A trace formula for Hecke operators
for modular groups
by Fabian Völz
Thesis
Submitted to the University of Warwick
for the degree of
Master of Science
Supervised by David Loeffler
Mathematics Institute
September 2012
ii
Acknowledgements
I would like to express my deepest gratitude to my supervisor David Loeffler for suggesting this thesis and supporting me throughout the year with regular meetings and
on the spot answers to almost all of my mails. I would also like to thank him for his
brilliant introduction to the theory of modular forms given as a lecture at the University
of Warwick during the academic year 2011/12, without which I might have missed this
fascinating area of mathematics.
Further, thanks go to the Mathematics Department of the University of Warwick
for offering a perfect work environment. In particular, I would like to thank Carole
Fisher for always being there for us MSc students. In addition, my thanks go to the
”Studienstiftung des Deutschen Volkes” for their financial support during the year.
Last but not least I would like to thank my parents for always supporting me spending
a year abroad. I also thank Joakim Skogholt and Kien Nguyen for all the time spent together at the Department working hard and playing ”Skat” during the breaks. Moreover,
I would like to thank Daniel Reker and Stefan Schmid for their valuable suggestions to
this work, and finally, I would like to express my gratitude to David Wegmann and
Patrick Tolksdorf for their fantastic last-minute support.
iii
The beginner should not be discouraged
if he finds that he does not have the prerequisites
for reading the prerequisites.
Paul Halmos
iv
Contents
1 Introduction
3
2 Fundamental concepts
2.1 Short introduction to modular forms . . . . . . . . . . . .
2.1.1 Some group actions . . . . . . . . . . . . . . . . . .
2.1.2 Modular groups . . . . . . . . . . . . . . . . . . . .
2.1.3 Modular forms . . . . . . . . . . . . . . . . . . . .
2.1.4 The Petersson inner product . . . . . . . . . . . . .
2.1.5 Hecke operators . . . . . . . . . . . . . . . . . . . .
2.2 Classification of elements in GL2 pRq . . . . . . . . . . . .
2.3 Introduction to reproducing kernel Hilbert spaces . . . . .
2.4 Some algebraic number theory . . . . . . . . . . . . . . . .
2.4.1 Number fields and their rings of integers . . . . . .
2.4.2 The discriminant of a number field . . . . . . . . .
2.4.3 Quadratic fields . . . . . . . . . . . . . . . . . . . .
2.4.4 Ideal class group and class number . . . . . . . . .
2.4.5 Orders of number fields . . . . . . . . . . . . . . . .
2.4.6 Table of class numbers of imaginary quadratic fields
3 The
3.1
3.2
3.3
3.4
3.5
reproducing kernel of Sk pΓq
Some function spaces on H . . . . . . . . . . .
Computation of the kernel of Hk2 pHq . . . . .
Interpretation of Sk pΓq as a reproducing kernel
Computation of the kernel of Sk pΓq . . . . . .
A first trace formula . . . . . . . . . . . . . .
4 Simplification of the trace formula
4.1 Interchanging summation and integration
4.2 Calculation of integrals . . . . . . . . . .
4.2.1 The scalar terms . . . . . . . . .
4.2.2 The elliptic terms . . . . . . . . .
4.2.3 The hyperbolic terms of type one
4.2.4 The hyperbolic terms of type two
4.2.5 The parabolic terms . . . . . . .
4.3 The final trace formula . . . . . . . . . .
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
5
5
6
6
7
8
8
12
14
14
15
16
17
18
20
. . . . . . . .
. . . . . . . .
Hilbert space
. . . . . . . .
. . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
21
21
23
31
33
39
.
.
.
.
.
.
.
.
41
41
53
54
55
56
59
63
67
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 A trace formula for the Hecke operators Tp acting on Sk pΓ0 pN qq
70
5.1 Motivating observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Hijikata’s trace formula for Γ0 pN q . . . . . . . . . . . . . . . . . . . . . . 72
6 Summary and outlook
81
Bibliography
84
2
1 Introduction
This thesis yields an introduction to the Eichler-Selberg trace formula, which is a formula
for the trace of Hecke operators acting on spaces of cusp forms. For the reader unfamiliar
with the area we quickly comment on the mentioned terms: Roughly speaking, a cusp
form is an holomorphic function on the upper half-plane H which behaves ”nicely” on
the closure of H in the Riemann sphere, and which is invariant under a certain weight
k action of some matrix group Γ. For a given weight k and a given group Γ, the set of
cusp forms is a finite dimensional vetor space, and Hecke operators are particular linear
operators acting on this space. (We give a more detailed introduction to the theory in
Section 2.1.)
The eigenvalues of Hecke operators are of interest as they describe Fourier coefficients
of modular forms, and these coefficients are important for applications to number theory.
For example the number of ways of representing an integer as a sum of four squares is
encoded in the Fourier coefficients of a certain modular form. (See Section 1.2 in [DS05]
for details on this matter.) Using traces of different Hecke operators it is possible to
recover their eigenvalues. (We refer to pages 266, 267 in [Miy06] for details.) This
motivates the study of trace formulae.
The first formulae were given by A. Selberg and M. Eichler, who studied trace formulae
simultaneously. In 1956 Selberg stated a trace formula in [Sel56] without proof for the
well-known Tn operators for the full modular group, and in 1957 Eichler proved a trace
formula for such Tn operators for the full modular group in the case that n is squarefree
in [Eic57]. He also states a trace formula for n not being squarefree. However, Eichler
had already been studying trace formulae during the past years (see [Eic55] and [Eic56]).
Since then a lot of different authors have contributed to the area. Therefore the term
Eichler-Selberg trace formula denotes a whole class of trace formulae being due to the
original formulae by Selberg and Eichler. In particular, we mention the paper [Hij74]
published by H. Hijikata in 1974, in which he proves a trace formula for Tn operators
acting on spaces of cusp forms of level Γ0 pN q with N being coprime to n.
The present work is mainly based on the first half of Chapter 6 of T. Miyake’s book
on modular forms ([Miy06]). We start by recalling the basics of the theory of modular
forms in Chapter 2. Afterwards we introduce two concepts which will be fundamental
in the course of the thesis:
• The theory of reproducing kernel Hilbert spaces, and
• the classification of elements in GL2 pRq into scalar, elliptic, parabolic and hyperbolic
elements.
3
The former concept is based on the beginning of the article [Aro50] by N. Aronszajn,
and the latter relies on Section 1.3 of [Miy06]. We end Chapter 2 by giving a brief
introduction to algebraic number theory based on [ST02], which will be necessary to
understand the mentioned trace formula by Hijikata in Chapter 5.
Chapter 3 and Chapter 4 are completely based on Miyake’s book. More precisely,
we deal with Section 6.1 to 6.3 of [Miy06] in Chapter 3 where a first trace formula is
developed by applying the theory of reproducing kernel Hilbert spaces to some function
spaces related to spaces of cusp forms. Subsequently, we simplify this trace formula in
Chapter 4 which covers Section 6.4 of [Miy06].
These two chapters provide the core of this thesis. Since trace formulae have been
studied intensely for years it has not been our goal to extend the theory as this would go
beyond the scope of this thesis. Instead we aim to give an easily accessible introduction.
The corresponding sections in Miyake’s book are often slightly vague, missing technical
details and structure. It has been our intention to improve on these points, which essentially meant providing proofs for statements trivial to the author (such as Proposition
4.1.9 and Proposition 4.1.11) and filling in details for existing proofs (such as Lemma
3.4.4, Theorem 3.4.5 and Theorem 4.1.13). Apart from that we claim the following to
be original work:
• On the pages 222 to 225 in [Miy06] Miyake uses Fourier analysis to develop a
precise formula for the reproducing kernel Kk of Hk2 pHq. We use a shorter and
more elementary method at this point. (This is the second half of our Section 3.2.)
• In the first part of Section 6.4. in [Miy06] one carefully interchanges summation
and integration as a step in the derivation of the trace formula. We point out that
one has to fix a fundamental domain first as for example the integrals in equation
(6.4.7) in [Miy06] will in general not be well-defined. This is an issue Miyake
ignores, though it turns out to be purely formal.
In Chapter 5 we present Hijikata’s trace formula though we cannot give a proof as the
gap between our final trace formula given in Section 4.3 and Hijikata’s formula is still
too big. Instead we explain the different terms appearing in Hijikata’s formula with the
help of two examples. Finally, we quickly summarise our results and give an outlook for
further studies in Chapter 6.
4
2 Fundamental concepts
In the present chapter we introduce some concepts that will be fundamental for this
thesis. Most of the results we present will not be proved, though we give references for
further reading. Section 2.1 and Section 2.2 are based on Miyake’s book [Miy06], Section
2.3 follows [Aro50] and Section 2.4 is mainly due to Stewart’s and Tall’s book [ST02].
2.1 Short introduction to modular forms
We start by quickly recalling the basic notation of modular forms and Hecke operators
used in this thesis. We mainly follow [Miy06], though we work in a less general setting
which sometimes simplifies things. We also mention [DS05] as a good introduction to
the theory of modular forms and Hecke operators.
2.1.1 Some group actions
We denote the upper half-plane in C by H. It is well known that the group
GL2 pCq acts
b
a b 8 a . In both
for
z
P
C
and
on the Riemann sphere C Y t8u via ac db z az
c d
cz d
c
cases we interpret the right-hand side as 8 if the corresponding denominator vanishes.
In particular, one can check that GL2 pRq acts on H. This gives rise to an action of
GL2 pRq on the space of functions f : H Ñ C via
pf |k αq pzq detpαqk1j pα, zqk f pαzq
where k is an arbitrary integer and j pα, z q cz d for α ac db . We call this the
weight k action. A function f : H Ñ C is called Γ-invariant of weight k for some
integer k P Z if f |k γ f for all γ P Γ. If the context determines the weight k, we call
such f simply Γ-invariant.
We collect some properties of the function j which can be checked easily: For α, β in
GL2 pCq and arbitrary z P C we have
j pαβ, z q j pα, βz qj pβ, z q and j pα1 , z q j pα, α1 z q1
as in equation (1.1.5), (1.1.6) on page 1,2 in [Miy06]. (For the second equation we
formally require j pα1 , z q 0.) Further, we have for α, β P GL2 pRq and z P H that
Impαz q detpαq Impz q |j pα, z q|2
as in equation (1.1.7) on page 3 in [Miy06]. We will use all of these properties without
further notice from now on.
5
2.1.2 Modular groups
Consider the group SL2 pZq and its finite index subgroups. The former is called the full
modular group, and the latter modular groups. Moreover, we define for N P N the
modular group of level N by
Γ0 pN q "
a b
c d
P SL2pZq : c 0 mod N
*
.
We will work with general modular groups up to Chapter 5, where we specialise to
modular groups of level N .
Let Γ be a modular group. Following Section 1.6 of [Miy06] we call F „ H a fundamental domain of Γ if the following conditions hold:
(i) H ”
P γF ,
γ Γ
U where U is the set of interior points of F , and
λU X U H for all γ P Γzt1u.
(ii) F
(iii)
Note that we do not require F to be connected. Theorem 4.1.2 on page 97 in [Miy06]
proves that the set
D : tz
P H : | Repzq| ¤ 1{2 and |z| ¥ 1u
is a fundamental domain for the full modular group
SL2 pZq. Let g1 , . . . , gl be coset
”
representatives for the quotient ΓzH, and put F lj 1 gj D. Then F is a fundamental
domain for Γ as one can check. In particular, this shows that every modular group has
a fundamental domain. Note that for a fixed integer k a function f : H Ñ C that is
Γ-invariant of weight k is completely determined by its values on a fundamental domain
for Γ.
One can easily check that the full modular group acts transitively on Q Y t8u. For
a modular group Γ we define the set of cusps of Γ as the set of Γ-orbits in Q Y t8u,
and denote it by C pΓq. Since Γ is of finite index in SL2 pZq the set of cusps is finite.
It turns out that the set of cusps of Γ contains exactly the elements missing for the
compactification of the quotient ΓzH. More precisely, one can show that the quotient
ΓzpH Y Q Y t8uq is a compact Riemann surface. (This is quite involved. For details
we refer to Section 1.7, Section 1.8, Theorem 1.9.1 and Theorem 4.1.2 in [Miy06].)
The topology we use for the Riemann surface is the one induced by the topology on
H Y Q Y t8u introduced in Section 1.7 of [Miy06], which has the normal open sets in H
and sets of the form σ ptz P H : Impz q ¡ δ uq with σ P SL2 pZq, δ ¡ 0 as a basis.
2.1.3 Modular forms
Next we want to introduce spaces of modular forms. Let Γ be a modular group and
let f : H Ñ C be Γ-invariant of weight k. We call f a modular form of weight k
6
and level Γ if f is holomorphic on H and well-defined as a function on the quotient
ΓzpH Y Q Y t8uq mapping to C. So roughly spoken, modular forms are holomorphic
functions on H that are Γ-invariant and behave ”nicely” at the cusps. We denote the
space of modular forms of weight k and level Γ by Mk pΓq. Further, we call a modular
form f a cusp form if it vanishes at all cusps, and we denote the space of cusp forms of
weight k and level Γ by Sk pΓq. (For a more detailed definition of these spaces and their
corresponding forms we refer to Section 2.1 of [Miy06].)
Clearly Mk pΓq is a vector space over C, and Sk pΓq is a linear subspace. Moreover, one
can show that these spaces are finite dimensional for every integer k and every modular
group Γ. (We refer to Theorem 2.5.2 on page 60, 61 in [Miy06] for a proof.) For k ¥ 3
the simplest example of a modular form is the Eisenstein series
¸
Gk,Γ,8 pz q P z
j pγ, z qk ,
z
P H,
γ Γ8 Γ
where Γ8 ΓXt 10 1 u. One can check that the sum defining Gk,Γ,8 converges absolutely
and uniformly on compact subsets of H, and that Gk,Γ,8 is a modular form of weight k
and level Γ. A detailed discussion of these functions is given in Section 2.6 of [Miy06].
In the case of Γ SL2 pZq we also refer to Lemma 4.1.6 on page 100 in [Miy06], since
Ek as given in the corresponding section equals Gk,SL2 pZq,8 up to a scalar multiple.
