Examensarbete i matematik, 30 hp Handledare: Wolfgang Staubach Examinator: Magnus Jacobsson Juni 2021
Department of Mathematics Uppsala University
Modular Forms and Related Topics
Linnea Rousu
This thesis is a study of modular forms and related topics. It begins with some necessary background information on the modular group, SL2(Z), and the fundamental domain. Following that, modular functions are presented. These
functions are invariant under the action of the modular group and are then generalized to modular forms. As an example of modular forms, the Eisenstein series is derived. Thereafter, Hecke operators are discussed. They are defined as
averaging operators over a suitable collection of double cosets with respect to a group. This chapter contains a section about Hecke operators on modular forms.
The final part of the thesis covers Dirichlet series, which are series of arithmetic functions. An interesting connection to modular forms is discovered.
First and foremost I would like to thank my supervisor, Wulf Staubach for all his help and guidance. I am very grateful that he has taken the time to supervise me.
I would also like to thank Alexander Söderberg for his support and the discussions we have had throughout the process. Additionally I would like to thank my family for all the support throughout the years. Furthermore I am grateful to all the teachers who have inspired me to study mathematics. I would also like to thank my classmates for making the years at university memorable.
Contents
1 Introduction 4
2 Modular group 5
3 Fundamental domain 7
4 Modular functions 9
5 Modular forms 12
5.1 Eisenstein series . . . . 17
6 Hecke operators 23 6.1 Introduction . . . . 23
6.2 Hecke operators . . . . 23
6.3 Hecke operators on periodic functions . . . . 25
6.4 Hecke operators on modular forms . . . . 28
7 Dirichlet series 32 7.1 Introduction . . . . 32
7.2 Dirichlet series and modular forms . . . . 33
7.3 The half plane of absolute convergence . . . . 35
7.4 The function defined by a Dirichlet series . . . . 36
7.5 Multiplication of Dirichlet series . . . . 38
7.6 Euler products . . . . 40
7.7 The half-plane of convergence of a Dirichlet series . . . . 42
7.8 Analytic properties of Dirichlet series . . . . 44
7.9 Dirichlet series with non-negative coefficients . . . . 45
7.10 Analytic continuation . . . . 46
7.11 Mean value formulas for Dirichlet series . . . . 48
7.12 An integral formula for the coefficients of a Dirichlet series 49
1 Introduction
Historically the study of modular functions (Swe: "modulära funktioner") started in 19th century where they first appeared in the theory of elliptic functions, more specifically as elements of the function field of an elliptic curve. The term goes back to Peter Gustav Lejeune Dirichlet, although the functions also occurred in the works of Carl Friedrich Gauss, Niels Henrik Abel and Carl Gustav Jacobi (in connection to his work on theta functions). Later, they also played a significant role in the works of Leopold Kronecker, Gotthold Eisenstein and Karl Weierstrass. Towards the end of 19th centry, modular functions became a crucial source of inspiration for Felix Klein and Henri Poincaré in the development of the theory of automorphic functions. Indeed the theory of Riemann surfaces became an important tool in this context, and it was Klein who used the term "Modulform" for the first time. A sys- tematic study of modular forms on SL2(Z) and its congruence subgroups, was made by Erich Hecke in 1925, and this established the modular forms as an independent discipline within function theory and analytic number theory.
Modular forms are used in several mathematical topics due to their geometrical, arithmetical and topological properties. For example topological modular forms is currently a topic of big interest in research. We will now present some applications of modular forms, based on [8].
In the proof of Fermat’s last theorem, Andrew Wiles uses modular forms exten- sively. Additionally, from this proof new techniques were developed to solve certain diophantine equations. These developments relied on having access to tables or software for computing modular forms.
Modular forms are actually used in cryptography and coding theory. More specif- ically, to construct elliptic curve cryptosystems one wants to count the number of points on the elliptic curves. In order to do so there are point counting algorithms which use modular forms. Furthermore, algebraic forms associated to modular forms are used in certain error-correcting codes.
This essay is a compendium of modular forms and related topics. Our goal is to make it as self-contained as possible and to include the details which explain the theory. In order to read this essay one should be familiar with abstract algebra, Fourier series, functional analysis and theory of integration on a basic level, as well as complex analysis on an advanced level.
We will begin this essay with section 2, covering the modular group, which is the integer subgroup of the special linear group. Most of the mathematics in the following will be done on this group, or on some subgroup. In section 3 we present the fundamental domain; given a topological space and a group Γ0 acting on it, a fundamental domain is a subspace (of the topological space) containing exactly one point of each Γ0−equivalence class. Section 4 covers modular functions which are functions invariant under the modular group and meromorphic on H ∪ {∞}. The
modular functions are then generalized to modular forms in section 5. These forms are, in the most basic case, modular functions holomorphic on H ∪ {∞}. Following that, we derive a famous example of modular forms, called Eisenstein series in 5.1.
In section 6 we discuss the properties of the so called Hecke operators. These act as averaging operators over a certain collection of double cosets with respect to a group. They arose from Hecke’s theory on classifying the modular forms having multiplicative Fourier coefficients. In this chapter we also discuss the properties of Hecke operators on modular forms. Section 7 covers the Dirichlet series, which are series of arithmetic functions. These series are central in the theory of analytic num- ber theory. For example, the famous Riemann zeta function is actually a Dirichlet series. The chapter begins with an interesting connection between modular forms and Dirichlet series. We show that there is a one-to-one correspondence between certain modular forms and Dirichlet series satisfying a specific functional equation.
2 Modular group
The following chapter is based on [5] and [7]. The special linear group SL2(R) =
a b c d
: a, b, c, d ∈ R, ad − bc = 1
acts on the complex upper half plane H = {z ∈ C : Im(z) > 0} by linear fractional transformations as γz = az+bcz+d for γ = a b
c d
∈ SL2(R). The action is indeed a group action. Firstly, since if γ =a b
c d
∈ SL2(R) and z ∈ H, then
γz = az + b
cz + d = (az + b)(cz + d) (cz + d)(cz + d)
= (bcz + adz) + (aczz + bd) (cz + d)(cz + d)
= ac|z|2+ bd + 2bc Re(z) + z
|cz + d|2 .
(1)
Therefore the imaginary part of γz is
Im (γz) = ad − bc
|cz + d|2Im (z) = Im (z)
|cz + d|2 > 0. (2) Thus SL2(R) × H → H.
Secondly, for the identity element I we have that
Iz = 1 · z 1 = z.
Thirdly, the action also satisfies the compatibility condition, i.e.
