Hecke Operators and Orthogonality on Γ1[N ]
Diplomarbeit
von
Christian Weiß
Universität Heidelberg
Fakultät für Mathematik
Dezember 2008
Betreuer: Prof. Dr. Winfried Kohnen
Danksagung
Vorab ein paar Worte auf Deutsch: Ohne die Hilfe und Unterstützung zahlreicher Personen wäre diese Arbeit nie in dieser Form entstanden: Zuallerst
möchte ich mich deshalb ganz herzlich bei Professor Dr. Winfried Kohnen
bedanken, der mir dieses Thema gestellt hat und der diese Arbeit in hervorragender Art und Weise betreut hat. Er hat sich trotz seines vollen Terminplans
immer Zeit für mich genommen und somit sehr viel zum Gelingen dieser Arbeit
beigetragen.
Mein wichtigster Dank gilt jedoch meinen Eltern, die mich mein ganzes Leben
und Studium hindurch begleitet haben und ohne deren Fürsorge und Unterstützung auf so vielen Ebenen ich nie so weit gekommen wäre. Doch ”leider
läßt sich eine wahrhafte Dankbarkeit mit Worten nicht ausdrücken.” (Johann
Wolfgang von Goethe)
Weiterhin danke ich Kilian Kilger, der mir einige Male für Gespräche zur
Verfügung stand und dessen Wissen mir stets weitergeholfen hat. Für das Korrekturlesen bedanke ich mich bei Clemens Kienzler, Nele Müller und Isabelle
Petrik. Außerdem danke ich Toni Gossmann für einige hilfreiche Hinweise.
Der bischöflichen Studienförderung Cusanuswerk danke ich für die finanzielle
Unterstützung meines Studiums, vor allem aber für zahlreiche Veranstaltungen, an denen ich mit Freude und Begeisterung teilgenommen habe und die
mein Leben sehr bereichert haben.
2
Contents
Introduction
5
1 Basic Concepts
1.1 The Modular Group . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7
8
2 Modular Forms for Congruence Subgroups
14
2.1 Congruence Subgroups . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Hecke Operators
3.1 The Double Coset Operator . . . . . . . . . . . . . . . . . . . .
3.2 Hecke Operators for Prime Numbers . . . . . . . . . . . . . . .
3.3 The Tn Operators . . . . . . . . . . . . . . . . . . . . . . . . . .
24
24
26
30
4 Petersson Scalar Product
32
4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Adjoints of the Hecke Operators . . . . . . . . . . . . . . . . . . 35
5 Orthogonality
37
5.1 Oldforms and Newforms . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Hecke Operators and Orthogonality . . . . . . . . . . . . . . . . 40
5.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 44
List of Symbols
45
3
Bibliography
47
4
Introduction
In this diploma thesis we analyze several orthogonality relations between cusp
forms and the connection to the theory of Hecke operators. More precisely, we
prove as a main result that in level N pr , the space of oldforms of Γ1 [N ] which
are orthogonal under the Petersson scalar product to the push forward of the
space of oldforms of level N under the map f (z) 7→ f (pz) is exactly the kernel
of the p-th Hecke operator Tp acting on the space of forms of level N . Although
we do not see any direct application of our result, we nevertheless think that
this is interesting from a theoretical point of view. The result and its proof
goes back to ideas of Winfried Kohnen in [KW2] who proved the above in the
special case of Γ0 [N ]. The main purpose of this diploma thesis is therefore to
generalize his result.
As for prerequisites, the reader is expected to be familiar with the main concepts of complex analysis and group theory. For the convenience of the reader,
we repeat some well-known material from the area of modular forms in the
first four chapters. There, we will mainly concentrate on those definitions and
theorems that are relevant for the further understanding of this diploma thesis.
In particular, modular forms for congruence subgroups and Hecke operators
will be dealt with here in detail. We will just repeat some of those proofs,
which are directly linked to our main results and thus we hope to avoid any
superfluous long proofs.
In the fifth chapter, we give a short introduction to the theory of oldforms and
newforms going back to Oliver Atkin and Joseph Lehner in [AL]. Before giving
the proof for our main result which was metioned above, we will explain how
the result is linked to that theory.
5
It would be an interesting topic for further reasearch to find a generalization
of our result in the case of level N pr , such as considering the push forward of
forms under the map f (z) 7→ f (pi z) for 1 ≤ i ≤ r − 1. So far we have not been
able to do this because the Hecke operators of non-prime level tend to become
rather complicated to treat. In particular, the main ideas of our proofs seem
not to work for Hecke operators of non-prime level.
6
Chapter 1
Basic Concepts
1.1
The Modular Group
We will denote the complex upper half-plane, i.e. {z ∈ C | Im(z) > 0} by H.
Furthermore we will as an abbreviation from now on write M ◦ z := az+b
for
cz+d
+
linear fractional transformations by matrices M ∈ GL2 (R).
)
(
a b
∈ GL+
Lemma 1.1. Let M =
2 (R). If z ∈ H then
c d
(
)
az + b
det M · Imz
Im
=
.
(1.1)
cz + d
|cz + d|2
Proof. See [FB], p 310.
Thus we observe that GL+
2 (R) operates on H by ◦. It is not complicated
at all to describe the main geometric property of these maps.
Proposition 1.2. A linear fractional transformation maps circles and lines
on C into circles or lines on C.
Proof. See [MT], p. 2.
Definition 1.3. We set Γ1 := SL2 (Z), i.e. the group of 2 × 2 matrices with
entries in Z and determinant 1. This group is called the modular group.
7
CHAPTER 1. BASIC CONCEPTS
For M ∈ Γ1 relation (1.1) becomes
Im (M ◦ z) =
Imz
.
|cz + d|2
Definition 1.4. By a fundamental domain F for the action of Γ1 on H
we mean an open subset of H such that
(i) For every z ∈ H there exists a M ∈ Γ1 with M ◦ z ∈ F̄.
(ii) If τ, τ ′ ∈ F and ∃M ∈ Γ1 with τ ′ = M ◦ τ , then M = ±E and hence
τ = τ ′.
{
}
(iii) ♯ M ∈ Γ1 |M ◦ F̄ ∩ F̄ ̸= ∅ < ∞.
We conclude this section by collecting some of the main results about the
modular group and fundamental domains:
{
}
Theorem 1.5. The open set F := z = x + iy ∈ H | |x| < 12 , |z| > 1 is a
fundamental domain. Let
(
)
(
)
1 1
0 −1
T =
and S =
.
0 1
1 0
Then:
(i) S and T generate Γ1 .
(ii) We have S 2 = (ST )3 = −E.
(iii) All points in H not equivalent to i or ρ :=
group in Γ1 / {±E}.
1
2
√
+ 12 i 3 have trivial isotropy
(iv) The groups generated by S and ST are the isotropy groups of i and ρ
respectively in Γ1 / {±E}.
Proof. See [KK], p. 124-128.
1.2
Modular Forms
The main interest of this diploma thesis are modular forms. In this first chapter
we will therefore shortly repeat some standard definitions and results.
8
CHAPTER 1. BASIC CONCEPTS
Definition 1.6. Let k ∈ Z. Then a function f : H → C is called a modular
function of weight k, if it satisfies the following properties:
(i) f is meromorphic in the upper half-plane H.
(ii) f (M ◦ z) = (cz + d)k f (z) for all M ∈ Γ1 .
(iii) f is meromorphic at ∞.
(i) By f is meromorphic
( at)∞, we mean that the Fourier
1 1
expansion of f (which exists since
∈ Γ1 and thus f (z +1) = f (z))
0 1
∑
has the form f (τ ) = m∈Z αf (m)·e2πimτ with αf (m) ̸= 0 for only finitely
many m < 0.
Remark 1.7.
(ii) Recall that because of Lemma 1.1 we have
side in (ii) is well-defined.
az+b
cz+d
∈ H and thus the left-hand
(iv) By inserting M = −E in (ii) of Definition 1.6, we immediately get that
there does not exist a modular function of odd weight with f ̸= 0. If not
otherwise mentioned, we will therefore in this chapter assume that k is
even.
We denote by Vk the vectorspace of modular functions of weight k. Furthermore since f · g is also meromorphic at ∞ if f and g are, we get the
inclusion Vk · Vl ⊂ Vk+l . Finally f1 ∈ V−k for 0 ̸= f ∈ Vk . We define K := V0 .
Instantly one sees:
Remark 1.8. K is a field, C ⊂ K.
We now introduce the stronger notion of modular forms.