2.1.4 The Petersson inner product
Identifying the upper half-plane H with the upper half-plane in R2 we define the measure
dν pz q on H by y 2 dpx, y q where z x iy, as in (1.4.2) on page 11 in [Miy06]. We are
interested in this measure since it is invariant under the action of GL2 pRq. Furthermore,
it is straightforward to check:
Lemma 2.1.1. Let α
exists. Then
P GL2 pRq, U „ H open and f : U Ñ C such that
»
U
f pz qdν pz q »
α 1 U
³
U
|f pzq|dν pzq
f pαz qdν pz q.
Let Γ be a modular group. The previous lemma shows that
»
z
Γ H
F pz qdν pz q
³
is well-defined if the function F is Γ-invariant of weight 0 and F |F pz q|dν pz q is finite for
some fundamental domain F of Γ. Hence it makes sense to determine the measure of
the quotient ΓzH. A direct calculation shows that ν pSL2 pZqzHq π {3. One may use
this to show:
Lemma 2.1.2. Let Γ be a finite index subgroup of SL2 pZq. The quotient ΓzH has finite
measure with respect to ν. More precisely,
ν pΓzHq dΓ π {3
where dΓ
rSL2pZq{t1u : Γ{pt1u X Γqs.
7
Moreover, we may define for f, g
xf, gyΓ P Mk pΓq
»
z
Γ H
f pz qg pz q Impz qk dν pz q.
One can check that the integral exists if at least one of f and g vanish at every cusp
of Γ. (Compare page 44 of [Miy06].) In particular, xf, g yΓ is well-defined for every
f, g P Sk pΓq. We call x, yΓ the Petersson inner product. One can check that it
indeed defines an inner product on the space of cusp forms Sk pΓq.
2.1.5 Hecke operators
Let Γ be a modular group. For an element g P GL2 pQq we let ΓgΓ denote the double
coset tγ1 gγ2 : γ1 , γ2 P Γu. One can show that for any such g P GL2 pQq the intersection
Γ X pg 1 Γg q has finite index in Γ and g 1 Γg. We
say Γ and g 1 Γg are commensurable.
—r
Hence there are α1 , . . . , αr such that ΓgΓ j 1 Γαj by Lemma 2.7.1 on page 69
in [Miy06]. (Note that the set Γ̃ given in the lemma equals GL2 pQq in our case by
the above observation.) Let RpΓq be the C-vector space with basis the symbols rΓgΓs
for each g P Γz GL2 pQq{Γ. Following Section 2.7 of [Miy06] one can show that RpΓq
equipped with a suitable multiplication is also a ring, so an algebra over C. It is called
the Hecke algebra of Γ.
For elements in RpΓq we define an action on the space of modular forms Mk pΓq via
f |k rΓgΓs r
¸
f |k αj
j 1
where α1 , . . . , αr P GL2 pQq such that ΓgΓ
in [Miy06] proves the following:
—rj1 Γαj .
Theorem 2.8.1 on page 74, 75
(1) The above definition is independent of the choice of representatives α1 , . . . , αr , and
thus well-defined.
(2) We have f |k rΓgΓs P Mk pΓq if f
P Mk pΓq, and f |k rΓgΓs P Sk pΓq if f P Sk pΓq.
(3) Mk pΓq and Sk pΓq are right modules over RpΓq.
We will mainly consider elements of RpΓq as linear operators acting on the spaces Mk pΓq
and Sk pΓq. In an abuse of notation we denote such operators by T ΓgΓ, and write
T pf q for f |k rΓgΓs. These operators are the so called Hecke operators.
2.2 Classification of elements in GL2 pRq
In this section we introduce the classification of elements in GL2 pRq following Section
1.3 and 1.5 of [Miy06]. The classification is essential for the understanding of the trace
formula presented in this thesis. It will be used first in Section 4.2.
8
Let α P GL2 pRq. We call α scalar if it is of the form a0 a0 for some a P R . For some
subset M of GL2 pRq we write Z pM q for the set of scalar elements in M , even though
coventionally Z pGq denotes the centre of a group G. However, a direct calculation proves
1 h1
that the
two
notations
agree
for
example
if
there
are
h
,
h
P
Z
zt
0
u
such
that
and
1
2
0 1
1 0
are
elements
of
M
.
This
is
in
particular
the
case
for
M
being
a
modular
group
h2 1
as one can check. We also note that Z pΓq Γ X t1u if Γ is a modular group.
For non-scalar α we say α is elliptic, parabolic or hyperbolic, if
Trpαq2
4 detpαq,
Trpαq2
4 detpαq
respectively. Since eigenvalues of α are given by λ1,2
see for non-scalar α:
or
Trpαq2
Tr2pαq 12
¡ 4 detpαq,
a
Trpαq2 4 detpαq, we
• The element α is elliptic if and only if the eigenvalues of α are complex conjugates
with non-zero imaginary part.
• The element α is parabolic if and only if α has only one eigenvalue which has algebraic multiplicity two and geometric multiplicity one. This eigenvalue is rational
if α P GL2 pQq.
• The element α is hyperbolic if and only if α has two distinct real eigenvalues.
These eigenvalues lie at worst in a quadratic extension of Q if α P GL2 pQq.
We note that an element α is elliptic, parabolic or hyperbolic if and only if all conjugates
of α are so, since trace and determinant are stable under
conjugation.
a
b
In the following we consider fixed points of α c d P GL2 pRq acting on the Riemann
sphere C Y t8u. If α is scalar, it acts trivially on C Y t8u, and thus fixes every point.
Suppose α is non-scalar. If c 0 then 8 is a fixed point of α and Trpαq2 4 detpαq pa dq2. Thus α cannot be elliptic, α is parabolic if and only if a d, and α is hyperbolic
if and only if a d. Moreover, since αx x for x P C is equivalent to pd aqx b, we
see that
8 is the unique fixed point of α, and
• α is hyperbolic if and only if the only fixed points of α are 8 and db a .
d 1 aTrpαq2 4 detpαq.
Now suppose that c 0. Then αx x is equivalent to x a2c
2c
• α is parabolic if and only if
Hence we have that
• α is elliptic if and only if the fixed points of α are complex conjugates with non-zero
imaginary part,
d is the only fixed point of α, and
• α is parabolic if and only if a2c
• α is hyperbolic if and only if the fixed points of α are two distinct real values. If
α P GL2 pQq then these fixed points are either both rational, or both non-rational.
In the latter case they lie in an quadratic extension of Q.
9
Combing these results we can characterise elements in GL2 pRq by means of their fixed
points on the Riemann sphere.
Corollary 2.2.1. Let α P GL2 pRq.
• The element α is elliptic if and only if there is z
unique fixed points of α.
P H such that z and z are the
• The element α is parabolic if and only if α has a unique fixed point in R Y t8u. If
α P GL2 pQq then this fixed point is in Q Y t8u.
• The element α is hyperbolic if and only if α has exactly two distinct fixed points
on R Y t8u. If α P GL2 pQq, either both fixed points lie in Q Y t8u, or they are
both irrational.
For a subgroup U of GL2 pRq we denote the stabilizer of z
Uz
tβ P U : βz zu.
P H we have
(
pGL2 pRqqz σ λ Φ : λ P R, Φ P SO2pRq σ1
where σ P SL2 pRq with σi z, and
a SO2 pRq denotes the special orthogonal group.
In particular, we may write α detpαq σΦσ 1 for any elliptic α P GL2 pRq with
fixed point z, where Φ is an element of SO2 pRq.
For x P R Y t8u we have
Lemma 2.2.2.
(2)
P C Y t8u in U by
(1) For z
α P pGL2 pRqqx : α parabolic or scalar
P SL2pRq with σ8 x.
For distinct x1 , x2 P R Y t8u we have
(
σ
"
a b
0 a
*
: a P R , b P R σ 1
where σ
(3)
pGL2 pRqqx X pGL2 pRqqx σ
1
where σ
2
"
a 0
0 d
P SL2pRq with σ8 x1 and σ0 x2.
We refer to Lemma 1.3.2 on page 8 in [Miy06] for a proof.
Lemma 2.2.3. For α P GL2 pRq we define the centralizer of α by
Z pαq tβ
(1)
P GL2pRq : αβ βαu.
If α is elliptic with fixed point z P H then Z pαq pGL2 pRqqz .
10
*
: a, d P R , ad ¡ 0 σ 1
(2) If α is parabolic with fixed point x P R Y t8u then
Z pαq tβ
P pGL2 pRqqx : β parabolic or scalaru.
P R Y t8u then
Z pαq X GL2 pRq pGL2 pRqqx X pGL2 pRqqx
and rZ pαq : Z pαq X GL2 pRqs 2.
(3) If α is hyperbolic with distinct fixed points x1 , x2
1
2
Again, we omit the proof, and refer to Lemma 1.3.3 on page 9 in [Miy06]. In Section
4.1 we will define Γpαq tγ P Γ : γα αγ u for some finite index subgroup Γ of SL2 pZq
and some α P GL2 pQq. Note that we have Γpαq Γ X Z pαq by definition. We use this
equality and the previous lemma to describe Γpαq:
Corollary 2.2.4. Let α P GL2 pQq and Γ be a finite index subgroup of SL2 pZq.
P H then Γpαq Γz .
(2) If α is parabolic with fixed point x P Q Y t8u then Γpαq Γx .
(3) If α is hyperbolic with fixed points x1 , x2 P Q Y t8u then Γpαq Z pΓq.
(4) If α is hyperbolic with fixed points x1 , x2 P RzQ then Γpαq Γx X Γx .
Proof. For α elliptic with fixed point z P H we see Γpαq Γ X Z pαq Γz using part
(1) If α is elliptic with fixed point z
1
2
(1) of Lemma 2.2.3, which proves (1). Similarly, (4) follows directly from part (3) of
Lemma 2.2.3. For (2) and (3) we claim that SL2 pZq does not contain any hyperbolic
elements with fixed points in Q Y t8u. Assume that γ is such an element with distinct
fixed points x, x1 P Q Y t8u. Let σ P SL2 pZq such that σ 8 x, then σ 1 γσ fixes 8,
and is therefore of the form 01 1 . But elements of this form are parabolic, so γ itself
has to be parabolic, which is a contradiction and thus proves the claim.
To show (2) let α be parabolic with fixed point x P Q Y t8u. By part (2) of Lemma
2.2.3 we have that Γpαq consists precisely of parabolic elements in Γ fixing x and scalar
elements in Γ. Therefore Γpαq is a subset of Γx , and the only elements that might be in
Γx but not in Γpαq, are hyperbolic elements in Γ fixing x. We proved that such elements
do not exist.
To see (3) note that we have for α hyperbolic Γpαq Γx X Γx1 by part (3) of Lemma
2.2.3. The right-hand side consists of hyperbolic elements in Γ fixing x and x1 and all
scalar elements in Γ. We proved that there are no hyperbolic elements with fixed points
in Q Y t8u.
We close this section with two more lemma. For the corresponding proofs see part (2)
of Lemma 1.3.5 on page 10 and Theorem 1.5.4 on page 18, 19 in [Miy06].
Lemma 2.2.5. If two distinct elements of GL2 pRq are either both elliptic or both parabolic, and if they are conjugate by a matrix in GL2 pRq of negative determinant, then
they are not conjugate in GL2 pRq.
11
Note that the corresponding lemma in [Miy06] is stated only for parabolic elements,
but the given proof works exactly the same for elliptic elements since we have for any
elliptic α P GL2 pRq that Z pαq „ GL2 pRq by part (1) of Lemma 2.2.3.
Lemma 2.2.6. Let Γ be a finite index subgroup of SL2 pZq.
P H, the stabilizer group Γz is finite.
For x P Q Y t8u, the quotient Γx {Z pΓq is isomorphic to Z. Moreover, we have
(1) For z
(2)
σ 1 Γ σ t1u "
x
1 hm
0 1
*
:mPZ
P SL2pRq with σ8 x and h is the width of the cusp rxs for Γ.
For distinct x1 , x2 P R Yt8u with Γx X Γx Z pΓq, the quotient pΓx X Γx q{Z pΓq is
isomorphic to Z. Moreover, there is u ¡ 0 such that for σ P SL2 pRq with σ 8 x1
and σ0 x2 we have
where σ
(3)
1
σ 1 pΓ
2
"
x1
X Γx qσ t1u 2
1
um
0
0 um
2
*
:mPZ .
2.3 Introduction to reproducing kernel Hilbert spaces
In this section we will give a short introduction to the theory of reproducing kernel
Hilbert spaces based on the beginning of [Aro50]. The concept will be fundamental for
the third chapter of this thesis.
Definition 2.3.1. Let X be an arbitrary set and let pH, x, yq be a Hilbert space consisting of complex valued functions on X. A function K : X X Ñ C is called the
reproducing kernel of H if
(1) the function K p, xq is an element of H for each fixed x P X, and
(2) for every function f
P H and every x P X we have f pxq xf, K p, xqy.
Property (2) is called the reproducing property of the kernel K. If such a function
K exists then H is called a reproducing kernel Hilbert space.
Following Section 1.2 of [Aro50] we will now prove some basic properties of reproducing
kernel Hilbert spaces. Throughout the section we assume pH, x, yq to be a Hilbert space
with H „ tf : X Ñ Cu where X is an arbitrary set. Further, we will sometimes write
kernel instead of reproducing kernel.
Proposition 2.3.2 (Uniqueness). If H is a reproducing kernel Hilbert space, then its
kernel K is unique.
12
Proof. Suppose K 1 : X X Ñ C is another reproducing kernel of H. Then we see for
any x P X using the reproducing property of K and K 1 that
}K p, xq K 1p, xq}2 xK p, xq K 1p, xq, K p, xq K 1p, xqy
xK p, xq K 1p, xq, K p, xqy xK p, xq K 1p, xq, K 1p, xqy
pK px, xq K 1px, xqq pK px, xq K 1px, xqq
0.
Here } } denotes the norm of H induced by x, y. Hence K K 1 as claimed.
Proposition 2.3.3 (Existence). The Hilbert space H has a reproducing kernel K if and
only if the evaluation functional Ex : H Ñ C, f ÞÑ f pxq is continuous for every x P X.