(γ ˜γ)z = (a˜a + b˜c)z + (a˜b + b ˜d)
(˜ac + d˜c)z + (˜bc + d ˜d) = γ(˜γ(z)) for all γ = a b
c d
, ˜γ = ˜a ˜b
˜ c d˜
∈ SL2(R) and z ∈ H. The transformations de- scribed above are also called Möbius transformations, which one may recognize from complex analysis. A particular discrete subgroup of SL2(R) plays a fundamental role in various branches of mathematics and physics and is called the modular group.
Definition 2.1. The modular group Γ (Swe: "den modulära gruppen") is the group of all matrices a b
c d
for a, b, c, d ∈ Z such that ad − bc = 1.
We also note that changing the sign of γ ∈ Γ does not change the action. Since, if γ =a b
c d
∈ Γ, then we have
−γz = −az − b
−cz − d = az + b
cz + d = γz. (3)
Theorem 2.2. The modular group Γ is generated by S = 0 1
−1 0
and T =
1 1 0 1
, i.e every A ∈ Γ can be expressed as A = Tn1STn2S . . . STnk for nj ∈ Z, j = 1, . . . , k.
Proof. Since S, T ∈ Γ we have that hS, T i ⊆ Γ where hS, T i is the span of S and T.
It remains to show that Γ ⊆ hS, T i. We have Tx=1 x 0 1
∈ hS, T i for x ∈ Z.
So N := 1 x 0 1
, x ∈ Z}
⊆ hS, T i. Since S−1T−yS = 1 0 y 1
we also have that N :=1 0
y 1
, y ∈ Z
⊆ hS, T i.Now leta b c d
be an arbitrary element of
Γ. We want to show thata b c d
∈ hS, T ias well. First note that S−1a b c d
S =
d −c
−b a
, and so without loss of generality we may assume that |a| 6 |d| . Next observe that a b
c d
1 x 0 1
= a ax + b c cx + d
. We may therefore assume that 0 6 b < |a| .
We also have that 1 0 y 1
a b c d
=
a b
ay + c by + d
. Hence we may assume that 0 6 c < |a| .
Since a b c d
∈ Γ we have that ad and bc are integers such that ad − bc = 1.
By the assumptions on the matrix elements we have that |ad| > a2 and |bc| < a2. Therefore we must have ad = 1 and bc = 0, which means thata b
c d
= ±1 0 0 1
∈ N ⊆ hS, T i.Thus an arbitrary element of Γ is in hS, T i, so Γ ⊆ hS, T i.
Now we would like to study the induced maps of the generators of Γ. For S = 0 1
−1 0
and T =1 1 0 1
we have that
Sz = −1
z and T z = z + 1 for all z ∈ H.
Since S and T generate the modular group, the induced maps generate the group of transformations defined by the modular group.
An important class of subgroups of the modular group is called congruence sub- groups, which we will now define.
Definition 2.3. A congruence subgroup (Swe: "kongruens delgrupp") of Γ is any subgroup of Γ = SL2(Z) containing Γ0(N ) = ker (SL2(Z) → SL2(Z \ N Z)). The smallest such N is the level of Γ0.
In this essay we will mainly make use of the congruence subgroup Γ0(N )defined below
Definition 2.4. Let N ∈ N, the modular group (Swe: "den modulära gruppen") Γ0(N ) of level N is
Γ0(N ) =a b c d
∈ Γ : c ≡ 0 mod N
.
Note that Γ0(1) = Γ so the modular group is actually the modular group of level 1.
3 Fundamental domain
The following is based on [5].
Definition 3.1. Let F ∈ H be a closed set with connected interior and let Γ0 ⊆ Γ be a subgroup of the modular group. F is called a fundamental domain (Swe:
"grundläggande domän") for Γ0 if
(i) Any z ∈ H is Γ0−equivalent to some point in F.
(ii) No two boundary points of F are Γ0−equivalent.
(iii) The boundary ∂F of F is a finite union of smooth curves in H ∩ F.
So given a topological space and a group acting on it, the fundamental domain is a subset of the space containing exactly one point from each equivalence class. For the modular group Γ the fundamental domain is given by the following proposition Proposition 3.2. The set
F =
z ∈ H : |Re (z)| 6 1
2, |z| > 1
is a fundamental domain for Γ.
Proof. (i) We want to show that for all z ∈ H there exists γ =a b c d
∈ Γ such that γz ∈ F. Recall that by equality (2) we have
Im (γz) = Im (z)
|cz + d|2
for c, d ∈ Z. So c and d cannot both be zero. Consequently |cz + d|2 attains a minimum as γ ranges over Γ. Therefore we may choose γ such that Im (γz) is maximal. To this end let T = 1 1
0 1
, then we have Tn =1 n 0 1
. Now choose n ∈ Z such that |Re ((Tnγ)z)| 6 12 and define γ0 = Tnγ for such n.
We would like to show that |γz| > 1. Towards contradiction, suppose not, and consider the point w = γ0z with w = u + iv. Here we observe that for S = 0 −1
1 0
we have
Sw = −1
u + iv = −u + iv u2+ v2. If |w| < 1 then
Im (Sw) = Im (w)
|w|2 > Im (w), i.e. Im (Sγ0z) > Im (γ0z) which implies
Im (Sγz) > Im (γz), contradicting maximality of Im (γz).
(ii) Suppose z and w are distinct points in the interior of F and γz = w for some γ =a b
c d
∈ Γ.We may assume that Im (w) > Im (z) (otherwise, just switch
z and w and replace γ with γ−1). Again, as in (2) we have Im (w) = Im (z)
|cz + d|2, and therefore |cz + d| < 1. In particular
|c| Im (z) = |Im (cz + d)| < 1.
But since z ∈ F we have that Im (z) > √23, which yields
|c| < 1
Im (z) < 2
√3. Thus since c ∈ Z we must have c ∈ {0, ±1}.
If c = 0, then the determinant condition gives ad = 1. Multiplying γ with
±1 if necessary (recalling (3), this does not change γ in Γ) we may assume a = d = 1. Hence we must have γ = Tn for some n ∈ Z, however Tnz 6∈ F.
Next suppose c = ±1. By multiplying γ with ±1 if necessary we may assume c = 1. Note that
1 > |cz + d|2 = Re (z + d)2+ Im (z)2 > Re (z + d)2+3 4,
which implies |Re (z + d)| < 12. Since |Re (z)| < 12 we must have d = 0. Hence γ = a −1
1 0
, thus γz = a − 1z which cannot be in F. To summarise, no two distinct points in F are in the same equivalence class of Γ.
(iii) The boundary
∂F =
z ∈ H : |Re (z)| = 1
2, |z| = 1
is obviously a union of smooth curves.
4 Modular functions
A modular function is a function that is invariant with respect to the modular group, and is meromorphic on H ∪ {∞}. The following section is based on [5] and [8].