Definition 1.9. Let k ∈ Z. Then a modular function f : H → C of weight k
is called a modular form of weight k, if it satisfies the following properties:
(i) f is holomorphic in the upper half-plane H.
(ii) f (M ◦ z) = (cz + d)k f (z) ∀M ∈ Γ1 .
(iii) f is holomorphic at ∞.
9
CHAPTER 1. BASIC CONCEPTS
Remark 1.10. By f is holomorphic
at ∞, we mean that f has a Fourier ex∑
pansion of the form f (τ ) = m≥0 αf (m) · e2πimτ .
Definition 1.11. A modular form f : H → C is called a cusp form if
αf (0) = 0.
It is obvious that the modular forms of weight k are a subspace of Vk ,
which we denote by Mk . We write Sk for the subpace of cusp forms of weight
k. Furthermore we immediately get Mk · Ml ⊂ Mk+l and Sk · Ml ⊂ Sk+l .
Now we will give a few examples of modular forms, which will turn out to
play an essential role:
Definition 1.12. The Eisenstein series 1 are defined as
∑
Gk (τ ) :=
(mτ + n)−k for k ∈ N, k ≥ 4, k even, τ ∈ H.
(1.2)
(m,n)̸=(0,0)
Lemma 1.13. For k ∈ N, k even, k ≥ 4, Gk is absolute and locally uniform
convergent. Thus Gk is a holomorphic function on H.
Proof. See [KK], p. 159.
Theorem 1.14. For k ∈ N, k ≥ 4 and all M ∈ Γ1 the equation
Gk (M ◦ τ ) = (cτ + d)k Gk (τ ), τ ∈ H
holds. For even k the Fourier expansion of Gk is given by
Gk (τ ) = 2ζ(k) + 2
with
ζ(k) =
∞
∑
∞
(2πi)k ∑
σk−1 (m) · e2πimτ
(k − 1)! m=1
m−k , k > 1 and σs (m) :=
m=1
∑
ds , s ∈ R
d∈N,d|m
and therefore Gk is a modular form of weight k.
1
Eisenstein series are named after the famous German mathematician Ferdinand Gotthold Max Eisenstein (1823-1852), who mostly lived and worked in Berlin and pioneered in
the field of elliptic functions.
10
CHAPTER 1. BASIC CONCEPTS
Proof. See [KK], p. 160-161 and p. 49-50.
Corollary 1.15. Mk = C · Gk ⊕ Sk .
Proof. If f ∈ Mk then f −
αf (0)
2ζ(k)
· Gk ∈ Sk .
The knowledge of the Fourier expansion of Gk allows us to define the normalized Eistenstein series, which have leading coefficient 1 in their Fourier
expansion by setting
Ek :=
1
· Gk .
2ζ(k)
Definition 1.16. The discriminant is defined as
∆ :=
E43 − E62
1728
(1.3)
Theorem 1.17. We have ∆ ∈ S12 . The Fourier expansion of the discriminant
is given by
∞
∑
∆(z) =
τ (m) · e2πimz , z ∈ H.
(1.4)
m=1
The coefficients τ (m) are all integers with τ (1) = 1. The function τ (m) is
called Ramanujan’s Tau function 2 . ∆ has no zeros on H.
Proof. See [KK], p. 162, p. 53, p. 41.
Our next aim is to find a possibility to describe the vectorspaces Mk of
modular forms of weight k in more detail. The subsequent so-called weight
formula will be essential for this. However, we have to define another notion
at first.
2
Srinivasa Ramanujan (1887-1920) was one of the most fascinating persons in the history
of mathematics ever. Born in a small town in southern India, Ramanujan did almost get
no formal education in mathematics. Nevertheless he made substantial contributions in
different mathematical areas. He is especially known for his joint work with Godfrey Harold
Hardy at Cambridge university.
11
CHAPTER 1. BASIC CONCEPTS
Definition 1.18.
(i) Let f ̸= 0, f ∈ Vk with Laurent-series
∑
f (τ ) =
γ(m) · (τ − w)m , γ(r) ̸= 0
m≥r
in w ∈ H. Then the order of f at w is defined by ordw f := r.
(ii) Let f ̸= 0, f ∈ Vk with Fourier expansion
∑
f (τ ) =
αf (m) · e2πimτ , αf (m0 ) ̸= 0.
m≥m0
Then the order of f at ∞ is defined by ord∞ := m0 .
(iii) The order of a point w ∈ F̄∗ := F̄ ∪ {∞} is defined by

 2 for w = i
3 for w = ρ .
ord w :=

1 else
Theorem 1.19. (Valence formula) Let f ∈ Mk , f ̸= 0. Then
∑
w∈F∪{∞}
1
1
1
k
· ordw f +
ordi f +
ordρ f = .
ord w
ord i
ord ρ
12
(1.5)
Proof. See [LS], p. 6-8.
When we apply the valence formula and use the following lemma, we can
get an important result about the structure of the Mk -:
Lemma 1.20. For k ≥ 4, k even:
Mk = ∆ · Mk−12 ⊕ Gk .
Proof. See [LS], p. 11.
Theorem 1.21. The functions G4 and G6 are algebraically independent, and
⊕
M :=
Mk = C[G4 , G6 ].
k even
12
CHAPTER 1. BASIC CONCEPTS
Proof. See [LS], p. 10-12.
Remark 1.22. Since Mk · Ml ⊂ Mk+l , M is a graded algebra.
By Lemma 1.20. and Corollary 1.15. we can now compute the dimension
of the Mk -spaces as follows.
Theorem 1.23. (Dimension formula) Let k ≥ 0. Then
 [k]
if k ≡ 2(mod 12)
 12
dimC Mk :=
.
 [k]
+ 1 if k ̸= 2(mod 12)
12
(1.6)
Proof. See [KK], p. 173-175.
The next example of a modular form was first analyzed by the mathematician Felix Klein.
Definition 1.24. The j-invariant is defined as
E43
.
∆
j :=
(1.7)
As another application of the weight formula we get:
Theorem 1.25.
(i) j : H → C is holomorphic and surjective.
(ii) j ∈ K = V0 .
(iii) Its Fourier expansion is given by
j(τ ) = e
−2πiτ
+
∞
∑
jm · e2πimτ , τ ∈ H
m=0
with jm ∈ N for all m.
(iv) j : F → C is a bijection.
Proof. See [KK], p. 165, p. 184, p. 54.
As a consequence it is possible to describe the structure of K in a very easy
way:
Theorem 1.26. K = C(j).
Proof. See [KK], p. 184.
13
Chapter 2
Modular Forms for Congruence
Subgroups
In Chapter 1 we saw that modular forms have certain transformation properties
for all matrices M ∈ Γ1 . To be more precise, we postulated that
f (M ◦ z) = (cz + d)k f (z) ∀M ∈ Γ1 .
(2.1)
So the following questions naturally arise: What happens if we look at a
subgroup of Γ1 instead of the whole group? How can we then generalize the
well-known concepts from Chapter 1? We will analyze these questions below.
2.1
Congruence Subgroups
Definition 2.1. Let N ∈ N. Two matrices L, M ∈ GL2 (Z) are conguent mod
N , i.e. L ≡ M (mod N ), if all their entries are equivalent mod N . We define
the principal congruence subgroup of level N by
Γ[N ] := {M ∈ Γ1 ; M ≡ E (mod N )} .
Theorem 2.2. The canonical projection
Φ : Γ1 → SL2 (Z/N Z), M 7→ M
14
CHAPTER 2. MODULAR FORMS FOR CONGRUENCE
SUBGROUPS
is a surjective homomorphism with kernel Γ[N ]. Γ[N ] is a normal subgroup of
finite index in Γ1 and the factor group Γ1 /Γ[N ] is isomorphic to SL2 (Z/N Z).
For N ≥ 2 we have
∏
♯SL2 (Z/N Z) = N 3 (1 − p−2 ),
p|N
where the product is taken over all prime divisors of N .
Proof. See [KK], p. 135-136.
Definition 2.3. A subgroup Γ of Γ1 is called a congruence subgroup, if
there exists a N ≥ 1 with Γ[N ] ⊂ Γ. In this case Γ we say thar a congruence
subgroup of level N.
Corollary 2.4. Every congruence subgroup has finite index in Γ1 .
Now we give two very important examples of congruence subgroups, which
are not principal congruence subgroups.
Example 2.5.
{(
Γ0 [N ] :=
a b
c d
{(
Γ1 [N ] :=
)
(
) (
)
}
a b
∗ ∗
∈ Γ1 ;
≡
(mod N )
c d
0 ∗
)
)
}
) (
(
1 ∗
a b
(mod N )
≡
∈ Γ1 ;
0 1
c d
a b
c d
Thus we have the following inclusions:
Γ[N ] ⊂ Γ1 [N ] ⊂ Γ0 [N ] ⊂ Γ1 .