Proof. Suppose that K is the kernel of H, and fix x P X. Then
a
|Expf q| |f pxq| |xf, K p, xqy| ¤ }f } xK p, xq, K p, xqy a
K px, xq}f }
by the Cauchy-Schwarz inequality and the reproducing property of K, so Ex is continuous. Conversely suppose that Ex is continuous for every x P X. Then every Ex is an
element of the dual space of H since it is clearly linear. By the Riesz representation
theorem (see Theorem 3.4 on page 13 in [Con97]) we find for every such Ex a unique
gx P H such that Ex x, gx y. Put K py, xq gx py q. Then K p, xq gx P H for all
x P X and
f pxq Ex pf q xf, gx y xf, K p, xqy .
Thus K is the reproducing kernel of H.
Corollary 2.3.4. If H is a reproducing kernel Hilbert space, then its kernel K has the
following properties:
(i) We have K px, xq
for all f P H.
P r0, 8q for all x P X, and K px, xq 0 if and only if f pxq 0
(ii) The kernel K is conjugate symmetric, that is K px, y q K py, xq for all x, y
P X.
Proof. Using the reproducing property of K we see K px, y q xK p, y q, K p, xqy. Hence
the first part of property (i) follows from x, y being a scalar product and thus positive
definite, and (ii) follows since x, y is also conjugate symmetric.
It remains to show that K px, xq 0 if and only if f pxq 0 for all f P H. Fix
x P X. Suppose that 0 K px, xq xK p, xq, K p, xqy. Then K p, xq 0 since the scalar
product is positive definite, and thus f pxq xf, K p, xqy 0. Conversly suppose that
0 f pxq for all f P H. Then K px, xq 0 follows trivially since K p, xq P H.
Proposition 2.3.5. Let H be a reproducing kernel Hilbert space with kernel K. If H is
a subspace of a larger Hilbert space J, then
πK : J
Ñ H, pπK f qpxq xf, K p, xqy
is a well-defined operator which projects J onto H.
13
Proof. Let f P J. Then there is a unique element g P H such that f g P H K where
H K denotes the orthogonal complement of H in J. (We refer to Section 1.2 of [Con97]
for details on this matter.) Hence
pπK f qpxq xg, K p, xqy xf g, K p, xqy xg, K p, xqy gpxq
since K p, xq P H and f g P H K . In particular, we have πK f f for all f P H.
Proposition 2.3.6. Let H be a reproducing kernel Hilbert space with kernel K and let
tej ujPJ be an orthonormal basis of H. Then
K px, y q ¸
P
ej pxqej py q.
j J
°
Proof. Since tej uj PJ is an orthonormal basis we have f j PJ xf, ej y ej for every f
(This is part of Theorem 4.13 on page 16 in [Con97].) Fix y P X. Then
¸
K p, y q P
P H.
xK p, yq, ej y ej
j J
and thus xK p, y q, ej y xej , K p, y qy ej py q gives the claimed statement.
2.4 Some algebraic number theory
Finally we recall some basic notation and facts about algebraic number theory. In
particular, we are interested in quadratic fields over Q and their orders, which will
appear in Chapter 5 of this thesis. The present section is based on [ST02], which we
follow closely. Throughout we require a ring to have a multiplicative identity element,
and a homomorphism of rings needs to map the unity of its domain to the unity of its
target.
2.4.1 Number fields and their rings of integers
We call a complex number α P C an algebraic number if it is algebraic over Q, so if
there is a polynomial p P Qrts, p 0, such that ppαq 0. Note that this is equivalent
to p having coefficients in Z as we can clear denominators. Define A to be the set of all
algebraic numbers in C. One can show that A is a subfield of C. (Compare Theorem
2.1 on page 36 in [ST02].) Note that the field A is an infinite field extension of Q. We
define a subfield K of C to be a number field if it is a finite extension of Q. Note that
any element in K is algebraic, so we have K „ A. Moreover, K is separable over Q
since Q and thus K have characteristic 0. Hence we have K Qpθq for some θ P K by
the primitive element theorem. (See for example Theorem 4.6 on page 243 in [Lan02].)
We call a complex number θ P C an algebraic integer if there is a monic polynomial
p P Zrts, p 0, such that ppθq 0, so
θn
an1 θn1
...
14
a1 θ
a0
0
for some a0 , . . . , an1 P Z. Define B to be the set of all algebraic integers in C. Then B
is a subring of A, and we have B X Q Z by Theorem 2.9 on page 43 and Lemma 2.14
on page 45 in [ST02]. Finally, we define for any number field K the ring of integers
of K by OK K X B. Note that OK is indeed a subring of K and Z „ OK . One can
check that for any α P K there is N P Z, N 0, such that N α P OK . Hence we have
OK b Q K. Further, this shows that we can always find θ P OK such that K Qpθq.
2.4.2 The discriminant of a number field
Let K Qpθq be a number field. We want to define the discriminant of K using
homomorphisms of the form σ : K Ñ C. Recall that a homomorphism of fields is always
injective, so such σ is an embedding. Further one can easily check that any such σ fixes
Q. Hence we are looking for possible extensions of the trivial embedding Q ãÑ C to the
number field K Qpθq. By Proposition 2.7 on page 233 in [Lan02] there are exactly
n embeddings σi : K ãÑ C where n is the number of distinct roots of the minimal
polynomial of θ, which agrees with the degree of K since K is separable over Q as
mentioned earlier. Further, one can check that the elements θi : σi pθq are precisely the
distinct zeros of the minimal polynomial of θ over Q.
Let now tα1 , . . . , αn u be a basis of K as a vector space over Q. Then we define the
discriminant of this basis to be
∆rα1 , . . . , αn s det
pσipαj qq1¤i,j¤n
2
.
If tβ1 , . . . , βn u is another basis of K over Q, then we can write βk
cj,k P Q, k 1, . . . , n, and hence
∆rβ1 , . . . , βn s det
pcj,k q1¤j,k¤n
2
°nj1 cj,k αj for some
∆rα1 , . . . , αn s.
(2.4.1)
The obvious choice of a basis for K over Q is t1, θ, . . . , θn1 u. To see that these elements
are indeed linearly independent we only have to note that the minimal polynomial of θ
has degree n. One can check that
∆ 1, θ, . . . , θn1
¹
¤ ¤
pθi θj q2 .
1 i j n
Here the right-hand side is rational and non-zero, and thus the discriminant of any basis
of K over Q is so, since the determinant of pcj,k q1¤j,k¤n is rational and non-zero as well.
Next we note that OK , the ring of integers of K, is an abelian group under addition,
and thus a Z-module. More precisely, Theorem 2.16 on page 46 in [ST02] proves that
OK is a free abelian group of rank n where n is the degree of K, and thus the Z-module
OK always has a basis. We call this basis an integral basis of K, which is reasonable
since any Z-basis of OK is also a Q-basis for K as OK b Q K.
Let tα1 , . . . , αn u and tβ1 , . . . , βn u be °
two integral bases of K. Since they are bases of
the Z-module OK we can write βk nj1 cj,k αj for some cj,k P Z, k 1, . . . , n, and
15
°
conversly αk nj1 c̃j,k βj for some c̃j,k P Z, k 1, . . . , n. Since the matrix pcj,k q1¤j,k¤n
is the inverse of pc̃j,k q1¤j,k¤n , and both matrices have integer entries, they both have to
have determinant 1. Therefore the discriminant of an integral basis is indepenent of
the choice of integral basis by equation (2.4.1), and we can define the discriminant of
the number field K, denoted by ∆K , as the discriminant of any integral basis of K.
2.4.3 Quadratic fields
We call a number field K a quadratic field if it is of degree 2 over Q. As noted at the
end of Subsection 2.4.1 we can write K Qpθq for some θ P OK . Let mθ t2 at b,
a, b P Q, be the minimal polynomial of θ. One can check that a and b have to be integers
since θ is an algebraic integer. (Compare Lemma 2.13 on page 45 in [ST02].) Further,
we can write
?
a a2 4b
.
θ
?2 2
Since a P Z we have K Qpθq Qp a 4b q. This is a quadratic extension of Q if
and only if a2 4b does not have a rational square root, which is the case if and only
if there is no integer r P N0 such that a2 4b r2 . Hence we can write a2 4b r2 d
for a unique
? squarefree, and thus
? pair of ?integers r P N and d P Zzt0, 1u with d being
K Qpr d q Qp d q. Conversly any field of the form Qp d q for some squarefree
integer d not equal to 0 or 1 is obviously a quadratic field, so we have shown:
?
Proposition 2.4.1. The quadratic fields are precisely the fields of the form Qp d q with
d being a squarefree integer and d 0, 1.
?
If such squarefree?d is positive, we call Qp d q a real quadratic field, and if d is
negative we call Qp d q an imaginary quadratic field. Next we determine the ring
of integers of a quadratic field.
Proposition
2.4.2. Let
? d 0, 1 be a squarefree integer and?K the quadratic field
?
d q{2s if d 1 mod 4.
Qp d q. Then OK Zr d s if d 2, 3 mod 4, and OK Zrp1
For a proof see Theorem 3.2 on page 62 in [ST02]. Note that d 0 mod 4 is not possible
since we assume d to be squarefree. We use this result to compute the discriminant of
a quadratic field.
Theorem 2.4.3. Let d 0, 1 be a squarefree integer and K
if d 2, 3 mod 4, and ∆K d if d 1 mod 4.
?
b dq a
?
Then ∆K
4d
Ñ C are the identity and the
?
?
b d q a b d.
Proof. Note that the two distinct embeddings σ1 , σ2 : K
conjugation, so for a, b P Q we have
σ1 pa
?
Qp d q.
?
b d and σ2 pa
ã
Suppose
? that d 2, 3 mod 4. By the previous proposition an integral basis of K is given
by t1, d u. Hence we can compute
∆K
∆r1,
d s det
?
2
σ1 p1q σ1 p?d q
σ2 p1q σ2 p d q
16
det
1
1
? 2
?d
4d.
d
Similarly we can ?
compute ∆K d if d
given by t1, p1
d q{2u in this case.
1 mod 4 using that an integral basis of K is
Further we want to describe the set of units of a ring of integers for some quadratic
field. Recall that the set of units of a ring R forms a group under multiplication. We
denote this group by U pRq.
Proposition 2.4.4. Let K
?
Qp d q be an imaginary quadratic field. Then
$
'
&
t1, iu,
U pOK q t1, e2πi{3 , e4πi{3 u,
'
%
t1u,
if d 1,
if d 3,
otherwise.
We refer to Proposition 4.2 on page 77 in [ST02] for a proof. The general case of
groups of units in quadratic fields is more complicated and known as the Dirichlet Units
Theorem. It is dealt with in the Appendix B of [ST02], and in Section 3.3 of [Ono90].
We will not need the general case in this thesis.
2.4.4 Ideal class group and class number
Let K be a number field as before. In this subsection we use the term ideal to denote a
: O zt0u.
non-zero ideal, and we put OK
K
We call an OK -submodule a of K a fractional ideal of OK if there is some c P OK
such that ca „ OK . Note that ca is still an OK -submodule of K. More precisely, it is
an OK -submodule of OK , and thus an ideal of OK . Hence the fractional ideals of OK
and b being an ideal of O . In particular, every ideal
have the form c1 b with c P OK
K
of OK is a fractional ideal. Conversly, a fractional ideal a is an ideal of OK if and only
if a „ OK .
Therefore we have generalised the concept of ideals of OK to fractional ideals. The
advantage of this generalisation is given by Theorem 5.5 on page 107 in [ST02]:
Theorem 2.4.5. The set of non-zero fractional ideals of OK is an abelian group under
multiplication with identity OK .
We omit the proof and denote the group of non-zero fractional ideals by F. Further,
we note that the set of ideals of OK itself is only a commutative semigroup, but not a
group as we are missing inverses in general.
Next we define a fractional ideal a to be principal if it comes from a principal ideal
and b being a principal ideal of O . Let
in OK , so if it is of the form c1 b with c P OK
K
P be the subset of F consisting of all principal fractional ideals. One can check that
P is a subgroup of F. Therefore we may define the ideal class-group of OK as the
quotient group
H : F {P.
Further, we define the class-number hpOK q as the order of the group H.
17
Theorem 2.4.6. The ideal class-group of a number field is a finite abelian group.
Hence the class-number is always a well-defined natural number. A proof of the
statement is given in [ST02]. The theorem itself can be found on page 157, but the
corresponding proof uses some more advanced techniques we haven’t developed here,
like Minkowski’s theorem. For a more elementary proof we refere to Section 2.10 on
page 74 to 76 in [Ono90] where a slightly different approach is used to introduce the
class-number:
such
For two ideals b, b1 of OK we define b b1 if there are elements c, c1 P OK
that xcyb xc1 yb1 . This gives an equivalence relation on the set of ideals of OK whose
equivalence classes are called ideal classes, denoted by rbs for some ideal b. Let J be
the set of ideal classes, then we can give J a group structure by defining rbsrb1 s rbb1 s.
(Compare Theorem 2.14 on page 77 in [Ono90].)
It turns out that this group is isomorphic to the ideal class-group as defined above,
and thus the number of ideal classes equals the class-number of K. We will explain this
correspondence in the following:
Define two fractional ideals a and a1 to be equivalent, denoted by a a1 , if they
represent the same coset in H, and denote such a coset by rrass. Write a c1 b with
and b being an ideal in O . Then b ca xcya, and thus a b, so every coset
c P OK
K
contains at least one proper ideal of OK . Hence we can choose a set of representatives
tb1, . . . , bhu for the quotient F {P where all bj are proper ideals of OK .
Now let b, b1 be ideals of OK that are equivalent as fractional ideals, so b b1 . We
claim that this implies b and b1 to be equivalent as ideals in OK as well. To see this let
. Then
p P P such that b pb1 and write p c1 xc1 y with c, c1 P OK
xcyb cb xc1yb1,
Conversly, b b1 clearly implies b b1 since all principal
and thus b b1 as claimed.
ideals of OK are elements of P.
This shows that rbs rb1 s if and only if rrbss rrb1 ss for ideals b, b1 of OK , and
thus the set tb1 , . . . , bh u is also a set of representatives for the set of ideal classes J.
This proves that the ideal class group H and J are indeed isomorphic since the group
operation is in both groups given by multiplication of ideals.