Definition 4.1. A weakly modular function (Swe: "svagt modulär funktion) of weight k is a meromorphic function f on H such that
f (γz) = (cz + d)kf (z) (4)
for all γ =a b c d
∈ Γ and all z ∈ H.
We will sometimes use the notation f|γ,k(z) = (cz + d)−kf (γz),for γ =a b c d
. Remark 4.2. Using the notation above, it is clear that the function f satisfies the weak modularity condition (4) if f|γ,k(z) = f (z) for all γ ∈ Γ and z ∈ H.
Example 4.1. The zero function is weakly modular of any weight k ∈ Z.
Example 4.2. The constant functions on H are weakly modular of weight 0.
Lemma 4.3. Suppose f and g are weakly modular functions of weight k and l respectively for the modular group Γ. Then
(i) For k = l and λ ∈ C, λf + g is a weakly modular function of weight k for Γ. Hence weakly modular functions of weight k form a complex vector space, which is denoted by Mk.
(ii) The product fg is a weakly modular function of weight k + l for Γ.
(iii) The fraction fg is a weakly modular function of weight k − l for Γ provided g 6≡ 0.
Proof. (i) We have
λf (z) + g(z) = (cz + d)−k(λf (γ(z)) + g(γ(z))).
Let Mk be the set of weakly modular functions of weight k. This is a subset of the vector space of meromorphic functions. Since 0 ∈ Mk (recall example 4.1) and λf +g ∈ Mk, Mkit is in fact a subspace. Hence Mk is itself a vector space.
(ii) We have
f (z)g(z) = (cz + d)−(k+l)f (γ(z))g(γ(z)).
(iii) We have
f (z)
g(z) = (cz + d)−(k−l)f (γz) g(γz).
Proposition 4.4. There is no non-zero weakly modular function of odd weight.
Proof. Assume f is weakly modular of odd weight k. Apply −I, then f (z) = (−1)−kf −z
−1
= −f (z).
Hence f must be 0.
Remark 4.5. When k is even, the condition (4) gives
f (γz) = (cz + d)kf (z) = d(γz) dz
−k2
f (z).
Hence (4) is equivalent to
f (γz)(d(γ(z)))k/2= f (z)(dz)k/2.
This means that the differential form f (z)(dz)k/2of weight k is fixed under the action of every element of the modular group Γ.
Theorem 4.6. Let f be a meromorphic function on H. Then f is a weakly modular function of weight 2k for k ∈ Z if and only if
f (z + 1) = f (z), f −1
z
= z2kf (z) for all z ∈ H.
Proof. Let f be a meromorphic function. First assume that f is a modular function of degree 2k for k ∈ Z. We have
f (z + 1) = f 1 · z + 1 0 · z + 1
= (0 · z + 1)2kf (z) = f (z).
We also have
f −1 z
= f 0 · z − 1 1 · z + 0
= (1 · z + 0)2kf (z) = z2kf (z), These yield
f (z + 1) = f (z) and f −1 z
= z2kf (z).
Next assume that f satisfies
f (z + 1) = f (z) and f −1 z
= z2kf (z).
If we can show that formula (4) holds for S = 0 1
−1 0
and T =1 1 0 1
then since S and T generates Γ (theorem 2.2), formula (4) holds for all γ ∈ Γ. Consequently f is a weakly modular function of weight 2k.
The condition f(z + 1) = f(z) gives
f (z + 1) = f 1 · z + 1 0 · z + 1
= (0 · z + 1)2kf (z).
Thus formula (4) holds for T =1 1 0 1
. Using f −1z = z2kf (z) we have
f −1 z
= f
0 · z + 1
−1 · z + 0
= ((−1) · z + 0)2kf (z) = z2kf (z).
Thus formula (4) holds for S = 0 1
−1 0
. Hence f is weakly modular of weight 2k.
Definition 4.7. A modular function (Swe: "modulär funktion) of weight k is a weakly modular function of weight k that is also meromorphic at ∞.
Example 4.3. The zero function is a modular function of every weight k ∈ Z.
Example 4.4. Every constant function on H is a modular function of weight 0.
Later on, in section 5.1, we shall see a less trivial example of modular functions.
5 Modular forms
A modular form is a holomorphic function on H satisfying a certain kind of functional equation with respect to the group action of the modular group. Modular forms appear in many areas of mathematics such as number theory and algebraic topology, and also in physics, in particular string theory [8]. The following chapter is based on [5] and [8].
Definition 5.1. A modular form (Swe:"modulär form") of weight k and level 1 is a modular function of weight k that is holomorphic on H ∪ {∞}.
Compared to modular functions of weight k we demand holomorphicity instead of meromorphicity on H ∪ {∞}.
Example 5.1. The zero function is a modular form of every weight k ∈ Z and level 1.
Example 5.2. Every constant function on H is a modular form of weight 0 and level 1.
These are of course trivial examples and in Section 5.1 we shall see an example of a non-trivial modular form represented as an Eisenstein series.
We may now generalise the modular forms to any level. In order to do so we need the following definitions.
Definition 5.2. Let Γ0 ⊆ Γ. The cusps (Swe: "spetsarna") of Γ0 are the set of Γ0−equivalence classes of Q ∪ {i∞}.
Example 5.3. Let a b c d
∈ Γ0(2) then c ≡ 0 mod 2 and a ≡ b ≡ d ≡ 1 mod 2.
We have
a b c d
i∞ = a c.
So i∞ is Γ0(2)−equivalent to a nonzero rational number ac if and only if c is even and a is odd. Similarly,
a b c d
0 = b
d.
Thus 0 is Γ0(2)−equivalent to a nonzero rational number db if and only if d is odd.
Hence Γ0(2) has exactly two cusps, represented by 0 and i∞.
Remark 5.3. Note that we can generalise the above example so that, for any prime p, Γ0(p) has exactly the two cusps represented by 0 and i∞.
Lemma 5.4. For all α, β ∈ Q ∪ {i∞} there exists γ ∈ Γ such that γα = β.
Proof. • For α = β = i∞, choose γ = I, then I(i∞) = i∞.
• For α = i∞, β = pq ∈ Q, choose γ =p b q d
∈ Γ. Then p b q d
i∞ = pq.
• For α = pq ∈ Q, β = i∞, choose a, b, c, d ∈ Z such that cp + dq = 0 and ad − bc = 1, then a b
c d
p
q = ap+bqcp+dq = i∞.
• For α = pq, β = rs ∈ Q, choose a, b, c, d ∈ Z such that ap + bq = r, cp + dq = s and ad − bc = 1. Then a b
c d
p
q = ap+dqcp+dq = rs.
Proposition 5.5. For any congruence subgroup Γ0(N ) the set of cusps is finite.