All of the missing indexes can easily be calculated as follows. The map
(
)
a b
Γ1 [N ] → Z/N Z,
7→ b(mod N )
c d
is evidently a surjection with kernel Γ[N ] and thus
Γ1 (N )/Γ[N ] ∼
= Z/N Z ⇒ [Γ1 [N ] : Γ[N ]] = N.
15
CHAPTER 2. MODULAR FORMS FOR CONGRUENCE
SUBGROUPS
Similary the map
(
)
a b
Γ0 [N ] → (Z/N Z) ,
7→ d(mod N )
c d
×
is obviously a surjection with kernel Γ1 [N ] and thus
Γ0 (N )/Γ1 [N ] ∼
= (Z/N Z)× ⇒ [Γ0 [N ] : Γ1 [N ]] = ϕ(N ).
where ϕ is the Euler totient function from number theory, namely
ϕ(n) =
∑
∏
1=n
m≤n, (m,n)=1
p prime, p|n
1
(1 − ).
p
(2.2)
Finally, by this calculation and by Theorem 2.2 we get
∏
[Γ : Γ0 [N ]] = N
p prime, p|n
1
(1 + ).
p
Next we intend to generalize the term fundamental domain (see Chapter
1):
Definition 2.6. Let Γ be a subgroup of Γ1 . An open subset F of H is called a
fundamental domain for the action Γ on H, if:
(i) For every τ ∈ H there exists a matrix M ∈ Γ with M ◦ τ ∈ F̄.
(ii) If τ, τ ′ ∈ F and ∃M ∈ Γ with τ ′ = M ◦ τ , then M = ±E and hence
τ = τ ′.
{
}
(iii) ♯ M ∈ Γ|M ◦ F̄ ∩ F̄ ̸= ∅ < ∞.
So F from Chapter 1 (see Theorem 1.5) is also a fundamental domain of
Γ1 in our ”new” sense. As we intended our new definition is a generalization
of the old one, as we intended. Now let M F := {M ◦ τ ; τ ∈ F} be the image
of F under the transformation τ 7→ M ◦ τ . We then immediately get:
Proposition 2.7. An open subset F is a fundamental domain for the action
of Γ on H, if and only if
16
CHAPTER 2. MODULAR FORMS FOR CONGRUENCE
SUBGROUPS
(i)’ H =
∪
M ∈Γ
M F̄.
(ii)’ F ∩ M F ̸= ∅ and M ∈ Γ imply M = ±E.
{
}
(iii)’ ♯ M ∈ Γ|M ◦ F̄ ∩ F̄ ̸= ∅ < ∞.
Our next aim is to explicitly find a fundamental domain for an arbitrary
subroup Γ ⊂ Γ1 of finite index. Applying what we already know, it can be
described very easily.
Theorem 2.8. Let F be the fundamental domain defined in Theorem 1.5.
Suppose that Γ has finite index in Γ1 and −E ∈ Γ. Let
∪
Γ′ Mν
Γ1 =
1≤ν≤[Γ1 :Γ]
be a disjoint partition of Γ1 into right cosets. Then
∪
F(Γ) :=
Mν F̄
(2.3)
1≤ν≤[Γ:Γ′ ]
is a fundamental domain of Γ.
Proof. See [KK], p. 133-134. Note that [Γ1 : Γ′ ] might be both, infinite or
finite, if Γ′ was an arbitrary subgroup.
2.2
Cusps
Before we are able to define modular forms for congruence subgroups, we have
to introduce the term cusp. In order to proceed in our considerations we need
to recall some standard definitions:
Definition 2.9. (i) Let G be a group, which acts on a set M from the left.
Let m ∈ M , then the set Gm := {gm; g ∈ G} is called the orbit of m
under G.
(ii) The subgroup Gm := {g ∈ G; gm = m} is called the stabilizer of m.
(iii) The set of all orbits G\M = {Gm|m ∈ M } is called the orbit space of
G on M or the quotient of the action.
17
CHAPTER 2. MODULAR FORMS FOR CONGRUENCE
SUBGROUPS
For a congruence subgroup Γ the idea is to adjoin not only ∞ (see Definition
1.9) but also the rational numbers Q to H. In the next step one can identify
some of these adjoined points under Γ-equivalence. This means that we mean
cusp-classes when we speak of cusps. This fact becomes clear by considering
the next definition:
Definition 2.10. (i) The set of cusps is P1 (Q) ∼
= Q ∪ {∞}, i.e. the proa
jective line of Q with 0 = ∞.
(ii) The group GL2 (Q) acts
(
a
c
on the set of cusps by
)
ar + b
r
ar + bs
b
· := rs
=
d
s
cs + d
cr + ds
where this means to take ∞ to
∞ if c = 0.
a
c
and
−d
c
to ∞ if c ̸= 0 and to take ∞ to
Remark 2.11. As the matrices are nonsingular no ” 00 expressions” will arise
and so everything is well-defined.
At first we get the following statement for the orbit space of Γ1 on P1 (Q):
Proposition 2.12. (i) The orbit space Γ1 \P1 (Q) (i.e. the cusp-classes)
consist of just one element.
(ii) The stabilizer of ∞ is
Γ1,∞
)
}
{ (
1 n
|n ∈ Z .
= ±
0 1
Proof. See [WG], p. 45.
This means that all rational numbers are Γ1 -equivalent to ∞ and thus Γ1
has only one cusp, which is represented by ∞.
Definition 2.13. Let Γ ⊂ Γ1 be a subgroup of finite index. Then the orbit
space Γ\P1 (Q) is called the set of cusp-classes of Γ.
Proposition 2.14. Let Γ ⊂ Γ1 be a subgroup of finite index.
18
CHAPTER 2. MODULAR FORMS FOR CONGRUENCE
SUBGROUPS
(i) Then the number of cusp-classes is smaller tham or equal to the index
[Γ1 : Γ].
(
)
a b
(ii) Let σ =
. Then the stabilizer of the cusp ac in Γ is given by
c d
⟨
(
)
⟩
1
1
−1
Γ ac = Γ ∩ ±E, σ
σ
.
0 1
Proof. See [WG], p. 46.
At last the denomination cusp still seems to be unclear. A geometric explanation of this is given by the next example:
Example 2.15. Let Γ := Γ0 [2]. The group Γ0 [2] has index 3 in Γ1 . A figure of
a fundamental domain of Γ0 [2] can be found in [KK], p.137. The cusps of the
upper considerations are now in one-to-one-correspondence to the geometric
cusps of the figure.
2.3
Modular Forms
(
)
a b
For a function f : H → C and a matrix M =
∈ GL+
2 (R) we will from
c d
now on use the slash-operator defined by
(f |k M )(τ ) = (det M )k/2 (cτ + d)−k f (M ◦ τ )
(2.4)
as an abbreviation. Note that for every modular function of weight k we have
f |k M = f for all M ∈ Γ1 .
Definition 2.16. Let Γ be a congruence subgroup of Γ1 and let k be an integer.
A function f : H → C is a modular form of weight k with respect to Γ
if
(i) f is holomorphic.
(ii) f |k L = f for all L ∈ Γ.
(iii) f |k M is holomorphic at ∞ for all M ∈ Γ1 .
19
CHAPTER 2. MODULAR FORMS FOR CONGRUENCE
SUBGROUPS
We now have to explain, what we exactly mean by property (iii): When Γ
is a proper subgroup of Γ1 , then not all cusps are Γ-equivalent and so Γ will
have other cusps than ∞ as well, represented by rational numbers. Writing
any s ∈ Q ∪ {∞} as s = γ · ∞ with γ ∈ Γ1 , holomorphy of f |k γ at ∞ means
that the corresponding Fourier expansion of f |k γ (which exists since Γ is a
congruence subgroup), extends holomorphically to 0.
When we speak of property (iii) we often say that f is holomorphic at all cusps.
Definition 2.17. If in addition α0 = 0 in the Fourier expansion of f |k M for
all M ∈ Γ1 , then f is called a cusp form of weight k with resprect to Γ.
There is still one important fact to be pointed out: Condition (iii) in Definition 2.16 does not associate a unique Fourier expansion of f to a rational
number q or even to q = ∞ as shown e.g. in [DS], p. 17-18. Nevertheless it
is well-defined if α0 is 0 and so the definition of cusp forms and the intuition
that a cusp form vanishes at all the cusps makes sense.