2.4.5 Orders of number fields
Next we introduce orders of number fields. As these are not treated in [ST02] we have
to use a different reference at this point, namely [Ste08].
Let K be a number field. We call a subring O of K an order in K, if O is a Z-module
which is free of rank n rK : Qs. Thus a Z-basis of an order O is always a Q-basis of
the corresponding number field, so O b Q K. Further, OK is clearly an order since
we remarked in Subsection 2.4.2 that OK is a free abelian group of rank n. We call an
order maximal if it is maximal with respect to inclusion. By Theorem 2.2 on page 213
of [Ste08] a subring O of K is an order if and only if O is of finite index in the ring of
integers OK . Thus OK is the unique maximal order of K.
18
Recall that we used fractional ideals of OK to introduce the ideal class-group H of
OK . Section 4 of [Ste08] shows how to generalise this concept to arbitrary orders in K:
Let O be an order in K. We may extend the definition of fractional ideals of OK
to fractional ideals of O, but in contrast to the former ones, non-zero fractional ideals
of O do in general not have a natural inverse. So to generalise Theorem 2.4.5 we only
consider invertible fractional ideals F pOq of O. Stevenhagen shows on page 216 that
these indeed form an abelian group under multiplication with identity O. Since in
particular all principal fractional ideals P pOq of O are invertible we may proceed as
before, namely define the ideal class-group of O as the quotient F pOq{P pOq and
the class-number hpOq as the order of this quotient group. Finally Corollary 10.6 on
page 238 of [Ste08] generalises Theorem 2.4.6 from the previous subsection, so hpOq is
a well-defined natural number.
We quote and explain a formula which lets us compute the class-number of an order in
an imaginary quadratic field directly. This will be very helpful in the course of Chaper
5. The formula is given in part (1) of Theorem 6.7.2 on page 257 in [Miy06]. We omit
the proof.
?
Theorem 2.4.7. Let K Qp d q be an imaginary quadratic field, so d a squarefree
and negative integer, and let O be an arbitrary order in K with rOK : Os n. Then the
class number of O is given by
hpOq
n hpOK q
rU pOK q : U pOqs
¹
p prime
pn
|
1
Lpd, pq
p
where Lpd, pq denotes the Legendre-Symbol.
The Legendre-Symbol Lpd, pq is introduced in the appendix of [ST02], more precisely,
on page 283. It is defined for an odd prime p and an integer d not divisible by p via
Lpd, pq #
1,
if there is m P Z such that m2
1, if there is no such m P Z.
d mod p,
If Lpd, pq 1 then d is called a quadratic residue modulo p. To calculate Lpd, pq we
can use Proposition A.15 on page 284 of [ST02] which states Lpd, pq dpp1q{2 mod
p for odd primes p and integers d not divisible by p. One may naturally extend the
Legendre-Symbol to integers d divisible by p via Lpd, pq 0. Further, Kronecker defined
Lpd, pq for p 2 via
$
'
if d 1 mod 8,
&1,
Lpd, 2q 0,
if d is even,
'
%
1, if d 3 mod 8.
(See for example [MV07] for details on this matter. In particular, the case p 2 is dealt
with on page 296.)
Next we quickly consider the quantity rU pOK q : U pOqs appearing in the formula. Let
d be a squarefree negative integer and let O be an order in the imaginary quadratic field
19
?
K Qp d q with rOK : Os n. By Proposition 2.4.4 we clearly have U pOq t1u if
d is neither 1 nor 3. Let d 1. If i P O then OK Qpiq „ O, so O OK .
Hence we have i P O if and only if n 1. Now let d 3. A similar argument shows
that one of e2πi{3 , e4πi{3 is in O if and only if O OK . Therefore we get
$
'
&4,
if d 1 and n 1,
|U pOq| '6, if d 3 and n 1,
%
2, otherwise,
$
'
&2,
and thus
if d 1 and n ¡ 1,
rU pOK q : U pOqs '3, if d 3 and n ¡ 1,
%
1, otherwise.
Finally, we consider discriminants of orders in imaginary quadratic fields. Let d be
a squarefree
? negative integer and let O be an order in the imaginary quadratic field
K Qp d q with rOK : Os n as before. By definition we have 1 P O, so Z „ O, and
O „ OK . Using Proposition 2.4.2 we get that
O
?
#
if d 2, 3 mod 4,
n d Z,
?
np1
d q{2 Z, if d 1 mod 4.
Z
Z
We may now define the discriminant of an order as the discriminant of the Z-basis of
the order. This is well defined, since such a basis is always a Q-basis for K. Hence we
can compute as in the proof of Theorem 2.4.3
?
∆pOq : ∆r1, n d s 4n2 d
?
and
∆pOq : ∆ 1, np1
if d 2, 3 mod 4,
d q{2
n2 d
if d 1 mod 4.
2.4.6 Table of class numbers of imaginary quadratic fields
We finish this section by quoting a small part of the table given on page 180 in Section
10.4 of [ST02].? It contains a list of class-numbers hpOK q for some imaginary quadratic
fields K Qp d q. By Theorem 2.4.7 this list also enables us to compute class-numbers
of arbitrary orders in these fields. We will use the list in Chapter 5 while calculating
examples.
d
hpOK q
1
2
3
5
6
7
10 11 13 14 15 17 19 21 22 23
1
1
1
2
2
1
2
1
2
4
2
4
1
Table 2.1: class-numbers of imaginary quadratic fields K
20
4
2
3
?
Qp d q
3 The reproducing kernel of Sk pΓq
At the end of this chapter we will be able to express the trace of a Hecke operator for
some modular group in terms of a kernel function of some reproducing kernel Hilbert
space, namely Hk2 pHq. We begin by introducing this space and some related function
spaces in the first section. In the second section we will determine its kernel Kk , which
plays a central role in this and the next chapter. Section three shows that the space of
cusp forms Sk pΓq for some modular group Γ is itself a reproducing kernel Hilbert space.
We use the kernel Kk to write down an expression for the kernel of Sk pΓq, which finally
leads to a first trace formula.
This chapter is based on the first part of Chapter 6 in [Miy06]. More precisely, we
deal with Section 6.1 and 6.3 in detail, quote an important result from Section 6.2, and
finish with Theorem 6.4.2.
3.1 Some function spaces on H
Throughout the following sections we assume k to be a fixed non-negative integer.
Definition 3.1.1. For p P r1, 8q and f : H Ñ C we define
}f }k,p »
f z Im z
pq pq
H
k 2 p
{
1{p
dν pz q
and
}f }k,8 ess supzPH |f pzq Impzqk{2|.
Moreover, we define Lpk pHq to be the space of measurable functions f : H Ñ C such that
}f }k,p 8
where we identify f, g P Lpk pHq with each other if }f g }k,p 0. Further, we
define Hkp pHq to be the subspace consisting of all holomorphic functions in Lpk pHq.
It can be easily checked that Lpk pHq is a normed space with respect to } }k,p for any
p P r1, 8s. In the case p 2 we can define
xf, gyk »
H
f pz qg pz q Impz qk dν pz q
for f, g P L2k pHq. Again it can be easily checked that this defines an inner product on
L2k pHq which induces the norm } }k,2 , so L2k pHq is an inner product space. As Hkp pHq is
a linear subspace of Lpk pHq it is also a normed space and in the case of p 2 an inner
product space.
21
Proposition 3.1.2. The space Lpk pHq is a Banach space for any p P r1, 8s. In particular, L2k pHq is a Hilbert space.
Proof. By the above observations it sufficies to show that Lpk pHq is complete. This is
clear for k 0 as Lp0 pHq is the usual Lp -space of functions on H with respect to the
measure dν pz q. But Lpk pHq and Lp0 pHq are isomorphic as normed spaces via the map
f pz q ÞÑ f pz q Impz qk{2 , so Lpk pHq is complete for any integer k.
We will see that Hkp pHq is a closed subspace of Lpk pHq for any p P r1, 8s, and thus also
complete, so a Banach space and if p 2 a Hilbert space. To prove this we need some
basic complex analysis:
Lemma 3.1.3. Let p P r1, 8q, z0 P H and ε ¡ 0 such that B : B3ε pz0 q is contained in
H. Moreover, let f : H Ñ C be holomorphic. Then there is C ¡ 0 depending on p, z0
and ε but not on f such that
sup
P p q
|f pzq| ¤ C
z Bε z0
»
B
f z Im z
k 2 p
pq pq
{
dν pz q
1{p
.
For a proof we refer to Theorem 2.6.1 on page 61 in [Miy06].
Corollary 3.1.4. Let p P r1, 8q and f : H Ñ C be holomorphic. For every z0
is Cz0 ¡ 0 such that |f pz0 q| ¤ Cz0 }f }k,p .
P U there
Proof. Let z0 P U and choose ε ¡ 0 such that B3ε pz0 q „ H. Then by Lemma 3.1.3 there
is C ¡ 0 depending on z0 but not on f such that |f pz0 q| ¤ C }f }k,p .
Corollary 3.1.5. Let p P r1, 8q and let pfn qn „ Hkp pHq be a Cauchy sequence with
respect to } }k,p . Then there is f : H Ñ C holomorphic such that fn Ñ f uniformly on
any compact subset of H.
Again we omit the proof as the statement is proven in detail in [Miy06], compare
Corollary 2.6.4 on page 63. The next proposition is Theorem 6.1.1 on page 220 in
[Miy06]. We give a proof since the one Miyake presents misses some details.
Proposition 3.1.6. For any p
ular, Hk2 pHq is a Hilbert space.
P r1, 8s the space HkppHq is a Banach space.
In partic-
Proof. We already know that Hkp pHq is a linear subspace of the Banach space Lpk pHq.
Therefore it sufficies to show that Hkp pHq is closed in Lpk pHq with respect to } }k,p .
Let pfn qn be a Cauchy sequence in Hkp pHq, then pfn qn is also a Cauchy sequence in
Lpk pHq and thus has a limit f P Lpk pHq. First suppose that p P r1, 8q. Then there is
h : H Ñ C holomorphic such that fn Ñ h uniformly on any compact subset of H by
Corollary 3.1.5. We have to show that f h almost everywhere.
By Theorem 5.2 on page 138 (p 1) and Theorem 5.2 on page 210 (1 p 8)
in [Lan93] a sequence pgn qn of some general Lp space that converges to some g P Lp
with respect to the corresponding norm }}p , has a subsequence which converges almost
22
everywhere to g. Put gn pz q : fn pz q Impz qk{2 and g pz q : f pz q Impz qk{2 . Then gn Ñ g
in Lp0 pHq which is a general Lp space, so there is a subsequence pgnl ql such that gnl Ñ g
almost everywhere. Hence we also have fnl Ñ f almost everywhere, and therefore f h
almost everywhere since fn Ñ h pointwise.
Now let p 8. Note that }g }k,8 supzPH |g pz q Impz qk{2 | for any continuous function
k {2
g P L8
as before, then gn is continuous since fn is, and
k pHq. Put gn pz q : fn pz q Impz q
}gn}8,H : sup |gnpzq| }fn}k,8.
P
z H
Thus pgn qn is a Cauchy sequence with respect to the uniform norm }}8,H on H as pfn qn
is a Cauchy sequence with respect to } }k,8 . Since the space of continuous functions
C pH, Cq is complete with respect to the uniform norm, there is g P C pH, Cq such that
gn Ñ g uniformly. Put hpz q : g pz q Impz qk{2 , then f h in L8
k pHq by construction.
Let K be a compact subset of H. Then
sup |fn pz q hpz q|
P
z K
¤
sup Impz qk{2 }gn g }8,H .
P
z K
Here the right-hand side goes to 0 as n Ñ 8 since the continuous function z ÞÑ Impz qk{2
is bounded on the compact set K and gn Ñ g uniformly. Therefore we have shown that
fn Ñ h uniformly on any compact subset of H, which implies that h is holomorphic.
Thus the sequence pfn qn has a limit in Hk8 pHq as claimed.
Theorem 3.1.7. The space Hk2 pHq is a reproducing kernel Hilbert space.
Proof. We showed in the previous proposition that Hk2 pHq is a Hilbert space. Fix z P H
and let Ez pf q : f pz q be the evaluation functional on Hk2 pHq. By Corollary 3.1.4 we
have |Ez pf q| |f pz q| ¤ Cz }f }k,2 for some Cz ¡ 0 depending on z but not on f . Hence
Ez is continuous for every z P H and therefore Hk2 pHq is a reproducing kernel Hilbert
space by Proposition 2.3.3.
Notation. We denote the kernel of Hk2 pHq by Kk .
Recall that by definition Kk is a function of the form H H Ñ C such that Kk p, wq
is an element of Hk2 pHq for every fixed w P H, and f pwq xf, Kk p, wqyk for every
f P Hk2 pHq, w P H. Using part (ii) of Corollary 2.3.4 we see
f pw q for any such f
»
H
f pz qKk pw, z q Impz qk dν pz q.
(3.1.1)
P Hk2pHq, w P H.
3.2 Computation of the kernel of Hk2pHq
We will now develop a precise formula for Kk . The following proposition starts characterising the kernel and will be very useful in the course of this thesis, too.
23
Proposition 3.2.1. For any α P GL2 pRq we have
Kk pαz, αwq
detpαqk j pα, z qk j pα, wq Kk pz, wq,
k
P H.
z, w
To prove this proposition we use a simple lemma:
P L2k pHq. Then
} f |k α }k,2 detpαqk{21}f }k,2.
Lemma 3.2.2. Let α P GL2 pRq and f
In particular, we have f P L2k pHq if and only if f |k α P L2k pHq, and similarly f
if and only if f |k α P Hk2 pHq.
P Hk2pHq
Proof. Let α P GL2 pRq and f P L2k pHq. Since elements in GL2 pRq act as automorphisms
on H we have αH H. Thus we see using Lemma 2.1.1
»
H
|f pzq|
2
Impz q dν pz q »
k
α 1 H
|f pαzq|2 Impαzqk dν pzq
detpαq »
2 k
H
|pf |k αqpzq|2 Impzqk dν pzq.
Hence }f }k,2 detpαq1k{2 } f |k α }k,2 . For the second part of the lemma we only have to
note that f is holomorphic on H if and only if f |k α is.
