automorphic form for Γ, as well as the existence of
global solutions to a certain second degree differential equation. The second
main result of this section is that the ρ−equivariant functions always exist.
This will follow from Theorem 2.5.9 and Theorem 3.3.5.
We end this section by constructing examples of ρ−equivariant functions,
specially when ρ is the monodromy representation of second degree ordinary
differential equations.
4.2.1 Differential equations
Let D be a domain in C and let f be a meromorphic function on D. Its
Schwarz derivative, S(f), is defined by
S(f) =
(f ′′
f ′
)′− 1
2
(f ′′
f ′
)2
.
This is an important tool in projective geometry and differential equations.
The main properties that will be useful to us are summarized as follows ( see
[40] for more details):
96
Proposition 4.2.1. We have
1. If y1 and y2 are two linearly independent solutions to a differential
equation y′′ + Qy = 0 where Q is a meromorphic function on D, then
S(y1/y2) = 2Q.
2. If f and g are two meromorphic functions on D, then S(f) = S(g) if
and only if f =ag + b
cg + dfor some
(a b
c d
)∈ GL(2,C).
3. S(f γ)(z) = (cz + d)4S(f) provided γ · z ∈ D, where γ =
(∗ ∗c d
).
In particular, we have
Proposition 4.2.2. If f is a ρ-equivariant for Γ, then S(f) is an unrestricted
automorphic form of weight 4 for Γ.
Now, consider the second order ordinary differential equation (ODE)
x′′ + Px′ +Qx = 0,
where P and Q are holomorphic functions on D. This ODE has two linearly
independent holomorphic solutions on D if D is simply connected. For a
fixed z0 ∈ D, set
y(z) = x(z) exp
(∫ z
z0
1
2P (w)dw
).
The above ODE reduces to an ODE in normal form
y′′ + gy = 0 , (4.2.1)
with
g = Q− 1
2P ′ − 1
4P 2.
When the domain D is not simply connected, we may not expect to find
global solutions to (4.2.1) on D. However, under some conditions on g, global
solutions do exist as it is illustrated in the following theorem which will be
crucial for the rest of this section.
97
Theorem 4.2.3. Let D be a domain in C. Suppose h is a nonconstant
meromorphic function on D such that S(h) is holomorphic in D, and let
g = 12S(h). Then the differential equation
y′′ + gy = 0
has two linearly independent holomorphic solutions in D.
Proof. Let Ui, i ∈ I be a covering of D by open discs with dimV (Ui) = 2
for all i ∈ I where V (Ui) denotes the space of holomorphic solutions to
y′′ + gy = 0 on Ui. Choose Li and Ki to form a basis for V (Ui). Using
property (1) of Proposition 4.2.1, we have S(Ki/Li) = 2g = S(h) on Ui.
Now, using property (2) of Proposition 4.2.1, we have Ki/Li = αi · h for
αi ∈ GL(2,C). On the other hand, on each connected component W of
Ui ∩ Uj, we have
(Ki, Li)t = αW (Kj, Lj)
t , αW ∈ GL(2,C) ,
since each of (Ki, Li) and (Kj, Lj) is a basis of V (W ). Hence, on W we have
Ki
Li= αW ·
Kj
Lj,
and therefore
αih = αWαjh .
It follows that
αiα−1j = αW
as h is meromorphic and nonconstant and thus it takes more than three
distinct values on the domain D. Therefore, αW does not depend on W .
Moreover, on Ui ∩ Uj we have
α−1i (Ki, Li)
t = α−1j (Kj, Lj)
t . (4.2.2)
If we define f1 and f2 on Ui by
(f1, f2)t = α−1i (Ki, Li)
t ,
98
then using (4.2.2), we see that f1 and f2 are well defined all over D and they
are two linearly independent solutions to y′′+ gy = 0 on all of D as they are
linearly independent over Ui.
4.2.2 The correspondence
The aim of this subsection is to prove the existence of ρ−equivariant functions
as well as their parametrization by unrestricted vector-valued automorphic
forms of multiplier ρ.
We start with the following proposition.
Proposition 4.2.4. Let F (z) = (f1(z), f2(z))t be a nonzero unrestricted
vector-valued automorphic form for ρ of a certain weight. If f2 is nonzero,
then hF := f1/f2 is a ρ−equivariant function for Γ.