We denote the vectorspace of the modular forms of weight k with respect
to Γ by Mk (Γ) and the vectorspace of the cusp forms by Sk (Γ). Evidently, we
see that Mk = Mk (Γ1 ) and Sk = Sk (Γ1 ) hold. Furthermore from Γ ⊂ Γ′ follows
that Mk (Γ′ ) ⊂ Mk (Γ). Because of this fact the study of modular forms with
respect to congruence subgroups can even be reduced to that of Mk (Γ(N )) for
some N ∈ N (as shown e.g. in [MT] p. 114).
We now try to better understand the space Mk (Γ1 (N )). To do this, we introduce the following definition:
Definition 2.18. A Dirichlet character modulo N is a group-homomorphism
χ : GN := (Z/N Z)× → C∗ .
Remark 2.19. (a) The set of Dirichlet characters modulo N is again a multiplicative group, called the dual group of GN , denoted Gˆn , if we define
the product character by the rule χψ(n) = χ(n)ψ(n).
(b) Since GN is a finite group the values taken by any Dirchlet character are
complex roots of unity, and so the inverse of a Dirichlet character is its
complex conjugate, defined by the rule χ̄(n) = χ(n) ∀ n ∈ GN .
20
CHAPTER 2. MODULAR FORMS FOR CONGRUENCE
SUBGROUPS
(c) Every Dirichlet character modulo N extends to a function χ : Z/N Z →
C, by setting χ(n) = 0 for all non-invertible elements n and further to a
function χ : Z → C where χ(z) = χ(z (mod N )) for all z ∈ Z.
(d) The trivial Dirichlet character is unambiguously denoted by χ0 .
21
CHAPTER 2. MODULAR FORMS FOR CONGRUENCE
SUBGROUPS
We are interested in Dirichlet characters because they decompose the vector space Mk (Γ1 (N )) into a direct sum of subspaces, which we can analyze
independently and often much easier than Mk (Γ1 (N )). For this purpose we
set:
Definition 2.20. For a Dirchlet charcter χ modulo N the spaces
Mk (N, χ) := {f ∈ Mk (Γ1 (N ))|f |k M = χ(d) · f ∀ M ∈ Γ0 [N ]}
(2.5)
Sk (N, χ) := {f ∈ Sk (Γ1 (N ))|f |k M = χ(d) · f ∀ M ∈ Γ0 [N ]}
(2.6)
and
are called the χ-eigenspaces of Mk (Γ1 [N ]) and Sk (Γ1 [N ]), respectively.
Note that Mk (N, χ0 ) = Mk (Γ0 [N ]). By inserting −E we immediately deduce:
Proposition 2.21. Let N ∈ N and let χ be a Dirchlet character mod N. Then
Mk (N, χ) = {0}, if χ(−1) ̸= (−1)k .
We point out, that unlike before this proposition does not exclude the
existence of χ-eigenforms of odd weight. Nevertheless one knows the following
non-existence result:
Proposition 2.22. A function f ∈ Mk (N, χ) with f ̸= 0 does not exist if
k < 0.
Proof. See [KK], p. 198.
Now we will show, how the spaces explicitly decompose Mk (Γ1 (N )). In the
following, these key identities will turn out be essential.
Theorem 2.23. We have the following identities:
⊕
Mk (Γ1 [N ]) =
Mk (N, χ),
(2.7)
χ
and
Sk (Γ1 [N ]) =
⊕
Sk (N, χ),
χ
where both sums are taken over all Dirichlet characters modulo N .
22
(2.8)
CHAPTER 2. MODULAR FORMS FOR CONGRUENCE
SUBGROUPS
Proof. This is a standard result from representation theory of finite groups
(see e.g. [FB], p. 384-385.). A direct proof is given in [KN], p. 137-138.
Remark 2.24. We will later show that this decomposition is even orthogonal
(with respect to the Petersson scalar product).
23
Chapter 3
Hecke Operators
The so-called Hecke operators are probably the most interesting set of operators on the modular forms. There are several possibilities to introduce them.
We have chosen to follow the method which is for example presented in [DS],
as we think it is the most useful approach for our application and as it does
avoid the use of the lattice concepts, which we do not want to focus on here.
Therefore we have to define the double coset operators first.
3.1
The Double Coset Operator
In general, we assume in this section that Γ and Γ′ are arbitrary congruence
subgroups of SL2 (Z). Our aim is to define an operator which takes elements
of Mk (Γ) to elements of Mk (Γ′ ). For this, we have to make the following
definition:
Definition 3.1. For α ∈ GL+
2 (Q) we set
ΓαΓ′ := {γ1 αγ2 |γ1 ∈ Γ, γ2 ∈ Γ′ }
(3.1)
and call it a double coset in GL+
2 (Q).
The group Γ acts on the double coset ΓαΓ′ by left multiplication, partition∪
ing it into orbits, such that the orbit space Γ\ΓαΓ′ is a disjoint union Γβj
with representatives β = γ1 αγ2 . We even get:
Proposition 3.2. The orbit space space Γ\ΓαΓ′ is finite.
24
CHAPTER 3. HECKE OPERATORS
Proof. See [DS], p. 164
Recall the definition of the slash operator (2.4). With its help we can
define:
Definition 3.3. For congruence subgroups Γ and Γ′ of SL2 (Z) and α ∈
′
GL+
2 (Q), the weight-k ΓαΓ operator or double coset operator takes
functions f ∈ Mk (Γ) to
∑
f |k βj
f [ΓαΓ′ ]k =
j
where {βj } are orbit representatives, i.e. ΓαΓ′ =
∪
j
Γβj is a disjoint union.
Remark 3.4. The double coset operator is well defined (see [DS], p. 409).
The next propostion shows that we constructed the double coset operators
in accordance with our aim.
Proposition 3.5. [ΓαΓ′ ]k is a function from Mk (Γ) to Mk (Γ′ ), which takes
cusp forms to cusp forms.
Proof. See [DS], p. 165-166.
Some important special cases are:
Remark 3.6. (a) Γ ⊃ Γ′ : Taking α = E turns the double coset operator
into f [ΓαΓ′ ]k = f , and hence is the natural inclusion of the subspace
Mk (Γ) in Mk (Γ′ ).
(b) α−1 Γα = Γ′ : Then ΓαΓ′ = Γαα−1 Γα = Γα and thus the double coset
operator means by definition f [ΓαΓ′ ]k = f |k α, i.e. the natural translation
from Mk (Γ) to Mk (Γ′ ), is even an isomorphism.
(c) Γ ⊂ Γ′ : Taking α = E and letting {γ2,j } be a set of coset representatives
∑
for Γ\Γ′′ turns the double coset operator into f [ΓαΓ′ ]k = j f |k γ2,j and
is a surjection.
25
CHAPTER 3. HECKE OPERATORS
3.2
Hecke Operators for Prime Numbers
First, we want to define the so-called diamond operator. For this we take
any α ∈ Γ0 [N ] and set Γ = Γ′ = Γ1 [N ]. Then we conisder the weight-k
double coset operator [ΓαΓ′ ]k . Since Γ1 [N ] is a normal subgroup of Γ0 [N ] with
Γ0 [N ]/Γ1 [N ] ∼
= (Z/N Z)× (see section 2.1), we are in case (b) of 3.6. This
means, that
f [Γ1 [N ]αΓ1 [N ]]k = f |k α, α ∈ Γ0 [N ]
and, that the weight-k double coset operator takes a function f ∈ Mk (Γ1 [N ])
again to
in Mk (Γ1 [N ]). Recall that the map Γ0 [N ] → (Z/N Z)×
( a function
)
a b
taking
to d (mod N ) is a surjective homomorphism with kernel Γ1 [N ].
c d
So the group Γ0 [N ] acts on Mk (Γ1 [N ]) and since its subsgroup Γ1 [N ] acts
trivially, this is really an action of the quotient (Z/N Z)× . These facts allow
us to make the following definition:
Definition 3.7. The diamond operator < d > is given by
< d >: Mk (Γ1 [N ]) → Mk (Γ1 [N ])
)
(
a b
∈ Γ0 [N ] with δ ≡ d (mod N ).
< d > f = f |k α, α =
c δ
The Hecke operator Tp (with p prime) is constructed very similar and is
also a weight-k double coset operator. Let again be Γ = Γ′ = Γ1 [N ].
)
(
1 0
Definition 3.8. Let p be a prime number and α =
. Then the Hecke
0 p
operator1 Tp is given by
Tp : Mk (Γ1 [N ]) → Mk (Γ1 [N ])
Tp f = pk/2−1 f [Γ1 [N ]αΓ1 [N ]]k .