Proof of Proposition 3.2.1. Let α P GL2 pQq and define
pαq pz, wq : detpαqk j pα, z qk j pα, wqk K pαz, αwq.
k
Kk
Using Proposition 2.1.1 one can easily check that
A
pαq
f, K p, wq
k
E
@
D
detpαqk1j pα, wqk f |k α1, Kk p, αwq
k
k
for any f P Hk2 pHq and any w P H. By Lemma 3.2.2 we know that f |k α1 is an element
of Hk2 pHq. Therefore we can use the reproducing property of the kernel Kk which yields
@
D
f |k α1 , Kk p, αwq
pαq p, wqy
Hence xf, Kk
f |k α1 pαwq detpαq1k j pα, wqk f pwq.
k
f pwq for all f P Hk2pHq, w P H. Moreover, we have
k
pαq pz, wq detpαq j pα, wqk pK p, αwq| αq pz q.
k
k
Kk
Since Kk is a reproducing kernel, Kk p, αwq is an element of Hk2 pHq for fixed w P H.
p αq
Thus Kk p, αwq|k α is an element of Hk2 pHq by Lemma 3.2.2, and therefore Kk p, wq
pαq
itself is an element of Hk2 pHq. Hence Kk is a reproducing kernel of Hk2 pHq, and thus
pαq
by uniqueness of the kernel (Proposition 2.3.2) we have Kk Kk .
24
?
Using Proposition 3.2.1 with α1 0a 1{0?a for some a ¡ 0, and α2 10 1b for some
b P R we directly get the following corollary, which describes the kernel further:
Corollary 3.2.3. We have for any a ¡ 0
Kk paz, awq ak Kk pz, wq,
and for any b P R
K k pz
bq Kk pz, wq,
b, w
z, w
P H,
z, w
P H.
The second part of this corollary is equation (6.1.7) on page 221 in [Miy06]. The
subsequent discussion on the same page of [Miy06] proves the following proposition,
which is Theorem 6.1.2 on page 222 in [Miy06].
Proposition 3.2.4. The reproducing kernel of Hk2 pHq is given by
Kk pz, wq ck
for some constant ck
zw
2i
k
P r0, 8q.
We present Miyake’s proof as the statement is essential for this section.
P H, z w P Hu and the function
pz, wq ÞÑ Kk pz, z wq.
Proof. Define the set Ω tpz, wq P C2 : z
h : Ω Ñ C,
One can check that if ϕpz q is holomorphic on an open domain U then ϕpz q is holomorphic on U . Hence hpz, wq is holomorphic in w since Kk is holomorphic in the first
argument and hpz, wq Kk pz w, z q as the kernel Kk is conjugate symmetric. Next
we fix w P H and consider hp, wq as a composition of the functions i : z ÞÑ pz, z q and
H : pz1 , z2 q ÞÑ Kk pz1 , z2 wq. The function H is holomorphic in both arguments by the
above considerations, and the components i1 , i2 are trivially holomorphic, too.
We are now going to use complex analysis of several variables to show that h is
holomorphic in z. First we note that i is holomorphic as a function from C to C2 by
part (5) of Proposition 1.2.2 on page 8 of [Sch05]. Further, H is partially holomorphic in
the sense of Definition 1.2.21 on page 13 of [Sch05]. This implies H to be holomorphic
by Hartogs’ theorem which is a deep result of the theory of complex analysis of several
variables. It is remarked in 1.2.28 on page 17 of [Sch05] and proven in Section 2.4 of
[Kra01]. Therefore we have that the composition hp, wq H i is holomorphic in the
usual sense by part (4) of Proposition 1.2.2 of [Sch05].
So we have shown that h is holomorphic in both arguments. Moreover, we have by
part (2) of Corollary 3.2.3 that
hpz
b, wq Kk pz
b, z w
bq Kk pz, z wq hpz, wq
for any pz, wq P Ω, b P R. We claim this implies hpz, wq hpz 1 , wq for any z, z 1 , w with
pz, wq, pz1, wq P Ω. To see this consider the map Φpτ q hpz τ, wq hpz, wq for τ P C
25
and fixed pz, wq P Ω. Then Φ is holomorphic on a neighbourhood of the real line as
h is holomorphic in the first argument, and Φ vanishes on the real line by the above
observation. Hence Φ vanishes everywhere. Consequently h is locally constant in z and
thus also globally as h is holomorphic in z. Therefore we can define
l : H Ñ C, w
ÞÑ hpzω , wq Kk pzω , zω wq
where zω is any element of the upper half-plane such that pzω , wq P Ω. (One can easily
see that there is such an element zω for every w P H.) The map l is holomorphic as h is
in w, and
for any z, w
Kk pz, wq Kk z, z pz wq
l pz w q
P H. Next we use part (1) of Corollary 3.2.3 which yields
lp2awq l aw apwq
Kk paw, apwqq ak Kk pw, wq ak lp2wq
for any a ¡ 0. In particular, taking w i{2 we get lpiy q y k lpiq for all y ¡ 0. Finally,
we define
z k
L : H Ñ C, z ÞÑ
lpiq.
i
Then L is obviously holomorphic on H and Lpiy q lpiy q for all y ¡ 0. So L and l
agree on the imaginary axis, and thus everywhere on H as they are both holomorphic.
Therefore we get
Kk pz, wq l pz wq L pz wq for any z, w P H. Put ck
Moreover, we have
zw
i
k
lpiq
2k lpiq then we see Kk pz, wq ck ppz wq{2iqk as claimed.
lpiq Kk i{2, i{2 i
Kk pi{2, i{2q P r0, 8q
by part (1) of Corollary 2.3.4, and thus also ck
P r0, 8q. Therefore we are done.
Next we want to compute the constant ck . Up to now we have been following [Miy06]
very closely. Though we filled in the details for some arguments, we kept the given
structure. The next part will differ from the book. On the pages 222 to 225 Miyake uses
Fourier analysis to calculate the constant ck . We will use a much simpler and shorter
argument, which does not describe the space Hk2 pHq as nicely as Miyake’s work does
(compare Theorem 6.1.6 and Corollary 6.1.7 on page 224), but gives the desired result
nevertheless.
The idea is to use the reproducing property of the kernel Kk (equation (3.1.1)) with
some explicit function f P Hk2 pHq. This reduces the problem of computing ck to
(1) finding an element of Hk2 pHq, and
26
(2) evaluating the corresponding integral.
Define
f0 : H Ñ C, z
iqk .
ÞÑ pz
We claim that f0 is an element of Hk2 pHq for any integer k ¥ 2. (Note that this is not a
limitation as Miyake proves in Corollary 6.1.7 that Hk2 pHq t0u for any integer k ¤ 1.)
We have
»
} } f0 2k,2
p0,8q
»
|px
iy q
x2
y2
R
p1,8q
i|2k y k
k
dpx, y q
y2
py 1qk2dpx, yq.
R
Clearly py 1qk2
¤ yk2 for y P p1, 8q, k ¥ 2. Hence
»
} } ¤
f0 2k,2
x2
p1,8q
»
y2
k
y k2 dpx, y q
R
¤
p1,8qp0,πq
»π
r2k pr sinpϕqqk2 rdpr, ϕq
sinpϕq dϕ
k 2
0
The last expression is obviously finite for k
may use equation (3.1.1) with f f0 and w
p2iqk
f0piq »
»H
1
¥ 2, so f0 P Hk2pHq as claimed.
i. We have
iqk p2iqk ck pi z qk Impz qk dν pz q.
H
iqk pi z qk
ck 4k
rpk 1q dr.
Thus we
f0 pz qKk pi, z q Impz qk dν pz q
pz
One can check that pz
equation (3.2.1) gives
»8
p1qk px2 p1
»
p0,8q
x
p1
2
yq
2
k
y q2 q
(3.2.1)
k where z x iy. Hence
1
y dpx, y q
k 2
.
(3.2.2)
R
It remains to compute this integral. Substituting s x{p1
»
p0,8q
x
2
p1
yq
2
k
y dpx, y q »
p1
k 2
R
R
s qk ds
2
»8
0
y q for x yields
p1
y q2k 1 y k2 dy. (3.2.3)
Let B pa, bq denote the so called beta function as defined on the bottom of page 20 in
[BW10]. By Theorem 2.1.2 on page 21 of the book we have
B pa, bq »8
0
ua1 p1
Γpaq Γpbq
uqpa bq du Γpa bq
27
for all a, b P C with positive real part. Here Γpz q denotes the gamma function as defined
on page 19 of the mentioned book. It has the well-known property Γpnq pn 1q! for all
positive integers n. For a further discussion of these two functions we refer to Chapter 2
of [BW10]. In particular, Section 2.1 covers the mentioned results. However, using the
stated identities, the second integral on the right-hand side of equation (3.2.3) solves to
»8
0
y q2k 1 y k2 dy
p1
B pk 1, kq pk p22kq! p2kq! 1q! .
(3.2.4)
Thus we are left with the first integral on the right-hand side of equation (3.2.3). Define
g pz q p1 z 2 qk , then g is a meromorphic function on C with poles of order k at i.
One can check that the residue of g at i is given by
p2k 2q! .
pk 1q! pk 1q!
Fix R ¡ 1. Put γ0 ptq Reπit and γ1 ptq Rp2t 1q for t P r0, 1s. By the residue theorem
Respg, iq 2i 4k we have
»
γ0 γ1
³
g pz qdz
2πi Respg, iq
1k
p4k π1q!p2kpk 21qq!!
(3.2.5)
³
Clearly γ1 g pz qdz Ñ R p1 s2 qk ds as R Ñ 8 which is the integral we want to compute.
On the other hand we have
»
γ0
g z dz pq
¤
»1
f Reπit Rπieπit dt
0
For sufficiently large R we have |1
»
γ0
pq
¤
2 R π
k
s2
1
k
R2 e2πit dt.
0
k
ds
R
»1
1 2k
dt.
0
Evidently the right-hand side goes to 0 as R
R Ñ 8 on both sides of equation (3.2.5)
1
Rπ
R2 e2πit | ¥ R2 {2, so
g z dz »
»1
Ñ 8.
Therefore we get taking the limit
1k
p4k π1q!p2kpk 21qq!! .
Combining this equality with equation (3.2.4) we are finally able to compute the integral
in (3.2.3). We have
»
p0,8q
x2
p1
y q2
k
R
and thus ck
y k2 dpx, y q
41k π
,
k1
pk 1q{p4πq by equation (3.2.2). Therefore we have shown:
28
Theorem 3.2.5. For any integer k
¥ 2 the reproducing kernel of Hk2pHq is given by
Kk pz, wq k1
4π
zw
2i
k
.
We end this section by studying some more properties of the kernel Kk , which will be
needed in Section 3.4 to determine the kernel of Sk pΓq.
Lemma 3.2.6. For any integer k
is an element of Hk1 pHq.
Proof. Let k ¥ 3 and fix w
δ Impwq. Then
}Kk p, wq}k,1 P H.
¥ 3 and any fixed w P H the kernel function Kk p, wq
We already know that Kk p, wq is holomorphic. Put
»
2k pk 1q
|
z w|k Impz qk{2 dν pz q
4π
H
»
k
2 pk 1q
|x iy|k py δqk{22dpx, yq
4π
Rpδ,8q
2k pk 1q
4π
»π»8
0
{ p q
δ sin ϕ
rk pr sinpϕq δ qk{22 rdrdϕ.
Substituting t r sinpϕq{δ 1 for fixed ϕ P p0, π q yields
»8
{ p q
δ sin ϕ
r k
1
pr sinpϕq δq { dr
k 2 2
δ k{2 sinpϕqk2
»8
0
tqk 1 tk{22 dt.
p1
For the integral on the right we may again use the identity of the beta function as given
in Theorem 2.1.2 on page 21 in [BW10]. We have
»8
0
p1
tqk 1 tk{22 dt
B
k
2
1, k2
,
and therefore
}Kk p, wq}k,1 2k pk 1q
Impwqk{2 B
4π
k
2
1, k2
» π
0
sinpϕqk2 dϕ.
Hence we have }Kk p, wq}k,1
8 since all the terms on the right are finite.
Lemma 3.2.7. For any integer k ¥ 2 and any fixed w P H the kernel function Kk p, wq
is an element of Hk8 pHq.
Proof. Let k ¥ 2 and fix w P H. Writing z x iy and w a ib one can check that
Kk z, w Im z
p
q pq
k {2 2k pk 1q
4π
29
y
px aq2 py
k{2
bq 2
.
Hence
2k pk 1q
4π
}Kk p, wq}k,8 Further, one can check that supy¡0 y py
k{2
y
¡ py bq2
bq2 p4bq1 , so
sup
.
y 0
1 Impwqk{2 8.
}Kk p, wq}k,8 k 4π
Since Kk p, wq is also holomorphic we are done.
Combining Lemma 3.2.6 and Lemma 3.2.7 we can prove part (3) of Theorem 6.2.1 on
page 226 in [Miy06]:
Corollary 3.2.8. For any integer k ¥ 3 and any fixed w
Kk p, wq is an element of Hkp pHq for all p P r1, 8s.
Proof. Let k ¥ 3 and fix w P H. We have Kk p, wq
and put f pz q Kk pz, wq Impz qk{2 . Then
}Kk p, wq} p
k,p
»
|f pzq| χtz : |f pzq|¤1upzqdν pzq
H the kernel function
P Hk1pHq X Hk8pHq.
»
Let p
P p1, 8q
|f pzq|pχtz : |f pzq|¡1upzqdν pzq
p
»H
P
»
H
¤ |f pzq|χtz : |f pzq|¤1upzqdν pzq }f }8 |f pzq| χtz : |f pzq|¡1upzqdν pzq
H
H
¤ }Kk p, wq}k,1 p1 }Kk p, wq}k,8q .