Proof. Consider any pq ∈ Q in reduced form. We may assume that pq is not Γ0(N )−
equivalent to i∞. There exists c0, d ∈ Z relative prime such that c0pN + dq = gcd(pN, q) which equals gcd(N, q) since gcd(p, q) = 1 . Furthermore c0pN + dq =
gcd(N, q) implies gcd(c, d) = 1 for c = c0N. Hence, there exists a, b such that
a b c d
∈ Γ0(N ). Therefore,
a b c d
p
q = ap + bq
cp + dq = ap + bq gcd(N, q).
Thus, replacing pq by something Γ0(N )−equivalent we may assume q|N. Since T =
1 1 0 1
∈ Γ0(N )we have Tn pq =1 n 0 1
p
q = p+nqq is also Γ0(N )−equivalent to pq so we may assume 0 6 p < q. This gives only a finite number of possible non-equivalent choices of pq.
In order to make sense of holomorphicity of a weakly modular function f for an arbitrary congruence subgroup Γ0 at any α ∈ Q we present the following lemma.
Lemma 5.6. Let f : H → C be a weakly modular function of weight k for a congruence subgroup Γ0. If δ ∈ Γ then f |δ,k is a weakly modular function for δ−1Γδ.
Proof. For s = δ−1γδ ∈ δ−1Γδ we have
(f |δ,k)|s,k = f |δs,k = f |γδ,k = f |δ,k. Hence f|δ,k is a weakly modular function if f is.
Now fix a weakly modular function f of weight k for a congruence subgroup Γ0 and suppose α ∈ Q.
If T = 1 1 0 1
∈ Γ0 then we would have f(z + 1) = f(z), but this is not the case for any congruence subgroup Γ0. Fortunately TN = 1 N
0 1
∈ Γ(N ), so any congruence subgroup of level N contains TN. Hence we have f(z + j) = f(z) for some positive j ∈ Z. Choose the minimal j > 0 such that 1 j
0 1
∈ δ−1Γ0δ where δ(i∞) = α. If the function f is holomoprhic at i∞ we obtain a Fourier expansion f (z) =P∞
n=0anqn/j where q1/j = e2πiz/j. For the other cusp α ∈ Q, lemma 5.4 gives some γ ∈ Γ such that γ(i∞) = α. So we consider f to be holomorphic at the cusp α if the weakly modular function f|γ,k is holomorphic at i∞.
Proposition 5.7. If a weakly modular function f is holomorphic at a set of repre- sentative elements for the cusps of Γ0(N ), then it is holomorphic at every element of Q ∪ {i∞}.
Proof. Let c1, . . . , cn ∈ Q ∪ {i∞} be representatives for the set of cusps of Γ0(N ) such that f is holomorphic at c1, . . . , cn ∈ Q ∪ {i∞}. If α ∈ Q ∪ {i∞}, then by lemma 5.4 there exists a γ ∈ Γ0(N ) such that α = γci for some i. If δ ∈ Γ is
such that δ(i∞) = ci then f|δ,k is holomorphic at i∞. Since f is a weakly modular function for Γ0(N ) we have
f |δ,k= (f |γ,k)|δ,k = f |γδ,k. (5) But γ(δ(i∞)) = γ(ci) = α so (5) makes f holomorphic at α.
Remark 5.8. Using proposition 5.7 we may generalize the discussion above. So that a weakly modular function f (z) is considered to be holomorphic at the cusps for a congruence subgroup Γ0(N ) if f |γ,k(z) is holomorphic at z = i∞ for all γ ∈ Γ.
To see this, let γ ∈ Γ be arbitrary. Assume the weakly modular function f |γ,k is holomorphic at i∞, then by proposition 5.7, f |γ,k(z) is holomorphic at every z ∈ Q ∪ {i∞}. Consider an arbitrary representative ci of the cusps. By proposition 5.5 we have i ∈ {1, . . . , n}. The equivalence class of ci is {γci : γ ∈ Γ0}. We have
f |γ,k(ci) = (˜cci+ ˜d)−kf (γci)
for all γ = ˜a ˜b
˜ c d˜
∈ Γ0. Thus, for f |γ,k holomorphic at ci, f is also holomorphic at γci for all γ ∈ Γ0.
Remark 5.9. There is a connection between a function being holomorphic at i∞
and it being bounded, which we will now explain. If f is a meromorphic periodic function of period 1 then f has a Fourier expansion
f (z) =X
n∈Z
anqn (6)
for some constants an and q = e2πiz. Consider the map z = x + iy 7→ e−2πye2πx, it is an holomorphic bijection from the vertical strip z ∈ H : 12 6 Im z < 12 in H to the punctured disk Dx = {z ∈ C : 0 < |z| < 1}. So we might think of f (z) as a function F (q) on Dx. As z → i∞ we recieve q → 0. Thus, for f (z) to be holomorphic at i∞
we need F (q) to be holomorphic at q = 0. If F (q) is holomorphic at q = 0 it must have a Laurent expansion which must be equal to the Fourier-expansion (6), meaning that an = 0 for all n < 0. This is equivalent to the condition that f (z) grows less than exponentially. i.e. there exists m such that |f (z)| < emy for y = Im z 0 (see [5] page 20 for the corresponding equivalence for f meromorphic at i∞).
Definition 5.10. Let Γ0 be a congruence subgroup of Γ of level N . A modular form f of weight k and level N for Γ0 is a weakly modular function f : H → C of weight k, holomorphic on H and at the cusps of Γ0.
We denote the space of modular forms of weight k for Γ0(N )by Mk(N ). Proposition 5.11. Mk(N ) is a complex vector space for even k > 0.
Proof. We will show that Mk(N ) is a subspace of the vector space of holomorphic functions on H and hence a vector space.
• For f ∈ Mk(N ) and c ∈ C we have cf ∈ Mk(N ).
• For f, g ∈ Mk(N )we have
f (z) + g(z) = (cz + d)−k(f (γz) + g(γz)) for all γ =a b c d
∈ Γ.
• If f and g are holomorphic at i∞ then f|γ,k and g|γ,k are bounded at i∞ for all γ = a b
c d
∈ Γ. Hence
(f + g)|γ,k = f |γ,k+ g|γ,k is bounded at i∞ for all γ =a b
c d
∈ Γ, which means that f + g is holomor- phic at the cusps.
Lemma 5.12. If f ∈ Mk(N ) and g ∈ Ml(N ) then f g ∈ Mk+l(N ).
Proof. By lemma 4.3, fg is a weakly modular function of weight k + l. Since both f and g are holomorphic at H, so is fg.