1
The operator is named after Erich Hecke (1887 - 1947). He received his PhD in Göttingen
under the supervision of David Hilbert, but left Göttingen for being professor in Hamburg.
Hecke was not only known for being a great mathematician; he also was a corrageous slasher
of the Nazi-regime.
26
CHAPTER 3. HECKE OPERATORS
Remark 3.9. Since double coset operators take cusp forms to cusp forms, so
do < d > and Tp .
We now want to give an explicit representation of Tp .
Proposition 3.10. Let N ∈ N. The operator Tp on
by:

(
)
∑p−1
1
j


 j=0 f |k

0 p

k/2−1
Tp f = p
(
)(
(
)


∑p−1
1 j
m n
p


+ f |k
 j=0 f |k
N p
0
0 p
Mk (Γ1 [N ]) is then given
if p|N
)
0
else, where mp-nN=1.
1
(3.2)
Proof. See [DS], p. 169-170 and note that our definition is slightly different
from that in [DS].
We shortly analyze the interaction of < d > and Tp .
Lemma 3.11. Let d, e ∈ (Z/N Z)× and p, q be prime numbers. Then
(i) < d > Tp = Tp < d >.
(ii) < d >< e >=< e >< d >.
(iii) Tq Tp = Tp Tq .
Proof. See [DS], p. 169 and p. 173.
The next theorem will be fundamental for the main result of this paper.
We could also have defined the Hecke operators by the next theorem ad hoc.
Nevertheless we think that the Hecke operators arised somehow more natural
by the way we introduced them.
Theorem 3.12. Let f ∈ Mk (Γ1 [N ]) with Fourier expansion
f (τ ) =
∞
∑
n=0
27
αf (n)q n .
CHAPTER 3. HECKE OPERATORS
Let χ0 : (Z/N Z)× → C× be the trivial character modulo N. Then Tp f has
Fourier expansion:
(Tp f )(τ ) =
∞
∑
n
k−1
αf (np)q + χ0 (p)p
n=0
∞
∑
α<p>f (n)q np .
(3.3)
n=0
That is
n
αTp f (n) = αf (np) + χ0 (p)pk−1 α<p>f ( ).
(3.4)
p
Here α( np ) = 0, if p does not divide n, and χ0 (p) = 1, if p does not divide N,
and χ0 (p) = 0 else.
Proof. Take 0 ≤ j < p and compute
(
)
∞
τ +j
1∑
1 j
k/2−1
p
(f |k
)(τ ) = pk−1 (0τ + p)−k f (
)=
αf (n)e2πin(τ +j)/p
0 p
p
p n=0
1∑
=
αf (n)qpn µnj
p
p n=0
∞
2πi/p
where qp = e2πiτ /p and µ∑
. We know from algebra that
thet the
p = e
∑for
p−1 nj
p−1 nj
geometric sum we have j=0 µp = p when p divides n, and j=0 µp = 0
when p does not divide n. Summing over j and switching sums gives:
)
(
p−1
∞
∑
∑
∑
1 j
n
αf (n)qp =
αf (np)q n .
(f |k
)(τ ) =
0 p
j=0
n≡0(p)
n=0
This equals Tp f (τ ) when p|N . Otherwise, Tp f (τ ) also includes the term (see
Proposition 3.10)
)
(
)(
m n
p 0
k/2−1
p
f |k
(τ )
0 1
N p
Note that mp − nN = 1. Thus we
(
p
k/2−1
... = p
(< p > f )|k
0
∞
∑
= pk−1
α<p>f (n)q np
get:
)
0
(τ ) = pk−1 (0τ + 1)−k (< p > f )(pτ )
1
n=0
This completes the proof.
28
CHAPTER 3. HECKE OPERATORS
Corollary 3.13. Let χ : (Z/N Z)× → C× be a Dirchlet character. If
f ∈ Mk (N, χ) (see (2.5)) then also Tp f ∈ Mk (N, χ), and its Fourier expansion
is given by
(Tp f )(τ ) =
∞
∑
n
k−1
αf (np)q + χ(p)p
n=0
That is
∞
∑
αf (n)q np .
(3.5)
n=0
n
αTp f (n) = αf (np) + χ(p)pk−1 αf ( ).
p
(3.6)
Proof. This follows immediately from 3.12 considering that
Tp (< d > f ) =< d > (Tp f ) and
Mk (N, χ) = {f ∈ Mk (Γ1 (N )); < d > f = χ(d) · f ∀ d ∈ (Z/N Z)× }.
Remark 3.14. It is important to note that all results from this chapter also
hold for Mk (Γ0 [N ]) or Sk (Γ0 [N ]) respectively, since Γ1 [N ] ⊂ Γ0 [N ] and thus
Mk (Γ0 [N ]) ⊂ Mk (Γ1 [N ]) (see Theorem 2.23).
Corollary 3.15. The discriminant ∆ is a simultaneous eigenform of all operators Tp .
Proof. We know, that all Tp restrict to Sk (Γ1 ). Since ∆ ∈ S12 (Γ1 ) and since
S12 (Γ1 ) is 1-dimensional (see Theorem 1.6) the statement follows instantly from
Theorem 3.12 (double coset operators take cusp forms to cusp forms).
At the end of this section, we introduce some useful and important notation:
Definition 3.16.
∞
∞
∑
∑
Vd (
an q n ) =
an q dn
n=0
∞
∑
Ud (
n=0
an q n ) =
(3.7)
n=0
∑
d|n, n≥0
an q n/d =
∑
adn q n
(3.8)
n≥0
By Corolllary 3.13 we get: Tp = Up + χ(p)pk−1 Vp . We will sometimes write
f |Tp instead of Tp f and use similar notation for Ud and Vd .
29
CHAPTER 3. HECKE OPERATORS
3.3
The Tn Operators
So far we have only dealt with Hecke operators for prime numbers. By Lemma
3.11 we know that the Tp -operators commute. We now want to transfer the
definition to all the natural numbers N and preserve this important property.
For n ∈ N with (n, N ) = 1, < n > is determined by n(mod N ). For (n, N ) > 1
we set < n >= 0. Then we have: < m >< n >=< mn > for all m, n ∈ N.
We next set T1 = 1. Let p be a prime number such that Tp is already defined.
For prime powers we inductively define:
Tpr = Tp Tpr−1 − pk−1 < p > Tpr−2
Because of the fundamental theorem of elementary number theory it is now
possible to define Hecke operators for arbitrary natural numbers:
∏
∏
Definition 3.17. Let n ∈ N, n = pri i then Tn := Tpri i .
This prime-power defintion seems to be rather mysterious and arbitrary.
We will give a short motivation, ∑
why we define Hecke operators exactly the
−s
be a generating function of the
way we do it: For this let g(s) := ∞
n=1 Tn n
Tn . Then the prime power definition, determining all the Tn in terms of Tp
and < p >, are encapsulated as a product expression for g,
∏
g(s) =
(1 − Tp p−s + < p > pk−1−2s )−1 ,
p
where the product is taken over all primes. This idea is closely linked to the
concept of L-functions (See [DS], p. 200-203). Moreover if one introduces
Hecke operators via lattices the multiplicative properties are rather calculations than defintions (see e.g. [KA], p. 242-250).
Lemma 3.18. For all n, m ∈ N
(i) Tn Tm = Tm Tn
(ii) Tnm = Tn Tm if (n, m) = 1.
hold.
Proof. See [DS], p. 179.
30
CHAPTER 3. HECKE OPERATORS
The Fourier coefficient formula in Theorem 3.12 now generalizes to:
Theorem 3.19. Let f ∈ Mk (Γ1 [N ]) have Fourier expansion
f (τ ) =
∞
∑
αf (m)q m
m=0
Then for all n ∈ N we have
∞
∑
(Tn f )(τ ) =
αTn f (m)q m
m=0
where
αTn f (m) =
∑
dk−1 α<d>f (mn/d2 )
d|(m,n)
In particular, if f ∈ M(N, χ) then
∑
χ(d)dk−1 αf (mn/d2 )
αTn f (m) =
d|(m,n)
Proof. See [DS], p. 180.
Again the discriminant ∆ plays an essential role:
Proposition 3.20. The discriminant ∆ is a simultaneous eigenform for all
Hecke operators with Tn ∆ = τ (n) · ∆.
We have already mentioned above, that Hecke operators preserve the property of being a cusp form. We want to conclude this chapter by giving an even
stronger result:
Proposition 3.21. The Tn operators preserve Mk (N, χ) and Sk (N, χ).