So }Kk p, wq}k,p 8 and thus Kk p, wq P Hkp pHq.
p
Now recall that by Proposition 2.3.5 the map
πk : L2k pHq Ñ Hk2 pHq, pπk f qpwq xf, Kk p, wqy
is a well-defined operator which projects L2k pHq onto its subspace Hk2 pHq. Let f P Lpk pHq
for any p P r1, 8s. Using Hölder’s inequality (compare Theorem 5.1 on page 209, 210 in
[Lan93]) we get
|xf, Kk p, wqy| ¤
»
H
»
f z Im z
p q p qk{2 Kk pz, wq Impzqk{2 dν pzq
f z Im z
k 2 p
{
¤
pq pq
H
}f }k,p}Kk p, wq}k,q
1{p »
dν pz q
H
Kk z, w Im z
p
k 2 q
q pq
{
dν pz q
1{q
where q is the well-known Hölder conjugate of p. Hence we may extend the operator πk
to any Lpk pHq space. Moreover, the following is true:
¥ 3 and any p P r1, 8s the operator
πk : Lpk pHq Ñ Hkp pHq, pπk f qpwq xf, Kk p, wqy
is well-defined and projects Lpk pHq onto its subspace Hkp pHq.
Theorem 3.2.9. For any integer k
30
Note that this is a very strong statement as it implies that the reproducing property of
Kk , namely f pwq xf, Kk p, wqy, does not only hold for f P Hk2 pHq, but for f P Hkp pHq
for all p P r1, 8s. We omit the corresponding proof as it is very involved and refer to
Section 6.2 in [Miy06] instead which deals with it in detail. In particular, the statement
is given by Theorem 6.2.2 on page 226.
Further, we note that we we will only need two special cases of this theorem in the
course of this theses, namely that πk maps L1k pHq to Hk1 pHq and that πk acts trivially
on Hk8 pHq. These facts will be essential for the proof of Theorem 3.4.5.
3.3 Interpretation of Sk pΓq as a reproducing kernel
Hilbert space
Throughout this and the following sections of the present chapter let Γ be a modular
group, so a finite index subgroup of SL2 pZq, and k ¥ 3 an integer. We start by defining
spaces of Γ-invariant functions similar to the ones introduced in Section 3.1. These will
be denoted by Lpk pΓq and Hkp pΓq, respectively. As in the case of Hk2 pHq it turns out that
Hk2 pΓq is a reproducing kernel Hilbert space, and one can easily check that Sk pΓq, the
space of cusp forms of weight k and level Γ, is contained in Hk2 pΓq. In fact, these spaces
agree for k ¥ 3. (This also explains why we only consider integers k ¥ 3 from now on.)
k{2
First note that for f being
k the map
z ÞÑ |f pz q Impz q | is
Γ-invariant of weight
Γ-invariant of weight 0, so f pγz q Impγz qk{2 f pz q Impz qk{2 for all γ P Γ. Thus the
following definition is well-defined:
Definition 3.3.1. For p
define
P r1, 8q and f : H Ñ C satisfying f |k γ f
}f }Γ,p »
f z Im z
z
Γ H
pq pq
k 2 p
{
for all γ
P Γ we
1{p
dν pz q
and
}f }Γ,8 ess supzPΓzH |f pzq Impzqk{2|.
Moreover, we define Lpk pΓq to be the space of measurable functions f : H Ñ C such that
f |k γ f for all γ P Γ and }f }Γ,p 8 where we identify f, g P Lpk pΓq with each other if
}f g}Γ,p 0. pFurther, we define HkppΓq to be the subspace consisting of all holomorphic
functions in Lk pΓq.
k {2
Clearly } }k,8 and } }Γ,8 agree on L8
| is Γ-invariant
k pΓq since z ÞÑ |f pz q Impz q
8
of weight 0 as mentioned earlier. In particular, Lk pΓq is a subspace of L8
k pHq. As in
Section 3.1 one can easily check that Lpk pΓq is a normed space with respect to } }Γ,p for
any p P r1, 8s, and similarly to x, yk we can put
xf, gyΓ »
z
Γ H
f pz qg pz q Impz qk dν pz q
for f, g P L2k pΓq. This is well-defined since z ÞÑ f pz qg pz q Impz qk is also Γ-invariant of
weight 0, and thus x, yΓ defines an inner product on L2k pΓq which clearly corresponds
31
to the norm } }Γ,2 . So L2k pΓq is an inner product space. As in Section 3.1, Hkp pΓq is
a normed space and in the case of p 2 an inner product space, since it is a linear
subspace of Lpk pΓq.
Proposition 3.3.2. The space Hkp pΓq is a Banach space for any p P r1, 8s. In particular,
Hk2 pΓq is a Hilbert space.
We only sketch the proof:
Proof. Similarly to Lpk pHq one may define the space Lpk pU q where we replace the upper
half-plane H by some open subset U of H. Let } }U,p denote the corresponding norm.
As in Proposition 3.1.2 one can check that this space is a Banach space.
Let U be the interior of some fundamental domain F of Γ. Then Hkp pΓq is a subspace
of Lpk pU q. Let pfn qn be a Cauchy sequence in Hkp pΓq with respect to }}U,p . Using similar
arguments as in Proposition 3.1.6 one can check that there is a holomorphic function
f P Lpk pU q such that fn Ñ f with respect to } }U,p . Clearly f is also Γ-invariant since
all fn are Γ-invariant and continuous.
Theorem 3.3.3. We have
Sk pΓq Hk8 pΓq Hk2 pΓq.
Moreover, Sk pΓq and Hk2 pΓq are isomorphic as Hilbert spaces.
The first equality of this theorem is exactly Theorem 2.1.5 on page 42 in [Miy06].
Further, Hk8 pΓq „ Hk2 pΓq is obvious since ΓzH has finite measure with respect to ν.
Hence it remains to check that Hk2 pΓq is contained in Sk pΓq, so that functions in Hk2 pΓq
vanish at the cusps of Γ. This is done in Theorem 6.1.3 on page 228, 229 in [Miy06].
We omit these slightly technical proofs here, as they are given in appropriate detail in
Miyake’s book. Finally, the second part of the theorem is clear since the Petersson inner
product of Sk pΓq coincides with the inner product of Hk2 pΓq.
The following theorem is almost a direct consequence of the previous identity.
Theorem 3.3.4. The space Sk pΓq is a reproducing kernel Hilbert space.
Proof. We already know that Sk pΓq Hk2 pΓq is a Hilbert space. Fix z P H and let
Ez pf q : f pz q be the evaluation functional on Sk pΓq. One can check that there is always
a finite number of fundamental domains F1 , . . . ,”
Fn such that the corresponding interiors
are pairwise disjoint and z lies in the interior of nj1 Fj . We call this interior U and may
choose ε ¡ 0 such that B : B3ε pz q „ U . By Lemma 3.1.3 there is C ¡ 0 depending on
z but not on f such that
|Ez pf q| |f pzq| ¤ C
2
2
»
f z Im z
2
C2
U
n »
¸
f z Im z
j 1 Fj
Hence Ez is continuous for every z
Hilbert space by Proposition 2.3.3.
P
p q p qk{22 dν pzq
p q p qk{22 dν pzq C 2n}f }2Γ,2.
H and therefore Sk pΓq is a reproducing kernel
32
3.4 Computation of the kernel of Sk pΓq
Next we want to determine the kernel of the reproducing kernel Hilbert space Sk pΓq
where k ¥ 3 is an integer and Γ is a modular group. Instead of characterising it step by
step as in Section 3.2, we will write down a guess for the reproducing kernel function,
and prove that this function has the desired properties.
We define for f P L1k pHq the function
f Γ pz q 1 ¸
|Z pΓq| γPΓ pf |k γ q pzq,
z
P H.
By definition f Γ is clearly Γ-invariant if it is well-defined, which is not obvious and will
be shown in the following proposition.
Proposition 3.4.1. Let f P L1k pHq. The sum defining f Γ converges absolutely almost
everywhere on H and f Γ is an element of L1k pΓq.
Note that this is part (1) of Theorem 6.3.2 on page 229 in [Miy06]. Since the corresponding proof is not very detailed, we will give a proof here.
Proof. Fix a fundamental domain F of Γ and note that
»
Γ f z Im z
F
pq
pq
{ dν pz q
k 2
¤
1
|Z pΓq|
» ¸
P
F γ Γ
|f pγzq| Impγzqk{2dν pzq.
We can interchange summation and integration since the integrand |f pγz q| Impγz qk{2 is
positive. (This follows for example from Corollary 5.13 on page 143 in [Lan93].) So
» ¸
¸»
{
|f pγzq| Impγzq dν pzq |f pγzq| Impγzqk{2dν pzq
F γ PΓ
γ PΓ F
¸»
|f pzq| Impzqk{2dν pzq
γ PΓ γ pF q
»
|Z pΓq| |f pzq| Impzqk{2dν pzq.
k 2
H
°
Therefore γ PΓ |f pγz qj pγ, z qk | Impz qk{2 is ν-integrable over any fundamental domain F
of Γ. Hence the sum defining f Γ is absolutely convergent almost everywhere on any
fundamental domain of Γ, so on H. Furthermore, we have shown that
Γ
f Γ,1
»
z
Γ H
Γ f z Im z
p qk{2dν pzq ¤ }f }k,1.
pq
Thus f Γ is a well-defined element of L1k pΓq.
Proposition 3.4.2. If f P Hk1 pHq then f Γ
convergent everywhere on H.
P Hk1pΓq and the sum defining f Γ is absolutely
33
This is part (2) of Theorem 6.3.2 on page 229 in [Miy06], and it follows directly from
the previous proposition and the following lemma, which is a special case of part (1) of
Theorem 2.6.6 on page 64 in [Miy06]. Considering the notation on the bottom of page
63 we chose Λ to be trivial and f to be holomorphic everywhere on H. We omit the
proof.
Lemma 3.4.3. Let f : H
of Γ. Put
Ñ C be holomorphic, and let tc1, . . . , cr u be the set of cusps
H1
Hz
r ¤
¤
P
γVi
i 1γ Γ
where Vi is any neighbourhood of the cusp ci . If f satisfies
»
H
|1 f pzq| Impzqk{2dν pzq 8
for all such H 1 , we define the Poincaré series of f as
Fk pz q ¸
P
pf |k γ q pzq.
γ Γ
The series defining Fk converges absolutely and uniformly on any compact subset of H.
In particular, Fk is Γ-invariant and holomorphic on H.
Now we define what will turn out to be the reproducing kernel of Sk pΓq. Recall that
Kk is the reproducing kernel of Hk2 pHq as determined in Section 3.2. We have shown in
Lemma 3.2.6 that Kk p, wq P Hk1 pHq for every fixed w P H. This allows us to define
KkΓ pz, wq 1 ¸
|Z pΓq| γPΓ pKk p, wq|k γ q pzq,
z, w
P H.
By Proposition 3.4.2 the sum defining KkΓ is absolutely convergent for every fixed pair
of elements pz, wq P H H, and KkΓ p, wq is an element of Hk1 pΓq for every fixed w in
H. Moreover, the following statement holds, which will be essential for the proof of
Proposition 4.1.9 in the next chapter:
Lemma 3.4.4. The sum defining KkΓ is uniformly convergent on any compact subset of
H H.
This is part (3) of Theorem 6.3.2 in [Miy06]. In the end of the corresponding proof
on page 230 Miyake refers to (his) Corollary 2.6.4. We remark that he probably means
(his) Theorem 2.6.1 instead. In the following we fill in the details for the proof Miyake
gives using Theorem 2.6.1 which corresponds to our Lemma 3.1.3.
Proof. Fix w
¸
P H and note that
z |pKk p, wq|k γ q p q| Γ,1 γ PΓ
»
¸
z
P
Γ Hγ Γ
34
|Kk pγz, wq| Impγzqk{2dν pzq.
We may interchange summation and integration for a fixed fundamental domain F of Γ
since |Kk pγz, wq| Impγz qk{2 is positive. So we see
¸
P
γ Γ
z |pKk p, wq|k γ q p q| Γ,1 ¸»
P
γ Γ F
|Kk pγz, wq| Impγzqk{2dν pzq
¸»
P
γ Γ γF
|Kk pz, wq| Impzqk{2dν pzq
»
|Z pΓq| |Kk pz, wq| Impzqk{2dν pzq.
H
Write w x iy for w P H and put σ y 1{2 01 yx . Then σ P SL2 pRq with σw
Further, we have j pσ, wq Impwq1{2 . We use Proposition 3.2.1 with α σ to get
i.
|Kk pz, wq| Impzqk{2 Impwqk{2|Kk pσz, iq| Impσzqk{2.
Therefore we have by Lemma 2.1.1 substituting z 1 σz in the integral that
¸
|p
Kk p, wq|k γ q pz q|
|Z pΓq| Impwqk{2 }Kk p, iq}k,1
γ PΓ
Γ,1
and thus
Γ
K , w k
Γ,1
p q
¤ }Kk p, iq}k,1 Impwqk{2
(3.4.1)
for any w P H.
Now let z0 P H be arbitrary. As in the proof of Theorem 3.3.4 we may choose a finite
number of fundamental domains F1 , . . . , Fn for Γ such
that the corresponding interiors
”n
are pairwise disjoint and z0 lies in the interior U of j 1 Fj . Further, we choose ε ¡ 0
such that B3ε pz0 q „ U . By Lemma 3.1.3 there is a constant C ¡ 0 depending only on
z0 and ε such that
sup KkΓ z, w p
P p q
z Bε z0
q ¤C
C
»
U
n
¸
Γ
K z, w Im z
k
»
q p qk{2 dν pzq
p
Γ
K z, w Im z
k
j 1 Fj
p
q p qk{2 dν pzq Cn KkΓp, wqΓ,1 .
for all w P H since all functions KkΓ p, wq, w P H, are holomorphic on H as remarked
earlier. Let w0 P H and put δ Impw0 q{2. Then Bδ pw0 q „ H and by equation (3.4.1)
sup
P p q
P p q
z B ε z0
w Bδ w0
Γ
K z, w k
p
q ¤
Cn}Kk p, iq}k,1 sup
P p q
w Bδ w0
Impwqk{2
8.
Therefore we have shown that for any pair pz0 , w0 q P H H there are ε ¡ 0, δ ¡ 0
such that the sum defining KkΓ is unformly convergent on Bε pz0 q Bδ pw0 q. In other
words, the sum is locally uniformly convergent, and hence also uniformly convergent on
compact subsets of H H.
35
We will now prove that KkΓ is indeed the reproducing kernel of Sk pΓq. Even though
the proof is given in [Miy06] fairly detailed (see Theorem 6.3.3 on page 230), we will
present it here, too, since the result is central for this thesis.