Proof. Suppose that F is of weight k, k ∈ Z, and that ρ(γ) =
(aγ bγcγ dγ
),
γ ∈ Γ. Since F |kγ = ρ(γ)F , we have
f1(γ) = Jkγ (aγ f1 + bγ f2) , (4.2.3)
and
f2(γ) = Jkγ (cγ f1 + dγ f2) . (4.2.4)
This implies thatf1(γ)
f2(γ)=
aγ f1 + bγ f2
cγ f1 + dγ f2
,
that is
hF (γ) =aγ hF + bγcγ hF + dγ
= ρ(γ) · hF .
The slash operator has the following useful property known as Bol’s iden-
tity.
99
Proposition 4.2.5. [27] Let r be a nonnegative integer, F (z) a complex
function and γ ∈ SL(2,C), then
(F |−rγ)(r+1)(z) = F (r+1)|r+2γ(z) .
As a consequence, we have
Corollary 4.2.6. Let r be a nonnegative integer, g an unrestricted automor-
phic form of weight 2(r+ 1) for Γ, and D a domain in H that is stable under
the action of Γ. Denote by Vr(D) the solution space on D to the differential
equation
f (r+1) + gf = 0 .
Suppose that f ∈ Vr(D). Then for all γ ∈ Γ, we have f |−rγ ∈ Vr(D).
Proof. Suppose that f ∈ Vr(D). Using Bol’s identity and the fact that
g|2(r+1)
= g, we have
(f |−rγ)(r+1) + g (f |−rγ) = f (r+1)|(r+2)
γ + g (f |−rγ) =
J−(r+2)γ
(f (r+1)(γ) + (g|
2(r+1)γ) f(γ)
)=(f (r+1) + g f
)|(r+2)
γ = (0) |(r+2)
γ = 0.
Corollary 4.2.7. The operator |−r provides a representation ρr of Γ in
GL(Vr(D)). Moreover, if f1, f2,. . . ,fr+1 form a basis of Vr (if the basis
exists), then
F = (f1, f2, . . . , fr+1)t
is an unrestricted vector valued automorphic form of multiplier ρr and weight
−r for Γ.
Recall from Definition 2.1.1 that the space of unrestricted vector-valued
automorphic forms for Γ of multiplier ρ and weight k ∈ Z, is denoted by
Gk(Γ, ρ). Our first main result of this section is the following.
100
Theorem 4.2.8. The map
G−1(Γ, ρ) −→ Eρ(Γ)
F 7→ hF
is surjective.
Proof. Suppose that h is a ρ−equivariant function for Γ. According Proposi-
tion 4.2.2, its Schwarz derivative S(h) is an unrestricted automorphic form of
weight 4 for Γ. Let g = 12S(h) and D the complement in H of the set of poles
of g. Then D is a domain that is stable under Γ since g is an unrestricted
automorphic form for Γ.
Using the same notation as in the previous section, we have, for r = 1,
S(f1/f2) = S(h) where f1, f2 are two linearly independent solutions in
V (D) provided by Theorem 4.2.3. Hence, by Proposition 4.2.1
f1
f2
= α · h , α ∈ GL(2,C) .
Also, using Corollary 4.2.7 with r = 1, we deduce that F1 = (f1, f2)t is a
vector-valued unrestricted automorphic form for Γ of multiplier ρ1 and weight
−1. Therefore,
α−1ρ1α = ρ .
Hence F = α−1F1 lies in G−1(Γ, ρ), and hF = h on D. Since g has only
double poles, then by looking at the form of the solutions near a singular
point, and using the fact that f1 and f2 are holomorphic and thus single-
valued, we see that f1 and f2 can be extended to meromorphic functions on
H.
Remark 4.2.9. If there exists an unrestricted automorphic form f of weight
k + 1 for Γ, then (f1, f2) −→ (ff1, ff2) yields an isomorphism between
G−1(Γ, ρ) and Gk(Γ, ρ), and therefore, the above surjection in the theorem
extends to Gk(Γ, ρ) whenever it is nontrivial.
101
We now prove the existence of ρ−equivariant functions. Recall that a
representation is called decomposable if it is the direct sum of two represen-
tations. If ρ is decomposable, then ρ = χ1 ⊕ χ2 for some two characters χ1
and χ2 on Γ. If k is a large enough integer, then by Theorem 2.5.9 and Theo-
rem 3.3.5, Mk(Γ, χ2) is nontrivial. Let f2 be a nonzero element of Mk(Γ, ρ2),
then for any f1 ∈ Mk(Γ, ρ1), we have F (z) = (f1(z), f2(z))t ∈ Mk(Γ, ρ).
Hence, by Proposition 4.2.4, hF = f1/f2 is a ρ−equivariant function for Γ.