Proof. See [KN], p. 160.
31
Chapter 4
Petersson Scalar Product
In order to proceed in our studies of the space of cusp forms Sk (Γ1 [N ]), we will
now define a scalar product on this vector space, which will be an integral similar to the L2 -scalar product. This construction will allow us to treat Sk (Γ1 [N ])
as Hilbert space, in which for example the concept of adjoint operators makes
special sense.
4.1
Definition
The Petersson scalar product is based on the hyperbolic measure:
Definition 4.1. For τ = x + iy ∈ H let dxdy be the 2-dimensional Lebesguemeasure on C ∼
= R2 . We define the hyperbolic measure by
dµ := dµ(τ ) := y −2 dxdy.
(4.1)
For every Lebesgue-measurable subset Ω ⊂ H and every continuous function
φ : H → C we define the hyperbolic integral by
∫
∫
φdµ :=
φ(τ )dµ(τ ).
(4.2)
Ω
Furthermore let µ(Ω) :=
∫
Ω
Ω
dµ be the hyperbolic measure of Ω.
The following lemma is very important.
32
CHAPTER 4. PETERSSON SCALAR PRODUCT
Lemma 4.2. The hyperbolic measure is invariant under all functions τ 7→
M ◦ τ with M ∈ GL+
2 (R).
Proof. See [KK], p. 228.
We now want to reconsider the concept of fundamental domains which we
presented earlier. It will turn out to be convenient to choose Ω = F, where F
is a fundamental domain, as integration domain. As a first result we get:
Lemma 4.3. Let F be the fundamental domain from Theorem1.5.
For every
∫
continuous and bounded function φ : H → C the integral F φdµ is absolute
convergent. Moreover, the hyperbolic measure of F is µ(F) = µ(F̄) = π3 .
Proof. See [KK], p. 229.
For arbitrary fundamental domains we state:
Proposition 4.4. Let Γ ⊂ Γ1 be a congruence subgroup and let F be any
fundamental domain of Γ. Furthermore let Γ = Γ/ ± {E} and Γ1 = Γ1 / {±E}
respectively. Then:
∫
(i) The integral µ(F) := F dµ converges and is independent of the choice
of F.
(ii) [Γ1 : Γ] = µ(F)/µ(F)
−1
−1
(iii) If α ∈ GL+
2 (Q) and α Γα ⊂ Γ1 , then [Γ1 : Γ] = [Γ1 : α Γα]
Proof. See [KN], p. 169.
Remark 4.5. By a simple calculation one sees that α−1 F is a fundamental
domain of α−1 Γα.
To define the Petersson scalar product, let f, g ∈ Mk (Γ) with at least one
of the two function f, g being a cusp form. Let us set
ϕf,g (τ ) := f (τ )g(τ )(Im(τ ))k
where the bar denotes the complex conjugation.
Lemma 4.6. The function ϕf,g is continuous, bounded and Γ-invariant.
33
(4.3)
CHAPTER 4. PETERSSON SCALAR PRODUCT
Proof. See [DS], p. 183-184.
We can now give the definition of the Petersson scalar product:
Definition 4.7. Let Γ ⊂ Γ1 be a congruence subgroup with fundamental domain F. Let f, g ∈ Sk (Γ) be cusp forms. Then the Petersson scalar product1
< ·, · >: Sk (Γ) × Sk (Γ) → C
is defined by:
1
< f, g >:=
[Γ1 : Γ]
∫
1
f (τ )g(τ )(Im(τ )) dµ =
[Γ1 : Γ]
F
∫
k
f (τ )g(τ )(Im(τ ))k−2 dxdy.
F
Remark 4.8. The Petersson scalar product is linear in f, conjugate linear in
g, Hermitian-symmetric, and positive definite.
By the following proposition we will ensure, that the definition makes sense:
Proposition 4.9. The Petersson saclar product is absolutely convergent, and
does not depend on the choice of the fundamental domain. If Γ′ is another
congruence subgroup such that f, g ∈ Mk (Γ′ ), then the definition is independent
of whether f and g are considered in Mk (Γ) or in Mk (Γ′ ).
Remark 4.10. The term
in the proposition.
1
[Γ1 :Γ]
is needed in order to have the second assertion
Proof. See [KN], p. 170.
Remark 4.11. The integral-expression of the Petersson scalar product also
exists if only one of the function f,g is a cusp form and the other is a modular
form of weight k. Therefore we can calculate < f, g > also in this case. This
fact will be used in the following.
1
The Petersson scalar product is named after Wilfried Hans Henning Petersson (19021984), who was a student of Erich Hecke in Hamburg. Later he was professor in Hamburg,
Praha, Strassbourg, Münster and Notre Dame (USA).
34
CHAPTER 4. PETERSSON SCALAR PRODUCT
4.2
Adjoints of the Hecke Operators
We can now show one main property of the Petersson scalar product:
Theorem 4.12. Let Γ ⊂ Γ1 be a congruence subgroup and let f, g ∈ Mk (Γ)
with f or g being a cusp form. Let α ∈ GL+
2 (Q). Then
< f, g >=< f |k α, g|k α > .
Proof. See [KN], p. 171.
It is crucial, that we now know how to switch matrix operators.
Corollary 4.13. Let Γ ⊂ Γ1 be a congruence subgroup, and let α ∈ GL+
2 (Q).
∗
−1
Set α := det α · α . If f, g ∈ Sk (Γ), then
< f |k α, g >=< f, g|k α∗ > .
Proof. See [KN], p. 171.
Since we know by Propostion 3.10 that the Hecke operators are just special
matrix operators we can calculate their adjoints.
Theorem 4.14. In the Petersson product space SK (Γ1 [N ]) with p not dividing
N the diamond operator < p > and the Hecke operator Tp have adjoints:
< p >∗ =< p >−1 and Tp∗ =< p >−1 Tp .
.Since they commute it follows that the operators are normal.
Proof. See [DS], p. 186.
This observation allows us to give a spectral theorem for Hecke operators:
Corollary 4.15. If (n, N ) = 1, the operator Tn is normal on SK (Γ1 [N ]) and
hence the space SK (Γ1 [N ]) has an orthogonal basis of simultanous eigenforms
for all Hecke operators Tn with (n, N ) = 1.
Proof. This is a standard result from functional analysis; see for example [AH],
p. 372.
35
CHAPTER 4. PETERSSON SCALAR PRODUCT
It is even possible to partly eliminate the condition (n, N ) = 1 as shown in
[DS]. As there is much additional theory needed to prove such results, we will
not do this here.
As a special case of the above theorem we get another corollary:
Corollary 4.16. Let Γ ⊂ Γ1 be a congruence subgroup and let f, g ∈ Mk (Γ)
with f or g being a cusp form.
(i) If f ∈ Mk (N, χ) then < Tn f, g >= χ(n) < f, Tn g > for (n, N ) = 1.
(ii) If f ∈ Sk (N, χ) is an eigenfunction of Tp , then for its eigenvalue λ the
equation λ = χ(p)λ holds.
Proof.
(i) See [KN], p. 171.
(i)
(ii) λ < f, f >=< Tp f, f >= χ(p) < f, Tp f >= χ(p)λ < f, f >
and so λ = χ(p)λ.
This means that for arbitrary p the eigenvalues of all Tp -eigenfunctions
f ∈ Sk (N, χ) geometrically lie on a line. However, it is not clear so far,
whether there exists a basis of simulataneous eigenfunction for all Tn inside
Sk (N, χ) or not. Indeed, this is the assertion of the following propostion.
Proposition 4.17. There exists a basis of Sk (N, χ) whose elements are eigenforms for all Tn with (n, N ) = 1.
Proof. See [KN], p. 173.
36
Chapter 5
Orthogonality
In this chapter we will use the presented theorems and propositions to analyze
some orthogonality relations between modular forms.
We will present two very simple orthogonality relations first. Recall from
Chapter 1 that Mk = C · Gk ⊕ Sk (Corollary 1.15). It is possible to show that
the direct sum is even orthogonal with respect to the Petersson scalar product.
This means:
Theorem 5.1. Let k ≥ 4 and f ∈ Sk . Then
< Gk , f >= 0.
Proof. See [KK], p. 232.
Another well-known result is:
Theorem 5.2. If f ∈ Sk (N, χ) and g ∈ Sk (N, χ′ ) with χ ̸= χ′ , then f and g
are orthogonal with respect to the Petersson scalar product.
Proof. See [LS], p. 113.