As in the proof of proposition 5.11, if f|γ,k, g|γ,k are bounded at i∞ for all γ =a b
c d
∈ Γ then
(f g)γ,k+l = (f |γ,k)(g|γ,l) is bounded at i∞ for all γ =a b
c d
∈ Γ. Hence fg is holomorphic at the cusps.
Another interesting feature of modular forms is their specific Fourier expansion in the upper half-plane. More precisely, if f is a modular form there are constants an such that
f (z) =
∞
X
n=0
anqn
for all z ∈ H and q = e2iπz. This series converges for all z ∈ H since f(q) is holomor- phic on the unit disk D so its Taylor series converges absolutely in D. This feature of modular forms is sometimes included in their definition see e.g. [8].
Now we define an important class of modular forms, the so-called cusp forms.
Definition 5.13. Let Γ0 be a congruence subgroup and let f ∈ Mk(Γ0). We define f as a cusp form (Swe: "spetsform") if f vanishes at the cusps, i.e. f (z) → 0 as z → i∞. Alternatively, we can view cusp forms as modular forms with the constant Fourier coefficient vanishing, a0 = 0. The space of cusps is denoted by Sk(Γ0) or Sk(N ) for Γ0 = Γ0(N ). As Mk(N ), this is also a vector space.
In Section 5.1 we define the Eisenstein series and in Section 6.4 we define an inner product on the space of modular forms, called the Petersson product, which is defined when one of the forms is a cusp form (since cusp forms vanish at the cusps).
It turns out that the Eisenstein series are the orthogonal complement of cusp forms with respect to the Petersson product.
5.1 Eisenstein series
We will now construct a non-trivial example of a modular form, called the Eisen- stein series, which were first studied by Gotthold Eisenstein. It is actually easier to construct a modular function by considering its derivatives. If f is a modular function, then
f(k)(γz) = (cz + d)2kf(k)(z) for γ =a b
c d
∈ Γ0 where Γ0 is a congruence subgroup of Γ. Therefore we would like to construct a function that satisfies the above transformation properties.
The idea is now to take a weighted average of some function g : H → ˆC over a congruence subgroup Γ0, where ˆC is the Riemann sphere C ∪ {∞}. To do so, given γ ∈ Γdefine the automorphy factor as
j(γ, z) : Γ × H → ˆC, j(γ, z) = cz + d for γ =a b c d
. For even k ∈ Z consider the average
f (z) =X
γ∈Γ
j(γ, z)−kg(γz). (7)
Then we have
f (γ0z) =X
γ∈Γ
j(γ, γ0z)−kg(γγ0z) (8)
for γ, γ0 ∈ Γ.Now we need the following lemma for the automorphy factor j.
Lemma 5.14. Let Γ be the modular group. For γ, γ0 ∈ Γ we have j(γγ0, z) = ±j(γ0, z)j(γ, γ0z).
Proof. We observe that (2) yields
Im (γz) = Im (z)
|j(γ, z)|2 and hence
Im (z)
|j(γγ0, z)|2 = Im (γγ0z) = Im (γ0z)
|j(γ, γ0z)|2 = Im (z)
|j(γ, γ0z)j(γ0, z)|2. (9) Thus the lemma holds up to absolute values. The result will now follow from the following claim:
Suppose f and g are two meromorphic functions such that
|f (z)|2 = |g(z)|2. (10)
Then f(z) = ζg(z) for ζ such that |ζ| = 1 when g 6≡ 0. Note that the claim says that
f ¯f = g¯g and f g =
f¯
¯ g. The image of f/g is contained in
z ∈ C : z = 1
¯ z
=z ∈ C : |z|2 = 1 .
By the open mapping theorem (restrict to an open set where f/g is holomorphic) this is impossible unless f/g is constant. Then the assumption (10) implies f(z) = ζg(z) for |ζ| = 1. Hence the denominators of (9) gives
j(γγ0, z) = ζj(γ0, z)j(γ, γ0z) for |ζ| = 1.
Finally the fact that j(γ, z) ∈ Z for z ∈ Z implies ζ = ±1.
By equality (8) and this lemma we have that f (γ0z) = j(γ0, z)kX
γ∈Γ
j(γγ0, z)−kg(γγ0z)
= j(γ0, z)kX
γ∈Γ
j(γ, z)−kg(γz)
= j(γ0, z)kf (z),
(11)
where we changed γγ0−1 to γ in the middle step.
Remark 5.15. The equality (11) means that if f converges and is meromorphic on H then f is a weakly modular function of weight k.
Taking the average over the whole congruence subgroup Γ0 will be too much in order for f(z) to converge (see [5] page 48). Instead we want to start with a periodic function and average it over just what we need in order for f to satisfy the weakly modular condition for Γ. Thus, consider the modular group Γ0(N )of level N which contains T =1 1
0 1
. Since T ∈ Γ0(N ), for a weakly modular function f we have f (T z) = f (z + 1) = f (z)so f is periodic. Let P be the subgroup of Γ0(N )generated by T =1 1
0 1
, i.e.
P = hT i =1 n 0 1
: n ∈ Z
⊆ Γ0(N ).
Remark 5.16. Note that for γ ∈ P we have j(γ, z) = 1 so any periodic function with period 1 satisfies the weak modularity condition (4) for T . Thus averaging over P \ Γ0(N ) should yield something weakly modular.
Let us now consider the coset space P \ Γ0(N ).
Lemma 5.17. A complete set of representatives for P \ Γ0(N ) is parametrized by the set
(c, d) ∈ Z2 : c ≡ 0 mod N, gcd (c, d) = 1 \ {±1}
where a coset representative with parameter (c, d) can be taken uniquely in the form
a b c d
∈ Γ0(N ) for 0 6 b < |d| or a b c d
= 0 1
−1 0
if (c, d) = ±(1, 0) and N = 1.
Proof. Let γ =a b c d
∈ Γ0(N ). We can replace γ with
1 n 0 1
a b c d
=a + cn b + dn
c d
which is in the same right P -coset of Γ0(N ) as γ. First suppose that d = 0, then
−bc = 1 so b = −c = 1 which implies N = 1. Choosing n = −a shows that γ ∈ P 0 1
−1 0
.
Secondly, suppose d 6= 0. Then we may uniquely choose n such that 0 6 b+dn <
|d| . Therefore we may assume 0 6 b < |d| . On the other hand, the determinant condition gives
a = 1 + bc
d ∈ Z i.e. bc ≡ −1 mod d.
This determines b and consequently also a uniquely.
We will now choose our modular form to be a periodic function over the coset space P \ Γ0(N ). Let k > 4 be an even positive integer. For z ∈ H define the normalized Eisenstein series of weight k and level N as
Ek,N(z) = X
γ∈P \Γ0(N )
1
j(γ, z)k = 1 2
X
(c,d)∈Z2 gcd (c,d)=1 c≡0 mod N
1 (cz + d)k.