Suppose that ρ is indecomposable. As usual, modulo a conjugation in
GL(2,C), the representation ρ is either even, odd, or a direct sum ρ = ρ+⊕ρ−of an even and an odd representations. Since ρ is indecomposable, the latter
case is excluded. Hence, by Theorem 2.5.9 and Theorem 3.3.5, there exists
k0 ∈ N such that for all k ≥ k0, Mk(Γ, ρ) is nontrivial.
We now show that for a certain integer k ≥ k0, there exists an element
F (z) = (f1(z), f2(z))t ∈ Mk(Γ, ρ) such that f2 is nonzero. Suppose the
converse is true, then for k ≥ k0, any nontrivial element F ∈ Mk(Γ, ρ)
has the form F = (f1, 0)t for some nonzero holomorphic function f1 on H.
According to (4.2.3), we have for all γ ∈ Γ
f1|kγ = aγ f1, (4.2.5)
where ρ(γ) =
(aγ bγcγ dγ
). Using the fact that |k is a linear operator, we deduce
that the map
α : Γ −→ C∗ γ 7→ aγ
is character on Γ. Thus f1 ∈ Gk(Γ, α). Also, from (4.2.4), we see that cγ = 0
for all γ ∈ Γ, and so F and f1 have the same cuspidal behavior with respect
to Γ. Since F ∈ Mk(Γ, ρ), we conclude that f1 ∈ Mk(Γ, α), and hence we
have an embedding of Mk(Γ, ρ) into Mk(Γ, α) given by F = (f1, 0)t −→ f1.
Consequently, for all k ≥ k0, we have
dΓ,ρ,k ≤ dΓ,α,k, (4.2.6)
where dΓ,ρ,k and dΓ,α,k respectively denote the dimensions of Mk(Γ, χ) and
102
Mk(Γ, α). But from Theorem 3.3.5, we know that
dΓ,ρ,k
k−→ mX , as k →∞,
anddΓ,α,k
k−→ mX
2, as k →∞.
where by see Remark 3.1.5
mX = 2gX − 2 + cX +∑P∈E
(1− 1/nP ) ≥ 1/42
Hence for k large enough, we have
dΓ,ρ,k
k>
dΓ,α,k
k,
that is
dΓ,ρ,k > dΓ,α,k,
which contradicts (4.2.6). Therefore, for a certain integer k ≥ k0, there exists
an element F (z) = (f1(z), f2(z))t ∈ Mk(Γ, ρ) such that f2 is nonzero. Then
by Proposition 4.2.4, hF = f1/f2 is a ρ−equivariant function for Γ.
Thus we have proved the following existence theorem for ρ−equivariant
functions.
Theorem 4.2.10. There exists a ρ−equivariant function hF = f1/f2, where
F (z) = (f1(z), f2(z))t ∈Mk(Γ, ρ) for a certain nonnegative integer k.
4.2.3 Examples
In this subsection, we shall illustrate our parametrization by constructing
examples of unrestricted vector valued automorphic forms, and their corre-
sponding ρ−equivariant functions.
Recall that the Γ−equivariant functions are simply the ρ0−equivariant
functions for Γ, with ρ0(γ) = γ for all γ ∈ Γ. One can easily see that
F0(z) =
(z
1
), z ∈ H
103
is an unrestricted vector-valued automorphic form for Γ of multiplier ρ0 and
weight −1, and
hF0(z) = z, z ∈ H.
As we mentioned before, if we take a weight k unrestricted automorphic
form f for Γ, then
hf (z) = z + kf(z)
f ′(z), ∈ H
is an equivariant function for Γ. This corresponds to the (k− 1) unrestricted
vector-valued automorphic form for Γ of multiplier ρ0 given by
F (z) =
(k f + z f ′
f ′
), z ∈ H. (4.2.7)
Our next example is the monodromy representation of a differential equa-
tion. Indeed, let U be a domain in C such that C \ U contains at least two
points. The universal covering of U is then H as it cannot be P1(C) because
U is noncompact and it cannot be C because of Picard’s theorem.
Let π : H −→ U be the covering map. We consider the differential
equation on U
y′′ + Py′ +Qy = 0, (4.2.8)
where P and Q are two holomorphic functions on U . This differential equa-
tion has a lift to Hy′′ + π∗Py′ + π∗Qy = 0 . (4.2.9)
Let V be the solution space to (4.2.9) which is a 2-dimensional vector space
since H is simply connected. Let γ be a covering transformation in Deck(H/U)
which is isomorphic to the fundamental group π1(U) and let f ∈ V . Then
γ∗f = f γ−1 is also a solution in V . This defines the monodromy represen-
tation of π1(U):
ρ : π1(U) −→ GL(V ) .
If f1 and f2 are two linearly independent solutions in V , we set F = (f1, f2)t.