This shows that the decomposition of Sk (Γ1 [N ]) into χ-eigenspaces from
theorem 2.23 is in fact orthogonal.
For further results we will need a more sophisticated technique, which we will
introduce in the next section. This theory of oldforms and newforms going
back to Oliver Atkin and Joseph Lehner [AL] asserts that the space Sk (Γ1 [M ])
decomposes into the direct sum of two subspaces.
37
CHAPTER 5. ORTHOGONALITY
5.1
Oldforms and Newforms
So far the theory has all taken place at one level N. This section concentrates
on results that move between levels, taking forms from lower levels up to higher
level. We start with a simple remark:
Remark 5.3. If M |N then Sk (Γ1 [M ]) ⊂ Sk (Γ1 [N ]). This is straightforward
since Γ1 [N ] ⊂ Γ1 [M ].
The following embedding result is less straightforward. It is very important
because the theory of oldforms and newforms, which we want to present here,
is based on it.
Proposition 5.4. Let f ∈ Mk (Γ1 [N ]), let m ∈ N be a divisor of M ∈ N.
Then the function
g(τ ) := f (mτ )
is in Mk (Γ1 [N M ]). The same holds for cusp forms.
Proof. It is clear, that g is holomorphic
on H since f is.( We check)the trans(
)
a b
a bm
formation law next, so let
∈ Γ1 [N M ]. Then
∈ Γ1 [N ],
c d
c/m d
since m|M . So
(
)
aτ + b
c
a(mτ ) + bm
a b
g(
τ ) = f (m
) = ( (mτ ) + d)k f (mτ )
) = f( c
c d
(mτ + d)
cτ + d
m
m
= (cτ + d)k g(τ ).
It remains
( to
a
Let η =
c
check
) holomorphy at the cusps.
b
∈ Γ1 and e := (c, ma). By the Euclidean
d
(
−r
that e = sc + rma with r, s ∈ Z. This shows that δ := c
e
(
) (
)
ma mb
α β
Then δ ·
=
with α, β, γ ∈ Z. Therefore
c
d
0 γ
(
)
(
ma mb
−1 α
(g|k η)(τ ) = (f |k
)(τ ) = (f |k (δ
c
d
0
ατ + β
(αβ)k/2
(f |k δ −1 )(
)
=
k
γ
γ
38
algorithm we get
)
−s
∈ Γ1 .
− ma
e
)
β
)(τ )
γ
CHAPTER 5. ORTHOGONALITY
Since f is a modular form limIm(τ )→∞ (f |k α−1 )(τ ) = G exists. So
(αβ)k/2
(αβ)k/2
−1 ατ + β
(f
|
δ
)(
)
=
G
k
Im(τ )→∞
γk
γ
γk
lim
The result now follows for modular forms and also for cusp forms since (in this
case we have G = 0).
This allows us to make a definition:
Definition 5.5. (i) A cusp form f ∈ Sk (Γ1 [N ]) is an old form, if it is a
linear combination of cusp forms gi (di τ ), with Mi > 1 is a proper divisor
of N and di ≥ 1 is a divisor of Mi and gi ∈ Sk (Γ1 [N/Mi ]. This vector
space is denoted Sold
k (Γ1 [N ]).
⊥
(ii) The space of newforms is defined by Sknew (Γ1 [N ]) := (Sold
k (Γ1 [N ])) .
As an important link to the theory of Hecke operators we see that those
operators respect the decomposition into oldforms and newforms.
new
Proposition 5.6. The subspaces Sold
k (Γ1 [N ]) and Sk (Γ1 [N ]) are stable under
the Hecke operators Tn .
Proof. See [DS], p. 188.
Considering the results from Chapter 4, we therefore have:
new
Corollary 5.7. The subspaces Sold
k (Γ1 [N ]) and Sk (Γ1 [N ]) have orthogonal
bases of eigenforms for the Hecke operators Tn with (n, N ) = 1 (see Corollary 4.15).
By Atkin-Lehner theory one then has a direct sum decomposition:
Theorem 5.8.
Sold
k (Γ1 [M ]) =
⊕
Sknew (Γ1 [t])|Vd .
(5.1)
td|M,t̸=M
Proof. The proof is very complicated (see [DS], p. 197-198). It uses Strong
Multiplicity One. The proof for the latter theorem can be found in [MT],
p. 153ff. The idea of Theorem 5.8 goes back to Oliver Atkin and Joseph
Lehner who showed the above for Γ0 [M ] instead of Γ1 [M ] in 1970 (see [AL],
p. 151-153).
39
CHAPTER 5. ORTHOGONALITY
5.2
Hecke Operators and Orthogonality
It is quite a natural question to ask to what extent orthogonality may hold for
(sums of) pieces on the right hand side of (5.1). Here we would like to address
the question in a simple case, namely when M = N pr with N, r ∈ N and p is
a prime not dividing N .
r
We will consider the sum Sk (Γ1 [N ]) ⊕ Sk (Γ1 [N ])|Vp ⊂ Sold
k (Γ1 [N p ]). In order
to find a result for Sk (Γ1 [N ]) we need to analyze the χ−eigenspaces at first.
Theorem 5.9. Let p be a prime number not dividing N ∈ N, let r ∈ N, let χ
be a Dirchlet character and let Vk,N,p be the maximal subspace of Sk (N, χ) that
is orthogonal to Sk (N, χ)|Vp inside Sk (N pr , χ). Then
Vk,N,p = ker Tp .
(5.2)
Proof. In this proof we will follow a method proposed by the advisor of this
diploma thesis, Winfried Kohnen (see [KW2]), who proved the above in the
special case of Γ0 [N ].
Since p does not divide N , the operator Tp is normal and Sk (N, χ) has an orthogonal basis {f1 , ..., fg } of eigenfunctions of Tp (see Proposition 4.17). Suppose that
fν |Tp = λν,p fν
for all ν ∈ {1, ..., g}.
Recall from Corollary 3.13 that by definition
f |Tp = f |Up + χ(p)pk−1 f |Vp
Then:
λν,p < fµ , fν >=< fµ , fν |Tp >=< fµ , fν |Up > +pk−1 χ(p) < fµ , fν |Vp >
By Proposition 3.10 we may write
fµ |Up = p
k/2−1
p−1
∑
j=0
40
(
)
1 j
fµ |k
0 p
CHAPTER 5. ORTHOGONALITY
and so by Corollary 4.13
< fµ , fν |Up >= p
k/2−1
p−1
∑
j=0
=p
k/2−1
p−1
∑
j=0
(
)
1 j
< fµ , fν |k
>
0 p
(
)
p −j
< fµ | k
, fν >
0 1
Furthermore (with τ ′ = pτ ) we see by definition that
(
)
(
)
p −j
1 −j
k/2
k/2
′
k/2
fµ |k
(τ ) = p fµ (pτ − j) = p fµ (τ − j) = p fµ |k
(τ ′ ).
0 1
0 1
Since fµ ∈ Γ1 [N ] we get
... = p
k/2
′
fµ (τ ) = p
k/2
(
)
p 0
fµ (pτ ) = fµ |k
(τ )
0 1
and so
k/2−1
p
p−1
∑
j=0
(
)
(
)
p −j
p 0
k/2
< fµ |k
, fν >= p < fµ |k
, fν >
0 1
0 1
= pk < fµ |Vp , fν >
Thus for all µ and ν we obtain
λν,p < fµ , fν >= pk−1 (p < fµ |Vp , fν > +χ(p) < fµ , fν |Vp >)
(5.3)
We now have to distinguish two different cases. Let us first consider the
case where ν ̸= µ. Interchanging the roles of µ and ν in (5.3) yields
λµ,p < fν , fµ >= pk−1 (p < fν |Vp , fµ > +χ(p) < fν , fµ |Vp >).
Since the basis is orthogonal we get
0 = p < fν |Vp , fµ > +χ(p) < fν , fµ |Vp > .
Taking complex conjugates on both sides leads to
41
CHAPTER 5. ORTHOGONALITY
0 = p< fν |Vp , fµ > + χ(p)< fν , fµ |Vp >
and hence to
0 = χ(p) < fµ |Vp , fν > +p < fµ , fν |Vp > .
On the other hand because of orthogonality on the lefthand side of (5.3) we
deduce
0 = p < fµ |Vp , fν > +χ(p) < fµ , fν |Vp > .
Observing that
(
)
χ(p)
p
det
= 1 − p2 ̸= 0
p
χ(p)
we then obtain
< fµ , fν |Vp >= 0 (µ ̸= ν).