The change of summands comes from lemma 5.17 and the 12−factor comes from that we allow both (c, d) and (−c, −d) in the rightmost series. We want to show that this series is a modular form.
Proposition 5.18. The Eisenstein series Ek,N is a modular form of weight k > 4 and level N .
Proof. Firstly, lemma 5.14 gives us
j(γ, γ0z)−k = j(γ0, z)kj(γγ0, z)−k, for γ ∈ Γ and γ0 ∈ Γ0(N )since k is even. Hence
Ek,N(γ0z) = X
γ∈P \Γ0(N )
j(γ, γ0z)−k = j(γ0, z)k X
γ∈P \Γ0(N )
j(γγ0, z)−k. (12)
Since right multiplying P \ Γ0(N )by γ0 simply permutes the cosets of P \ Γ0(N )we can replace γ by γγ0−1 in the rightmost sum of equality (12), which gives
Ek,N(γ0z) = j(γ0, z)kEk,N(z). (13) We will now show absolute convergence of Ek,N. Since Ek,N is a sum of strictly smaller sets of terms than Ek,1 it suffices to show convergence of Ek,1.
Consider the fundamental domain for Γ;
F =
z ∈ H : |Re z| 6 1
2, |z| > 1
. Now let z ∈ F and let ζ3 = e(2πi)/3.
We have
|cz + d|2 = (cz + d)(c¯z + d) = c2|z|2+ 2cdRe z + d2. (14) Since z ∈ F, we have |z| > 1 and Re z 6 12. Hence (14) becomes
|cz + d|2 > c2− |cd| + d2 = |cζ + d|2 for cd > 0
|cζ3− d|2 for cd < 0 .
So z ∈ F gives
|Ek,1(z)| 6 X
c,d>0 gcd (c,d)=1
1
|cζ3+ d|k + X
c>0, d60 gcd (c,d)=1
1
|cζ3 − d|k
= 2 X
c,d>0 gcd (c,d)=1
1
|cζ3+ d|k.
We want to check how many (c, d) ∈ Z2 there are such that |cζ3 + d| < r for some r ∈ R or equivalently, such that cζ3+ d ∈ Dr when Dr is a disk centered at 0 with radius r.
Firstly, consider the case when |c| > 2r. Then we have
|Im (cζ3+ d)| = |Im (cζ3)| = |c|
√3 2 > r.
Thus cζ3+ d ≮ r. Secondly, consider the case when |c| < 2r. In order to have
|Re (cζ3+ d)| = −c
2+ d < r we need |d| < 2r.
This means that the number of (c, d) ∈ Z2 such that cζ3 + d ∈ Dr is less than 4r2 and the number of points (c, d) ∈ Z2 such that cζ3+ d ∈ Dr+1− Dr is less than 4(r + 1)2.
Hence for all z ∈ F we have
|Ek,1(z)| 6 X
(c,d)∈Z2 (c,d)6=(0,0)
1
|cζ3+ d|k 6 X
(c,d)∈Z2 r6|cζ3+d|<r+1
1 rk <
∞
X
r=1
4(r + 1)2
rk (15)
which converges for k > 4.
Thus Ek,1 converges absolutely on F and hence so does Ek,N for N ∈ Z.
For γ ∈ Γ, equality (13) gives
|Ek,1(γz)| =
j(γ, z)kEk,1(z)
. (16)
Hence (15) gives us absolute convergence of Ek,1 at any point z ∈ H. However, using (15) we also see that Ek,1 is a bounded function on F, which by Weierstrass M-test implies uniform convergence of Ek,1(z) and hence also of Ek,N(z)on F. For Ek,1 this implies compact convergence on H by the property (16) since, for fixed γ ∈ Γ, j(γ, z) is uniformly bounded on compact sets. Since the tail of a series for Ek,N(z) can be bounded by the tail of a series for Ek,1(z), Ek,N is also uniformly bounded on compact subsets of H and hence compactly convergent on H.
So Ek,N is a series of holomorphic functions which converges absolutely and com-
pactly on H and is therefore, by Montel’s theorem holomorphic on H. Furthermore, by remark 5.15 it is a weakly modular function.
Now it remains to show that the series is holomorphic at the cusps. Let τ =
r s t u
∈ Γ. Then we have
Ek,n|τ,k(z) = 1 2(tz + u)k
X
(c,d)∈Z2 gcd (c,d)=1 c≡0 mod N
1 crz+utz+u + dk
= 1 2
X
(c,d)∈Z2 gcd (c,d)=1 c≡0 mod N
1
(c(rz + s) + d(tz + u))k
= 1 2
X
(c,d)∈Z2 gcd (c,d)=1 c≡0 mod N
1
((cr + dt)z + (cs + du))k
= 1 2
X
(c,d)∈Z2 gcd (c,d)=1 c≡0 mod N
1 (c0z + d0)k
for c0 = cr + dt and d0 = cs + du. Now consider Tτ =r t s u
∈GL2(Z), it acts on
Z × Z and gives Tτc d
=c0 d0
. Hence the final sum runs over a certain subset of pairs (c0, d0) ∈ Z × Z \ {(0, 0)} . Consequently we have
|Ek,N|τ,k(z)| 6 1 2
X
(c0,d0)∈Z2\{(0,0)}
1
|c0z + d0|k
which we have shown is bounded for all z ∈ F. As z → i∞ we recieve 1
2
X
(c0,d0)∈Z2\{(0,0)}
1
|c0z + d0|k → 1 2
X
d∈Z\{0}
1
|d|k =
∞
X
d=1
1
dk = ζ(k).
This means that Ek,N|τ,k(z)is bounded at i∞, and hence holomorphic at i∞. Thus Ek,N(z) is holomorphic at the cusps.
6 Hecke operators
6.1 Introduction
Erich Hecke developed a general theory in order to classify the modular forms that have multiplicative Fourier coefficients {an}∞n=1 (i.e. amn = aman when gcd (m, n) = 1). In this theory he defined the Hecke operators, which we will define later on in this section. There are several different ways to define these operators. One can define them as functions acting on lattices, as operators acting on cusp forms or as double coset operators. In this thesis we will do the latter.
Generally, Hecke operators are averaging operators over a suitable collection of double cosets with respect to a group. This means that much of the Hecke theory belongs to linear algebra. But when studying the spectral analysis of these operators, things become more interesting. There are many results known only for arithmetic groups and therefore we will focus on the modular group.
Hecke operators commute and are self-adjoint and hence the modular forms which are eigenvectors with respect to all Hecke operators form a basis of the space of cusp forms.
The following section is based on [4].
6.2 Hecke operators
Consider the set
Gn=a b c d
: a, b, c, d ∈ Z, ad − bc = n
.