Theorem 3.4.5. The reproducing kernel of Sk pΓq is given by
KkΓ pz, wq 1 ¸
|Z pΓq| γPΓ pKk p, wq|k γ q pzq.
Proof. By definition we have to check that
(1) KkΓ p, wq P Sk pΓq for every fixed w
P H, and
(2) f pwq f, KkΓ p, wq Γ for every f P Sk pΓq and every w P H.
We start with (1). Fix w P H. Recall that KkΓ p, wq P Hk1 pΓq by Proposition 3.4.2.
In particular, KkΓ p, wq is Γ-invariant and holomorphic. Thus it sufficies to show that
KkΓ p, wq is an element of L8
k pHq, as this would imply
}KkΓp, wq}Γ,8 }KkΓp, wq}k,8 8,
which then gives KkΓ p, wq P Hk8 pΓq Sk pΓq as desired.
To show that KkΓ p, wq P L8
k pHq we need to recall some basic functional analysis. Let
X be a Banach space over C. A sequence px1n qn in the dual space X 1 of X is called weakly* convergent if the sequence px1n pxqqn converges in C for every fixed x P X. Similarly we
say that the sequence px1n qn is weakly-* convergent to x1 P X 1 if x1n pxq Ñ x1 pxq in C for
every x P X. The Banach-Steinhaus Theorem (see for example Theorem 14.6 on page
@
D
96/97 in [Con97]) implies that every weakly-* convergent sequence has a weakly-* limit,
and one can easily check that this limit is unique.
Now recall that the usual L8 -space is isomorphic to the dual space of L1 via the
isomorphism
1
Φ : L8 Ñ L1 , f ÞÑ rg ÞÑ xf, g ys .
(For a proof of this see for example Theorem 2.2 on page 188 in [Lan93].) Hence L8
k pHq
p
p
1
is isomorphic to the dual space of Lk pHq since Lk pHq is isomorphic to L0 pHq which is
the usual Lp -space with respect to the measure ν.
1
Consider a sequence pfn qn „ L8
k pHq. Then every fn corresponds to some xn in
1
1
1
1
pLk pHqq where xn is the linear functional given by g ÞÑ xfn, gyk , g P Lk pHq. Suppose
that pxfn , g yk qn is convergent as a sequence in C for every fixed g P L1k pHq. Then the
sequence px1n qn is weakly-* convergent, and thus has a unique limit x1 P pL1k pHqq1 by the
above observations. Corresponding to x1 there is a unique element f P L8
k pHq such that
1
1
1
1
x pg q xf, g yk for g P Lk pHq, and since x is the weakly-* limit of pxn qn we have that
xfn, gyk Ñ xf, gyk in C for every fixed g P L1k pHq.
The idea is now to interpret the sum defining KkΓ p, wq as the weakly-* limit of its
partial sums. Denote the partial sums of KkΓ p, wq by fn . Then each fn is up to a scalar
factor a finite sum of terms of the form Kk p, wq|k γ for different γ P Γ. Hence each fn is
36
in L8
k pHq as Kk p, w q is by Lemma 3.2.7. Thus pfn qn can be identified with a sequence
in pL1k pHqq1 . Suppose that this sequence is weakly-* convergent, then it has a unique
limit in pL1k pHqq1 , which again can be identified with some element f P L8
k pHq. In the
Γ
following we will check that Kk p, wq and f agree almost everywhere on H, meaning that
Γ
8
they denote the same element in L8
k pHq, so Kk p, w q P Lk pHq as desired. Afterwards we
will prove that the sequence of partial sums is indeed weakly-* convergent as assumed
earlier.
By the above considerations there is f P L8
k pHq such that xfn , g yk Ñ xf, g yk in C for
1
every fixed g P Lk pHq. We want to show that f and KkΓ p, wq agree almost everywhere
on H. Suppose this is not the case, then there is a compact set K „ H such that f and
KkΓ p, wq do not agree almost everywhere on K. Put N tz P K : f pz q KkΓ pz, wqu,
then N is measurable with 0 ν pN q ¤ ν pK q 8. Recall that the sequence of partial
sums fn converges pointwise to KkΓ p, wq on H, and note that each fn is obviously
measurable as it is continuous. Therefore we may use Egorov’s Theorem (see for example
Theorem 4.4 on page 33 in [SS05]) which tells us that we can find a closed set A „ N
such that ν pN zAq ¤ ν pN q{2 and fn converges to KkΓ p, wq uniformly on A.
Now let g P L1k pHq be a function with compact support in A, then we see
xf, gyA,k xf, gyk nlim
xf , gy lim xf , gy Ñ8 n k nÑ8 n A,k
@
KkΓ p, wq, g
D
A,k
.
Here x, yA,k denotes the restriction of the scalar product introduced in Section 3.1 to
A, and the last equality holds since fn Ñ KkΓ p, wq uniformly on A. Define
Gpz q sign f pz q KkΓ pz, wq
χApzq
where χA denotes the characteristic function of A which is 1 on A and 0 otherwise. Then
G P L1k pHq as ν pAq is finite, and G has compact support in A. Therefore we get
»
A
f z
p q
KkΓ
z, w Im z
p
q
pq
{ dν pz q »
k 2
@A
f pz q KkΓ pz, wq Gpz q Impz qk{2 dν pz q
D
f KkΓp, wq, G
0.
A,k
Therefore f and KkΓ p, wq agree almost everywhere on A. But this is a contradiction
since A „ N and ν pAq ¥ ν pN q{2 ¡ 0. Hence f and KkΓ p, wq agree almost everywhere
on H, and thus KkΓ p, wq P L8
k pHq as claimed. It remains to prove that the sequence of
partial sums is indeed weakly-* convergent.
The functional in pL1k pHqq1 associated to some fn is given by g ÞÑ xfn , g yk , g P L1k pHq.
We want to show that xfn , g yk is convergent as a sequence in C for every fixed g P L1k pHq.
Fix such a g. Then
xfn, gyk » H
¸
1
p
Kk p, wq|k γ q pz q g pz q Impz qk dν pz q
|Z pΓq| γPA
n
37
where An is a finite subset of Γ with An Ñ Γ. Since the sum is finite we may interchange
summation and integration. Fix γ P Γ. Using Proposition 3.2.1 with α γ we get
pKk p, wq|k γ q pzq j pγ 1, wqk Kk pz, γ 1wq,
»
so
H
pKk p, wq|k γ q pzq gpzq Impzqk dν pzq j pγ 1, wqk
@
Kk p, γ 1 wq, g
D
k
.
Let πk be the projection operator as defined in Theorem 3.2.9. Then
@
Kk p, γ 1 wq, g
and therefore
lim xfn , g yk
Ñ8
n
D
nlim
|Z pΓq|1
Ñ8
|Z pΓq|1
¸
P
k
pπk gqpγ 1wq,
¸
P
k
j pγ 1 , wq
pπk gqpγ 1wq
γ An
ppπk gq|k γ 1q pwq
γ Γ
|Z pΓq|1
¸
P
ppπk gq|k γ q pwq.
γ Γ
Hence we have limnÑ8 xfn , g yk pπk g qΓ pwq which is absolutely convergent for every
w P H by Proposition 3.4.2 and since πk maps g P L1k pHq to πk g P Hk1 pHq by Theorem
3.2.9. So the sequence of functionals associated to the partial sums of KkΓ is indeed
weakly-* convergent as claimed. Thus we are
with
@ done
D (1).
Γ
It remains to prove (2), so that f pwq f, Kk p, wq Γ for every f P Sk pΓq and every
w P H. Fix such f and w. Recall that we have KkΓ pz, wq KkΓ pw, z q by part (ii) of
Corollary 2.3.4. So
@
f, KkΓ
D
p, wq Γ »
z
Γ H
f pz qKkΓ pw, z q Impz qk dν pz q
1
|Z pΓq|
»
¸
z
P
Γ Hγ Γ
f pz qKk pγw, z qj pγ, wqk Impz qk dν pz q.
(3.4.2)
We want to interchange summation and integration. Note that by Proposition 3.2.1 and
since Kk pz, wq Kk pw, z q
Kk γw, z j γ, w
p
qp
k k
1
1
q Kk pw, γ zqj pγ , zq |Kk pγ 1z, wq||j pγ 1, zq|k .
Now fix a fundamental domain F of Γ. Then
¸»
P
z
γ Γ Γ H
f z Kk γw, z j γ, w
pq p
¤
qp
sup f z Im z
P
z H
pq pq
qk Impzqk dν pzq
»
{ ¸ |K pγ 1 z, wq| Impγ 1 z qk{2 dν pz q
k
γ PΓ F
k 2
}f }Γ,8 |Z pΓq| }Kk p, wq}k,1
8
38
since f P Sk pΓq Hk8 pΓq and Kk p, wq
and integration in equation (3.4.2), so
@
f, KkΓ
D
p, wq Γ 1 ¸
|Z pΓq| γPΓ
|Z p1Γq|
Since f
@
»
F
P L1k pHq.
Thus we may interchange summation
f pz qKk pγw, z qj pγ, wqk Impz qk dν pz q
¸»
f pγz qKk pγw, γz qj pγ, wqk Impγz qk dν pz q.
1
γ Γ γ F
P
P Sk pΓq we have f pγzq j pγ, zqk f pzq. Using again Proposition 3.2.1 we get
f, KkΓ
D
p, wq Γ 1 ¸
|Z pΓq| γPΓ
|Z p1Γq|
»
H
»
¸»
f pz qj pγ, z qk Kk pw, z qj pγ, z q Impγz qk dν pz q
k
γ 1 F
1
γ Γ γ F
P
f pz qKk pw, z q Impz qk dν pz q
f pz qKk pz, wq Impz qk dν pz q.
So xf, KkΓ p, wqyΓ xf, Kk p, wqy pπk f qpwq where πk still denotes the projection operator defined in Theorem 3.2.9. Note that f P Sk pΓq Hk8 pΓq „ Hk8 pHq as remarked
earlier. Therefore we have
@
D
f, KkΓ p, wq
Γ
pπk f qpwq f pwq
8
8
since πk projects L8
k pHq onto Hk pHq and thus acts trivially on Hk pHq itself. So we are
done.
3.5 A first trace formula
Let’s recall some basic linear algebra: Let pV, x, yq be a finite dimensional inner product
space, and let B tb1 , . . . , bn u be an orthonormal basis of V . Further, let ϕ be a linear
operator on V . Then we can write ϕ as a matix A paij q in terms of the basis B, and
the trace of A is given by the sum of the diagonal entries of A, so TrpAq a11 . . . ann .
One can show that the trace of A does not depend on the choice of basis B, so we can
define the trace of the operator ϕ as the trace of one of its matrix representations.
Keeping notation we have
xϕpbl q, bl y xa1l b1
...
anl bn , bl y ¸n
ajl xbj , bl y all
j 1
since B is an orthonormal basis. Therefore we can write
Trpϕq ¸n
xϕpbl q, bl y .
j 1
(3.5.1)
We use this simple identity to write down a first trace formula for Hecke operators.
39
Theorem 3.5.1. Let k ¥ 3 be an integer and Γ be a finite index subgroup of SL2 pZq.
Further, let T ΓgΓ be a Hecke operator acting on Sk pΓq where g P GL2 pQq. Then
TrpT
ü
detpg qk1
|Z pΓq|
Sk pΓqq »
¸
z
Kk pαz, z qj pα, z qk Impz qk dν pz q.
P
Γ Hα T
Proof. Let tf1 , . . . , fn u be an orthonormal basis of the finite dimensional Hilbert space
Sk pΓq. By equation (3.5.1) we have
TrpT
ü
n
¸
Sk pΓqq xT fj , fj yΓ .
j 1
Let g1 , . . . , gd
P GL2 pQq such that T —dj1 Γgj . Then
pT f qpzq d
¸
d
¸
pf |k giq pzq i 1
for any f
TrpT
detpgi qk1 j pgi , z qk f pgi z q
i 1
P Sk pΓq. Note that detpαq detpgq for all α P T . Hence we have
ü
Sk pΓqq n »
¸
z
d
¸
j 1 Γ H
detpgi qk1 j pgi , z qk fj pgi z q fj pz q Impz qk dν pz q
i 1
detpgqk1
»
d
¸
z Γ Hi 1
n
¸
fj pgi z qfj pz q j pgi , z qk Impz qk dν pz q.
j 1
We can replace
the sum in the brackets according to Proposition 2.3.6, which says that
°n
Γ
Kk pz, wq j 1 fj pz qfj pwq. Hence
TrpT ü Sk pΓqq detpg q »
d
¸
k 1
»
detpg q k 1
d
¸
z Γ Hi 1
KkΓ pgi z, z qj pgi , z qk Impz qk dν pz q
¸
Kk pγgi z, z qj pγ, gi z qk j pgi , z qk Impz qk dν pz q
|Z pΓq|
ΓzH i1 γ PΓ
»
d ¸
¸
detpg qk1
|Z pΓq|
Kk pγgi z, z qj pγgi , z qk Impz qk dν pz q
ΓzH i1 γ PΓ
k 1 »
¸
det|ZppgΓqq|
Kk pαz, z qj pα, z qk Impz qk dν pz q.
ΓzH αPT
The last equality follows from T
—dj1 Γgj .
40
4 Simplification of the trace formula
Throughout this chapter we assume k ¥ 3 to be a fixed integer, Γ to be a modular group
and T ΓgΓ to be a Hecke operator acting on Sk pΓq where g is an element of GL2 pQq.
The goal of this chapter is to simplify the trace formula given in Theorem 3.5.1 following
Section 6.4 of [Miy06].
In Section 4.1 we will try to interchange summation and integration in the formula,
which turns out to be quite involved. Afterwards we will calculate different types of
integrals in Section 4.2, before we summarise our results in Section 4.3.
A further simplification can be found in the following chapter, where we focus on
Tp -operators acting on Sk pΓ0 pN qq.
We also note that this chapter and the corresponding section in Miyake are based on
[Shi63], Section 2 and 3. In particular, the formula presented in Section 4.3 (including the
two finishing lemmata) is Theorem 1 in Shimizu’s paper for n 1 and trivial character.