(5.4)
Now let µ = ν. Then since < fν |Vp , fν >= < fν , fν |Vp > equation(5.3)
becomes
λν,p < fν , fν >= pk−1 (p< fν , fν |Vp > + χ(p) < fν , fν |Vp >).
(5.5)
Taking complex conjugates on both sides and taking into account that
< fν , fν >= < fν , fν > we get
λν,p < fν , fν >= pk−1 (p < fν , fν |Vp > +χ(p)< fν , fν |Vp >).
(5.6)
We know from Corollary 4.16 that λν,p = χ(p)λν,p . Inserting this into (5.6)
and then using (5.5) we obtain
λν,p < fν , fν >= χ(p)λν,p < fν , fν >= pk−1 (χ(p)p < fν , fν |Vp > +< fν , fν |Vp >).
(5.7)
Thus from the equations (5.5) and (5.7) we have the linear equation system:
42
CHAPTER 5. ORTHOGONALITY
λν,p /pk−1 < fν , fν >= χ(p) < fν , fν |Vp > +p< fν , fν |Vp >
λν,p /pk−1 < fν , fν >= χ(p)p < fν , fν |Vp > +< fν , fν |Vp >
(5.8)
We now compute the determinant on the righthand side as:
(
χ(p) p
det
χ(p)p 1
)
= χ(p) − χ(p)p2 ̸= 0
This means that λν,p = 0 implies < fν , fν |Vp >= 0. On the other hand if
< fν , fν |Vp >= 0 then of course < fν , fν |Vp > = 0 and so
< fν , fν |Vp >= 0 ⇔ λν,p = 0.
(5.9)
Considering that λν,p = 0 ⇔ fν ∈ ker Tp we obtain by (5.4) and (5.9):
< fν , fi |Vp >= 0 ∀i ∈ {1, ..., g} ⇔ fν ∈ ker Tp
(5.10)
Since {f1 , ..., fg } is a basis for Sk (N, χ), {f1 |Vp , ..., fg |Vp } is a basis for
Sk (N, χ)|Vp . Our claim that Vk,N,p = ker Tp now easily follows. Indeed, if
{f1 , ..., fτ } (τ ≤ g) is a basis of ker Tp , then from (5.10) we immediately
deduce that {f1 , ..., fτ } (τ ≤ g) is a basis of Vk,N,p , too.
Corollary 5.10. Let p be a prime number not dividing N ∈ N, let r ∈ N
and let Vk,N,p be the maximal subspace of Sk (Γ0 [N ]) that is orthogonal to
Sk (Γ0 [N ])|Vp inside Sk (Γ0 [N pr ]). Then
Vk,N,p = ker Tp .
(5.11)
Proof. This follows immediately from Theorem 5.9 since Sk (N, χ0 ) = Sk (Γ0 [N ]).
We can now state the corresponding result for Sk (Γ1 [N ]).
43
CHAPTER 5. ORTHOGONALITY
Corollary 5.11. Let p be a prime number not dividing N ∈ N, let r ∈ N
and let Vk,N,p be the maximal subspace of Sk (Γ1 [N ]) that is orthogonal to
Sk (Γ1 [N ])|Vp inside Sk (Γ1 [N pr ]). Then
Vk,N,p = ker Tp .
(5.12)
Proof. This follows from Theorem 5.9 since the space Sk (Γ1 [N ]) decomposes
into χ-eigenspace (Theorem 2.23) and since by Theorem 5.2 this decomposition
is orhtogonal. A very similar calculation as in Theorem 5.9 then shows our
claim.
This corollary shows that in level N pr the space of the oldforms of level N ,
which are orthogonal to the push forward of the space of level N under the
map f (z) 7→ f (pz) is exactly the kernel of the p-th Hecke operator Tp acting
on the space of forms of level N .
5.3
Concluding Remarks
Our main result can be understood in two ways: On the one hand it is a
statement about a orthogonality relation in the space of oldforms. On the
other hand it is a characterization of the kernel of the p-th Hecke operator on
a certain space. It is surely not very surprising that the result does not only
hold for Γ0 [N ], as shown by Winfried Kohnen, but also for Γ1 [N ] (although
both spaces are in fact rather different). The proof of our result is not too
complicated and primarly relies on the key relation < f |k α, g >=< f, g|k α∗ >
of the Petersson scalar product, which allows one to ”simplify” the Hecke
action considerably, as well as some rather basic calculations. Finally, we
would once again like to stress that there might still be some scope for further
generalization providing an interesting topic for further research.
44
List of Symbols
αf (n) . . . . . . . .
χ ............
χ0 . . . . . . . . . . .
∆ ............
< d > ........
dµ . . . . . . . . . . .
Ek . . . . . . . . . . .
F ............
F ............
F(Γ) . . . . . . . . .
f [ΓαΓ′ ]k . . . . .
f |k α . . . . . . . . .
< f, g > . . . . . .
Γ ............
Γ[N ] . . . . . . . . .
Γ0 [N ] . . . . . . . .
Γ1 . . . . . . . . . . .
Γ1 [N ] . . . . . . . .
Gk (τ ) . . . . . . . .
GL2 (Q) . . . . . .
nth Fourier coefficient of f , 9
Dirichlet character, 20
Trivial Dirichlet character, 20
Discriminant function, 11
Diamond operator, 25
Hyperbolic measure, 31
Normalized Eistenstein series of weight k, 11
Fundamental domain, 8
Standard fundamental domain, 8
Fundamental domain of Γ, 17
Weight-k double coset operator, 23
Slash operator, 19
Petersson scalar product, 33
Congruence subgroup, 15
Principal congruence subgroup of level N , 14
A certain congruence subgroup, 15
Modular group, 7
A certain congruence subgroup, 15
Eisenstein series of weight k, 10
General linear group of 2-by-2 matrices with rational entries,
18
GL2 (Z) . . . . . . General linear group of 2-by-2 matrices with integer entries,
14
GL+
(R)
.
.
.
.
.
.
General
linear group of 2-by-2 matrices with positive determi2
nant and real entries, 7
Gm . . . . . . . . . . Orbit of m, 17
45
G\M . . . . . . . .
Gm . . . . . . . . . .
H ............
j .............
K ............
M ............
Mk . . . . . . . . . . .
Mk (Γ) . . . . . . .
Mk (N, χ) . . . .
µ(Ω) . . . . . . . . .
(n, N ) . . . . . . . .
ord w . . . . . . . .
ordw f . . . . . . . .
P1 (Q) . . . . . . . .
ϕ ............
ρ .............
σs (m) . . . . . . . .
Sk . . . . . . . . . . . .
Sk (Γ) . . . . . . . .
Sk (N, χ) . . . . .
Snew
k (Γ1 [N ]) . .
Sold
k (Γ1 [N ]) . . .
SL2 (Z) . . . . . . .
τ (n) . . . . . . . . .
Tn . . . . . . . . . . .
Tp . . . . . . . . . . . .
Ud . . . . . . . . . . .
Vd . . . . . . . . . . . .
Vk . . . . . . . . . . .
ζ(k) . . . . . . . . . .
Orbit space of G on M , 17
Stabilizer of m, 17
Complex upper half-plane, 7
j-invariant, 13
V0 , 9
Union of modular forms of all weights, 10
Modular forms of weight k, 10
Modular forms of weight k with respect to Γ, 20
χ-eigenspace of Mk (Γ1 [N ]), 21
Hyperbolic measure of Ω, 31
Greatest common divisor of n and N , 29
Order of a point w, 12
Order of f at point w, 12
Projective line of Q, 18
Euler totient
function, 16
√
1
1
+ 2 i 3, 8
2
Arithmetic function, 10
Cusp forms of weight k, 10
Cusp forms of weight k with respect to Γ, 20
χ-eigenspace of Sk (Γ1 [N ]), 21
Newforms at level N, 38
Oldforms at level N, 38
Group of 2-by-2 matrices with entries in Z and determinant
1, 8
Ramanujan’s Tau function, 11
Hecke operator, n any positive integer, 29
Hecke operator, p prime, 25
Ud -operator, 28
Vd -operator, 28
Modular functions of weight k, 9
Riemann zeta function, 10
46
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48
Erklärung
Hiermit erkläre ich, dass ich meine Arbeit selbstständig unter
Anleitung verfasst habe, dass ich keine anderen als die
angegebenen Quellen und Hilfsmittel benutzt habe, und dass ich
alle Stellen, die dem Wortlaut oder dem Sinne nach anderen
Werken entlehnt sind, durch die Angabe der Quellen als
Entlehnungen kenntlich gemacht habe.
Heidelberg, den 8. Dezember 2008
Christian Weiß