Note that G1 = Γ.
Lemma 6.1. A complete set of right coset representations of Gn modulo Γ is
∆n =a b 0 d
: ad = n, 0 6 b < d
.
This means there is a disjoint partition Gn= [
ρ∈∆n
Γρ.
Proof. Let ρ = a ∗ c ∗
∈ Gn. There are integers γ, δ such that γa + δc = 0 and gcd (a, c) = 1 (Let γ = gcd (a,c)c , δ = −gcd (a,c)a ). So there exists a matrix τ =∗ ∗
γ δ
∈ Γ giving τρ =∗ ∗ 0 ∗
.
In order for τρ to be in ∆n one may need to change the sign of τ, then we have τ ρ =a b
0 d
for ad = n, d > 0.
Multiplying on the left by 1 n 0 1
for n ∈ Z gives 0 6 b + nd < d (since the matrix product should also be in ∆n) so we may assume 0 6 b < d.
As ρ ranges over ∆n the cosets Γρ are disjoint, since
α β γ δ
a b 0 d
=a0 b0 0 d0
for α β γ δ
∈ Γ
gives
α β γ δ
=1 0 0 1
.
Remark 6.2. The number of elements of ∆n is σ(n) =P
d|nd.
Lemma 6.3. There exists a one-to-one correspondance between ∆n× Γ and Γ × ∆n. Namely for any ρ ∈ ∆n and τ ∈ Γ there exist unique τ0 ∈ Γ and ρ0 ∈ ∆n such that
ρτ = τ0ρ0. (17)
Proof. We begin by showing uniquness. Let
ρ =a b 0 d
, ρ0 =a0 b0 0 d0
, τ =α ∗ γ δ
, τ0 =α0 ∗ γ0 δ0
.
Then (17) becomes
αa + γb ∗
γd δd
=α0a0 ∗ γ0a0 γ0b0+ δ0d0
. (18)
The determinant condition gives gcd(α0, γ0) = 1 and hence a0 = gcd(αa + γb, γd),
α0 = (αa + γb)/ gcd(αa + γb, γd), γ0 = γd/ gcd(αa + γb, γd),
d0 = n/ gcd(αa + γb, γd)
because a0d0 = n. Moreover, the determinant condition gives α0δ0 ≡ 1 mod |γ0| so δ0 is given modulo |γ0|. Then, by equating the lower right entries of (18) we find b0 mod d. Restricting b0 to 0 6 b0 < d0 determines b0 uniquely, and hence also δ0.
Note that the argument for uniqueness also proves existence of ρ0, τ0 satisfying (17).
Let χ : Z → C be any function, this yields a function χ on GL2(Z) defined as
χ(ρ) = χ(a) for ρ =a ∗
∗ ∗
∈ GL2(Z). (19)
Definition 6.4. Given χ and k ∈ Z we define the Hecke operator of f : H → C as Tnf = nk2−1 X
ρ∈∆n
χ(ρ)f |ρ,k.
By lemma 6.1 we, more explicitly, have
Tn = 1 n
X
ad=n
χ(a)ak X
06b<d
a b 0 d
(20)
and
Tnf (z) = 1 n
X
ad=n
χ(a)ak X
06b<d
f az + b d
. (21)
Definition 6.5. We define the Hecke algebra as the ring of Hecke operators on Γ.
This is indeed an algebra, see [3] page 43.
6.3 Hecke operators on periodic functions
In the following we assume that
χ(1) = 1and χ(−1) = (−1)k. (22)
Consider the group
Γ∞=
±1 n 0 1
: n ∈ Z
.
From now on we consider functions automorphic to the group Γ∞, which means f |τ,k = χ(τ )f for all τ ∈ Γ∞. (23) Taking τ =1 1
0 1
∈ Γ∞ shows that f is periodic of period 1.
Theorem 6.6. The operator Tn maps periodic functions to periodic functions.
Proof. Consider
ρ =a b 0 d
, ρ0 =a b0 0 d
, τ =1 u 0 1
, τ0 =1 v 0 1
for any ρ, ρ0 ∈ ∆0n and τ, τ0 ∈ Γ∞. Then (19) and (22) yields χ(ρ) = χ(ρ0) = χ(a) and χ(τ0) = χ(τ ) = 1.
Consequently we have
χ(a) = χ(ρ)χ(τ0) = χ(ρ0)χ(τ ). (24) Furthermore the relation ρτ = τ0ρ0 (lemma 6.3) gives
f |ρτ = f |τ0ρ0 = f |ρ0 = χ(τ0)f |ρ0. (25) Combining (24) and (25) gives
(Tnf )|τ = nk2−1 X
ρ∈∆n
χ(ρ)f |ρτ
= nk2−1 X
ρ0∈∆n
χ(τ )χ(ρ0)f |ρ0 = χ(τ )Tnf.
Thus by (23), Tnf is periodic of period 1.
Now we want to study the action of the Hecke operator on the coefficients of a Fourier series.
Proposition 6.7. Let p(z) = e2πiz and consider f (z) = P∞
m=0a(m)p(mz), which converges absolutely in H. Then
Tnf (z) =
∞
X
m=0
an(m)p(mz)
with coefficients
an(m) = X
d| gcd (m,n)
χ(d)dk−1a(mnd−2).
Proof. By (21) we have
Tnf (z) = 1 n
∞
X
m=0
a(m) X
ad=n
χ(a)ak
d−1
X
b=0
p
maz + b d
.
Let
S = 1 n
∞
X
m=0
a(m)
d−1
X
b=0
p
maz + b d
.
We have
d−1
X
b=0
p
maz + b d
= p maz
d
Xd−1
b=0
p
mb
d
.
Note that
d−1
X
b=0
p
mb
d
6= 0 if and only if m ≡ 0 mod d.
Further use that n = ad, this gives
S = 1 ad
X
m=0 mod d
a(m)p maz
d
d.
Hence
Tnf (z) =
∞
X
l=0
X
ad=n
χ(a)ak−1a(dl)p(alz)
=
∞
X
m=0
X
al=mad=n
χ(a)ak−1a(dl)
p(mz),
which proves the theorem.
Corollary 6.8. For m, n> 1 we have an(m) = am(n).
From now on we assume that χ is completely multiplicative,
χ(mn) = χ(m)χ(n).
We will also show that the operator Tn has a multiplicativity property.
Theorem 6.9. For m, n> 1, we have TmTn= X
d| gcd (m,n)
χ(d)dk−1Tmnd−2.
Proof. By (20) we have
mnTmTn= X
a1d1=m a2d2=n
χ(a1a2)(a1a2)k X
b1 mod d1
b2 mod d2
a1 b1 0 d1
a2 b2 0 d2
.