Then we have
F γ = ρ(γ)F,
104
that is F ∈ G0 (π1(U), ρ). Therefore, the quotient f1/f2 is a ρ−equivariant
function on H for the group π1(U), which is a torsion-free Fuchsian group.
105
Bibliography
[1] M.F. Atiyah, Vector bundles on an elliptic curve, Proc. London Math.
Soc. 7 (1957) 414–452.
[2] W. Barth, C. Peters: A. Van de Ven, Compact Complex Surfaces,
Ergebnisse der Mathematik und ihrer Grenzgebiet 3. Folge. Band 4 A
Series of Modem Surveys in Mathematics, Springer-Verlag, 1984.
[3] P. Bantay, The dimension of spaces of vector-valued modular forms
of integer weight, Lett. Math. Phys (2013) 103:1243–1260.
[4] P. Bantay and T. Gannon. Vector-valued modular functions for the
modular group and the hypergeometric equation, Commun. Number
Th. Phys. 1 (2008) 637–666.
[5] A. F. Beardon, The geometry of discrete groups, Springer-Verlag,
New-York, 1983.
[6] R. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimen-
sions, Duke Math. J. 97, no. 2, 219–233 (1999), Correction: Duke
Math. J. 105, no. 1, 183–184 (2000).
[7] R. Borcherds, Reflection groups of Lorentzian lattices, Duke Math. J.
104, no. 2, 319–366 (2000).
106
[8] C. Dong, H. Li, G. Mason, Modular-invariance of Trace Functions
in Orbifold Theory and Generalized Moonshine, Comm. Math. Phys.
214 (2000), 1–56.
[9] W. Eholzer, N.-P. Skoruppa, Modular invariance and uniqueness of
conformal characters, Comm. Math. Phys. 174 No. 1 (1995), 117–136.
[10] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Prog. Math.
55 (Birkhauser, Boston, 1985).
[11] A. Elbasraoui, A. Sebbar, Equivariant forms: Structure and geometry.
Canad. Math. Bull. Vol. 56 (3), (2013) 520–533.
[12] A. Elbasraoui, A. Sebbar, Rational equivariant forms. Int J. Number
Th. 08 No. 4(2012), 963–981.
[13] O. Forster, Lectures on Riemann surfaces, Graduate Texts in Mathe-
matics, 81. Springer-Verlag, New York, 1991.
[14] J. Fischer, An approach to the Selberg trace formula via the Selberg
zeta function, Lecture Notes in Mathematics, 1253, Springer Verlag,
Berlin, Heidelberg, New York (1987)
[15] E. Freitag, Riemann surfaces: Sheaf Theory, Riemann Surfaces, Au-
tomorphic Forms. CreateSpace Independent Publishing Platform, 1
edition (2014)
[16] J. H. Gallier, Logarithms and Square Roots of Real Matrices (2008).
Technical Reports (CIS). Paper 876.
[17] T. Gannon, The theory of vector-valued modular forms for the mod-
ular group, arXiv:1310.4458
[18] F.R. Gantmacher, The Theory of Matrices, Vol. I. AMS Chelsea, First
edition, 1977.
107
[19] P. Griffiths, J. Harris, Principles of algebraic geometry. Wiley Classics
Library. John Wiley & Sons, Inc., New York, 1994.
[20] R. Gunning, Introduction to holomorphic functions of several vari-
ables (volume I) Function Theory. Wadsworth & Brooks/Cole, 1990.
[21] R. Gunning, Introduction to holomorphic functions of several vari-
ables (volume III) Homological Theory. Wadsworth & Brooks/Cole,
1990.
[22] D. A. Hejhal, The Selberg trace formula for PSL(2,R), Vol. 1. Lecture
Notes in Mathematics 548, Springer-Verlag, Berlin-New York, 1976.
[23] D. A. Hejhal, The Selberg trace formula for PSL(2,R), Vol. 2. Lecture
Notes in Mathematics 1001, Springer-Verlag, Berlin-New York, 1983.
[24] S. Katok, Fuchsian groups. The University of Chicago Prees, Chicago
and London, 1992.
[25] W. Kohnen, G. Mason, On generalized modular forms and their ap-
plications. Nagoya Math. J. 192 (2008), 119–136.
[26] W. Kohnen, Y. Martin, On product expansions of generalized modular
forms. Ramanujan J (2008) 15: 103–107.
[27] M. Knopp, Polynomial automorphic forms and nondiscontinuous
groups, Trans. Amer. Math. Soc. 123 (1966), 506-520. MR 34:341.