The proof is done by reducing the matrix Pb1 mod d1
b2 mod d2
a1 b1 0 d1
a2 b2 0 d2
and then rearranging the terms of the sum, see [4] page 96.
Corollary 6.10. The Hecke operators commute as TmTn = TnTm.
6.4 Hecke operators on modular forms
We observe that since by Lemma 6.1 the collection ∆nforms a complete set of right cosets of Gn modulo Γ, then by restricting Tn to Mk(N ) one can exchange every ρ ∈ ∆n with anything in the coset Γρ.
Thus the Hecke operator on a modular form is Tnf = nk2−1 X
ρ∈Γ\Gn
f |ρ,k (26)
since
f |τ ρ,k = f = f |ρ,k for τ ∈ Γ and f ∈ Mk(N ). (27) Since a modular form is a periodic function (by theorem 4.6) all results from section 6.3 hold for modular forms. So for example, modular forms are commutative.
Theorem 6.11. The Hecke operator Tn maps a modular form to a modular form and a cusp form to a cusp form. I.e.
Tn: Mk(N ) → Mk(N ) Tn: Sk(N ) → Sk(N )
Proof. For f ∈ Mk(N ) meromorphic on H, Tnf is meromorphic on H. The correla- tion ρτ = τ0ρ0 and (27) gives
f |ρτ,k = f |τ0ρ0,k = f |ρ0,k and hence
(Tnf )|τ,k = nk2−1 X
ρ∈∆n
f |ρτ,k = nk2−1 X
ρ0∈∆n
f |ρ0,k = Tnf
for all τ ∈ Γ so Tnf is a modular function (by remark 4.2).
If f|ρ,k is holomorphic on H for ρ ∈ ∆n then Tnf is holomorphic on H.
Now we shall prove holomorphicity at the cusps. Let τ ∈ Γ and consider (f |ρ,k)|τ,k = f |ρτ,k for ρτ =a b
c d
. As Im z → ∞ we have
f |ρτ,k → lim
Im z→∞
1
(cz)kfa c
which is finite since f is holomorphic at the cusp a/c ∈ Q. Consequently, each f|ρτ,k
is holomorphic at i∞ and hence Tnf is holomorphic at the cusps.
Applying the argument of holomorphicity on cusps on cusp forms shows that if f vanishes at the cusps, then so does Tnf.
Corollary 6.12. The space of cusp forms Sk(N ) is spanned by the Poincaré series Pm(z) = X
τ ∈Γ∞\Γ
j(τ, z)−kp(mτ z). (28)
Proof. For proof see [4] page 53.
Hence TnPm is a linear combination of various Poincaré series.
Theorem 6.13. For k > 2, m> 0, n > 1 we have TnPm= X
d| gcd (m,n)
n d
k−1
Pmnd−2.
Proof. Combining expressions for Tn (26) and Pm (28) gives TnPm(z) = nk−1 X
g∈Γ∞\Gn
j(g, z)−kp(mgz).
Let H and G be any sets of right cosets representatives of Γ∞\ Γ and Γ \ Gn
respectively. Then both HG and GH are sets of right coset representatives. Hence
TnPm(z) = nk−1X
ρ∈G
X
τ ∈H
j(ρτ, z)−kp(mτ z)
= nk−1 X
ad=n
d−k X
b mod d
X
τ ∈H
j(τ, z)−kp aτ z + b d
= nk−1 X
ad=n d|m
d1−kX
τ ∈H
j(τ, z)−kpam d τ z
= X
d| gcd (m,n)
n d
k−1 X
τ ∈Γ∞\Γ
j(τ, z)−kp mnd−2τ z
= X
d| gcd (m,n)
n d
k−1
Pmnd−2(z)
For m = 0 theorem 6.13 gives us the Eisenstein series,
TnP0 =X
d|n
n d
k−1 X
τ ∈Γ∞\Γ
j(τ, z)−kp(0) = X
τ ∈Γ∞\Γ
j(τ, z)−k
= 1 2
X
gcd (m,n)=1
(cz + d)−k = Ek(z).
Actually, theorem 6.13 implies that the Eisenstein series Ek is an eigenfunction of all the Hecke operators Tn with eigenvalue σk−1(n). I.e.
TnEk = σk−1(n)Ek. Corollary 6.14. For m, n> 1 we have
mk−1TnPm = nk−1TmPn.
A similar symmetry occurs for the inner product hTnf, Pmi. To see this, we shall first define the inner product on Mk(N ) × Sk(N ).
Definition 6.15. We define the Petersson inner product h·, ·i : Mk(N )×Sk(N ) → C (after Hans Petersson) as
hf, gi = Z
F
f (z)g(z)(Im z)kdν(z),
where F is the fundamental domain for Γ and ν(z) = dxdyy2 for z = x + iy is the hyperbolic volume form.
Proposition 6.16. For m, n> 1 and f ∈ Mk(N ) we have mk−1hTnf, Pmi = nk−1hTmf, Pni . Proof. For verification, see [4] page 99.
Theorem 6.17. The Hecke operators Tn acting on Sk(N ) are self adjoint, meaning that
hTnf, gi = hf, Tngi for all f, g ∈ Sk(N ). (29) Proof. Since Sk(N ) is spanned by the Poincare series (28), it suffices to check (29) for the Poincare series. Corollary 6.14 and proposition 6.16 gives
hTnPm, Pli = hTnPl, Pmi = hPm, TnPli
since the Fourier coefficients of the Poincare series, and hence the inner products are real.
Now we will use a well known result from linear algebra.
Proposition 6.18. Let A ⊆ Mν(C) be a commuting family of normal matrices.
There exists a unitary matrix U ∈ Mν(C) such that U−1AU is diagonal for every A ∈ A.
This proposition can be stated in terms of linear operators.
Proposition 6.19. Let S be a finite dimensional Hilbert space over C and T a commuting family of normal operators T : S → S. Then there exits an orthonormal basis F of S consisting of common eigenfunctions of all the operators in T .
Proof. Let T = {fi}i6ν be an orthonormal basis of S. Then T fi =X
j
λij(τ )fj
with some λij(τ ) ∈ C. In this way T is represented by a matrix Λ(T ) = (λij(T )) which commutes with its adjoint Λ(T )∗ = Λ(T )0. This means that Λ(T ) is a normal matrix. These matrices commute, so by a suitable linear transformation with a unitary matrix the system T can be diagonalized.
Now we will state an important theorem on Hecke eigenfunctions.
Theorem 6.20 (Hecke). In Sk(N ) there exists an orthonormal basis F consisting of eigenfunctions of all the Hecke operators Tn.
Proof. Apply proposition 6.19 on the Hecke algebra.