[28] M. Knopp, G. Mason, Generalized modular forms. J. Number Theory
99 (2003), no. 1, 1–28.
[29] M. Knopp, G. Mason, Parabolic generalized modular forms and their
characters. Int. J. Number Theory 5 (2009), no. 5, 845–857.
[30] M. Knopp, G. Mason, Vector-valued modular forms and Poincare se-
ries, Illinois J. of Math. 48 (2004), 1345–1366.
108
[31] M. Knopp, G. Mason, Logarithmic vector-valued modular forms, Acta
Arith. 147 (2011) 261–282.
[32] M. Knopp, J. Lehner and W. Raji, Eichler cohomology for generalized
modular forms, Int. J. Number Theory 5 (2009) 1049–10.
[33] M. Knopp, W. Raji, Eichler cohomology for generalized modular
forms II. Int. J. Number Theory Vol. 6, no. 5 (2010) 1083–1090
[34] M. Kuga, G. Shimura, On vector differential forms attached to auto-
morphic forms, J. Math. Soc. Japan 12 (1960), 258–270.
[35] C. Marks, Irreducible vector-valued modular forms of dimension less
than six, Illinois J. Math. 55 (2011) 1267–1297.
[36] C. Marks, G. Mason, Structure of the module of vector-valued mod-
ular forms, J. London Math. Soc. 82 (2010)32–48.
[37] G. Mason, Vector-valued modular forms and linear differential oper-
ators, Int J. Number Th. 3 (2007) 377–390.
[38] F. Martin, R. Emmanuel, Formes modulaires et periodes, Seminaires
Et Congres, collection SMF, num 12.
[39] C. Meneses, On vector-valued Poincare series of weight 2.
arXiv:1408.6494v2 [math.CV].
[40] J. McKay; A. Sebbar Fuchsian groups, Automorphic functions and
Schwarzians. Math. Ann. 318 (2), (2000) 255–275.
[41] M. Miyamoto, Modular invariance of vertex operator algebras satis-
fying C2-cofiniteness, Duke Math. J. 122 No. 1 (2004), 51–91.
[42] Y. Choie, M. H. Lee, Symmetric tensor representations, quasimodular
forms, and weak Jacobi forms. arXiv:1007.4590v1 [math.NT].
109
[43] V. B. Mehta, C. S. Seshadri, Moduli of vector bundles on curves with
parabolic structures, Math. Ann. 248 (1980), 205–239.
[44] W. Raji, Fourier coefficients of generalized modular forms of negative
weight. Int. J. Number Theory 5 (2009), no. 1, 153–160.
[45] W. Raji, Construction of generalized modular integrals, Funct. Ap-
prox. Comment. Math. 41(2) (2009) 105–112.
[46] R. Rankin, Modular Forms and Functions, Cambridge Univ. Press,
Cambridge, 1977.
[47] W. Roelcke, Das Eigenwertproblem der automorphen Formen in der
hyperbolischen Ebene I, Math. Ann. 167(1966) pp.292-337.
[48] W. Roelcke, Das Eigenwertproblem der automorphen Formen in der
hyperbolischen Ebene II, Math. Ann. 168(1967) pp.261-324.
[49] A. Selberg, On the estimate of Fourier coefficients of modular forms,
Proc. Sympos. Pure Math VIII (Amer. Math. Soc. Providence, 1965)
pp. 1–15.
[50] H. Saber, A. Sebbar, Vector-valued automorphic forms and vector
bundles. arXiv:1312.2992v3 [math.NT].
[51] H. Saber, Equivariant functions for the Mobius subgroups and appli-
cations. arXiv:1412.8100 [math.NT].
[52] H. Saber, A. Sebbar, Equivariant functions and vector-valued modular
forms. Intl J. Number Th. 10, 949 (2014)
[53] H. Saber, A. Sebbar, On the critical points of modular forms. J. Num-
ber Theory 132 (2012) 1780–1787.
[54] A. Sebbar, A. Sebbar, Equivariant functions and integrals of elliptic
functions. Geom. Dedicata 160 (1), (2012) 37–414.
110
[55] G. Shimura, Introduction to the Arithmetic Theory of Automorphic
Functions (Princeton University Press, 1971).
[56] G. Shimura, Sur les integrales attaches aux formes automorphes, J.
Math. Soc. Japan 11 (1959), 291–311.
[57] N.-P. Skoruppa, Uber den Zusammenhang zwischen Jacobiformen und
Modulformen halbganzen Gewichts, Ph.D. Thesis, Universitat Bonn
(1984).
[58] G. Springer, Introduction to Riemann surfaces, Addison-Wesley,
Reading, 1957.
111