4.1 Interchanging summation and integration
We start by recalling the trace formula shown at the end of the previous section. We
have
TrpT
where
ü
Sk pΓqq detpg qk1
|Z pΓq|
»
¸
z
P
Γ Hα T
κpz, αqdν pz q
(4.1.1)
κpz, αq Kk pαz, z qj pα, z qk Impz qk .
We want to simplify the integral in equation (4.1.1) by interchanging summation and
integration. Therefore we divide the integral into an integral on a compact set which
behaves nicely and an integral on neighbourhoods of cusps where we have to argue more
carefully.
First we need to note that κpz, αq will in general not be Γ-invariant, and thus the
integral
»
z
Γ H
κpz, αqdν pz q
might not be well-defined. Therefore we have to fix a fundamental domain F of Γ while
we interchange summation and integration. Note that the sum of all integrals will not
depend on the choice of F since
the trace of T acting on Sk pΓq as in (4.1.1) is unique.
”l
Hence we may choose F j 1 gj D where g1 , . . . , gl are fixed coset representatives of
ΓzH, and
D tz P H : | Repz q| ¤ 1{2 and |z | ¥ 1u
41
is the usual fundamental domain of SL2 pZq. We will be able to replace F by some
appropriate quotient at the end of this section.
Further, we remark that Miyake does not fix such a fundamental domain in the corresponding section in his book which causes some formal problems. For example equation
(6.4.7) on page 235 will in general not be well-defined for the mentioned reason. However, this is a purely formal issue, as we will show that all the arguments Miyake is
using still work when we use a fixed fundamental domain. Moreover, we will be able to
recover Miyake’s notation in (our) Theorem 4.1.13 which corresponds to Theorem 6.4.8
in [Miy06].
Notation. Recall that C pΓq denotes the set of Γ-orbits in Q Y t8u, which is usually
called the set of cusps of Γ. In addition, one may define the total set of cusps to be
Q Y t8u itself. As we will be mainly working with the latter in this section, we fix the
following notation to avoid confusion: A cusp x will denote a single element of Q Y t8u
and a cusp c rxs will denote a Γ-orbit of x in Q Y t8u.
Definition 4.1.1. For any cusp x of Γ we define
tα P T : αx xu, Ux σxU8 and Fx F X Ux
where σx P SL2 pZq with σx 8 x and U8 tz P H : Impz q ¡ δ u for some δ ¡ 1.
Note that U8 is a neighbourhood of the cusp 8 and thus Ux is a neighbourhood of the
cusp x. We have Uγx γσx U8 γUx for any γ P Γ. Recall that we denote the stabilizer
of a cusp x in Γ by Γx and note that Γx σx Γ8 σx1 . The neighbourhoods Ux are stable
Tx
under Γx since U8 is stable under Γ8 . Furthermore, the following two Lemmata hold:
Lemma 4.1.2. For any cusps x y of Γ we have Ux X Uy
Proof. Suppose there is z P Ux X Uy , then there are u, v
so u σx1 σy v. Let τ σx1 σy , then
H.
P U8 such that z σxu σy v,
1 Impuq Impτ v q Impv q|j pτ, v q|2 .
Further we have |j pτ, v q|2 ¥ pcτ Impv qq2
of the matrix τ . If cτ 0 then
¥ c2τ Impvq where cτ denotes the lower left entry
1 Impv q|j pτ, v q|2
gives a contradiction since cτ
Hence y σx 8 x.
¤ cτ 2
P Z, so cτ 0, and thus 8 τ 8 σx1σy 8 σx1y.
Lemma 4.1.3. For all but finitely many cusps x P Q Y t8u of Γ the set Fx is empty.
Proof. Let D be the usual fundamental domain of SL2 pZq. By construction D
X Ux is
”
non-empty for some cusp x P Q Y t8u if and only if x 8. We have F lj 1 gj D
with g1 , . . . , gl being the fixed coset representatives of ΓzH. So there is a unique cusp
xj for every j such that pgj Dq X Uxj is non-empty, namely xj gj 8. Therefore Fx is
non-empty if and only if x gj 8 for some j P t1, . . . , lu.
42
³
We will now consider the integral
°
P κpz, αqdν pz q for some cusp x.
α T
Fx
Proposition 4.1.4. We have for any cusp x of Γ
»
¸
P z
Fx α T T
x
κpz, αqdν pz q
¸
»
P z
Fx
α T Tx
κpz, αqdν pz q.
We need two lemmata to prove this proposition. (These correspond to Lemma 6.4.3
and Lemma 6.4.4 in [Miy06].) Moreover, we quickly refer to Corollary 5.13 on page 143
in [Lan93] at this point, which gives a sufficient condition to interchange summation and
integration in a general setting. We will use this corollary several times, but since it is
very well-known, we will use it silently without further notice of the statement or the
reference.
Notation. For α a b
c d
we write cα for the entry c of α and dα for the entry d.
Lemma 4.1.5. The sum
¸
P zp z q{
α Γ8 T T8 Γ8
| cα | k
is convergent.
Proof. Note that the sum is well-defined since firstly cα 0 if and only if α P T8 , and
secondly |cα | |cβ | for all β P Γ8 αΓ8 . Let h be the width of the cusp r8s for Γ, and
let A be a set of double coset representatives for Γ8 zpT zT8 q{Γ8 such that |dα | |hcα |
for all α P A. This is possible since
1 0
α
0 1
p1q
m
1 hm
0 1
p1q
m
cα hcα m
dα
.
Then we see for α P A that |j pα, z q| ¤ |cα z |
|dα| |cα|p|z| |h|q and hence
|cα|k p|h| |z|qk |j pα, zq|k .
°
Therefore it sufficies to show that the sum αPA |j pα, z q|k is convergent for some z P H.
Note that α P A representing the double coset Γ8 αΓ8 also represents the coset Γ8 α,
and that α, α1 P A, α α1 , represent different cosets in Γ8 zT , since Γ8 α Γ8 α1 would
imply Γ8 αΓ8 Γ8 α1 Γ8 . Hence
¸
¸
¸
|j pα, zq|k ¤
|j pα, zq|k ¤
|j pα, zq|k .
P
P z
Let g1 , . . . , gd
PT
such that T
¸
P zp—dj1 Γgj q
P z
α Γ8 T
α A
α Γ8 T
—d
Γgj , then we can write the above sum as
j 1
|j pα, zq|k ¸
d
¸
P z γ Γ8 Γ j 1
α Γ8
d
¸
|j pγgj , zq|k
|j pgj , zq|k ¸
P z
γ Γ8 Γ
j 1
43
|j pγ, gj zq|k .
°
Finally recall that the sum Gk,Γ,8 pz q γ PΓ8 zΓ j pγ, z qk converges absolutely for any
z P H. Thus we are done since the remaining sum is finite.
Lemma 4.1.6. For h ¡ 0 and l ¡
¸
pa
P
1
2
nhq2
there is a constant Ch,l
b2
l
Ch,l |b|2l
¡ 0 such that
|b|2l
1
n Z
for all a, b P R.
l
Proof. Let h ¡ 0, l ¡ 12 and a, b P R. Define f pxq rpa hxq2 b2 s for x P R. Note
that f ¥ 0, and that f 1 pxq ¡ 0 for x a{h, f 1 pxq 0 for x a{h and f 1 pxq 0 for
x ¡ a{h. Thus there is N P Z such that
¸
P f pnq 1 ¤
»
f pxqdx R
n Z,n N
1
h
»
b2 ql dy.
py2
R
Here we used the substitution x ³ py aq{h. To estimate the integral on the right, we
divide it into two parts. We see |y|¤|b| py 2 b2 ql dy ¤ 2|b| |b|2l and
»
|y|¡|b|
py
b ql dy ¤ 2
2
2
Finally we note that f pN q rpa
¸
P
f pnq ¤
»
1
h
n Z
»8
|b|
y 2l dy
2l 2 1 |b|2l
1
.
l ¤ |b|2l . Therefore
hN q2
b2 s
py 2
b2 ql dy
f pN q
R
¤ h2 |b|2l
1
2
|b|2l
hp2l 1q
1
|b|2l
which gives the claimed estimate.
We will now use the previous lemmata to interchange summation and integration
in Proposition 4.1.4. Our argumentation follows the proof of Theorem 6.4.5 on page
233/234 in [Miy06], but we include more details for the generalisation to arbitrary cusps.
Proof of°Proposition 4.1.4. We will proof the statement first for the cusp x 8. Put
S pz q αPT zT8 |κpz, αq|. We have to show that the sum defining S converges for every
z P F8 and that S is integrable on F8 .
—
Let A be a set of coset representatives of Γ8 zT , so T αPA Γ8 α. One can check
that γ8 α P T8 for γ8 P Γ8 , α P T , if and only if α P T8 . Thus —
we have for α P A either
Γ8 α „ T8 or Γ8 α X T8 H. Put A0 AzT8 , then T zT8 αPA0 Γ8 α. Hence
S pz q ¸
¸
P
P
α A0 γ8 Γ8
Impzq
k
|Kk pγ8αz, zq| |j pγ8α, zq|k Impzqk
¸
P
|j pα, zq|k |Z pΓq|
α A0
¸
P
m Z
44
|Kk pαz
hm, z q|.
where h is the width of the cusp
the inner sum:
¸
P
|Kk pαz
hm, z q| m Z
r8s for Γ as usual.
k1 k ¸
|2i| |αz
4π
mPZ
2 pk4π 1q
We use Lemma 4.1.6 to estimate
hm z |k
¸
k
pRepαz zq
P
hmq2
k{2
m Z
2 pk4π 1q Ch,k | Impαz zq|k
k
Moreover, Impαz z q Impαz q
Impαz z q2
1
| Impαz zq|k
.
Impz q ¥ Impz q, and thus
S pz q CΓ,k p1
¸
Impz qq
P
|j pα, zq|k .
α A0
Now we choose a set A1 of double coset representatives for Γ8 zT {Γ8 and adjust the
choice of A such that A „ A1 Γ8 . To see that this is indeed possible, suppose that there
1
is α P A with α R A1 Γ8 . As α P T there is α1 P A1 such that α P Γ8 α1 Γ8 , so α γ8 α1 γ8
1 P Γ8 . Thus we can replace the representative α by γ 1 α α1 γ 1 which
for some γ8 , γ8
8
8
still represents the coset Γ8 α but—is now also an element of A1 Γ8 . As before we can put
A10 A1 zT8 , such that T zT8 α1 PA1 Γ8 α1 Γ8 . Hence
0
¸
P
|j pα, zq|k ¤
α A0
¸
¸
α1 A10 γ8 Γ8
P
¸
α1 A10
P
P
|j pα1γ8, zq|k
|Z pΓq|
¸
P
|cα1 pz
hmq
dα1 |k .
m Z
Also as before we use Lemma 4.1.6 to estimate the inner sum:
¸
P
|cα1 pz
hmq
k { 2
2
¸
dα1
k
2
|cα1 |
Repz q hm
Impz q
cα 1
mPZ
|cα1 |k Ch,k Impzqk 1 Impzqk
dα1 |k
m Z
Finally we can use Lemma 4.1.5 since A10 is a set of double coset representatives for
Γ8 zpT zT8 q{Γ8 , so
S pz q
1 Impz qk p1
CΓ,k
Impz qq2
¸
α1 A10
P
|cα1 |k ¤
2 Impz qk p1
CΓ,k
Impz qq2 .
The estimate on the right is bounded for z P F8 since we assume k ¥ 3, so S is
convergent and bounded on F8 , and thus also integrable on F8 since ν pF8 q ¤ ν pF q is
finite. Therefore we have shown that
»
¸
F8 α T T
8
P z
|κpz, αq|dν pzq 8,
45
so we can interchange the order of summation and integration as claimed.
It remains to generalise to arbitrary cusps. Let x be a cusp of Γ and σ P SL2 pZq such
that σ 8 x. Put Γ1 σ 1 Γσ, g 1 σ 1 gσ and T 1 Γ1 g 1 Γ1 σ 1 T σ. Then F 1 : σ 1 F
is a fundamental domain for the action of Γ1 on H, and thus σ 1 Fx σ 1 pF X σU8 q F81 . Hence
»
»
¸
¸
|κpz, αq|dν pzq 1
F8
α T Tx
P z
Note that Tx σT81 σ 1 and T zTx
|κpσz, αq|dν pzq.
P z
σpT 1zT81 qσ1. Therefore we get using Proposition
3.2.1 with α σ and the fact that j pσα1 σ 1 , σz q j pσ, α1 z qj pα1 , z qj pσ, z q1 :
Fx α T T
x
¸
P z
|κpσz, αq| α T Tx
¸
1
α1 T 1 T8
P z
¸
1
α1 T 1 T8
P z
¸
1
α1 T 1 T8
P z
|κpσz, σα1σ1q|
|Kk pσα1z, σzq| |j pσα1σ1, σzq|k Impσzqk
|Kk pα1z, zq| |j pα1, zq|k Impzqk .
Thus we have shown that
»
¸
P z
Fx α T T
x
|κpz, αq|dν pzq »
¸
1 1 1 1
F8
α PT zT8
|κpz, α1q|dν pzq.
Here the right-hand side is finite, as shown in the first part of the proof, so we can
interchange the order of summation and integration also for arbitrary cusps.
Originally we wanted to study the integral
»
¸
P
Fx α T
κpz, αqdν pz q
for some cusp x of Γ, but so far we have only discussed the pT zTx q-part of the sum. The
following proposition deals with the remaining part, which needs special treatment:
Proposition 4.1.7. We have for any cusp x of Γ
»
¸
P
Fx α Tx
κpz, αqdν pz q
lim
s
¸ »
×0 αPT
x
Fx
κpz, αq Impz qs |j pσx1 , z q|2s dν pz q.
Here s × 0 means that s Ñ 0 monotonically and s ¡ 0, and σx is any element of SL2 pZq
with σx 8 x.
We start with a small lemma that will help us to generalise from the cusp
arbitrary cusps during the proof of this proposition.
Lemma 4.1.8. For any cusp x of Γ, the subgroup Γx is of finite index in Tx .
46
8
to
Proof. Let α, β P Tx such that α β in ΓzT , then there is γ P Γ such that γα β, so
γx βα1 x x since α, β P Tx . Thus γ P Γx and hence α β in Γx zTx . Therefore we
have |Γx zTx | ¤ |ΓzT | 8.
Proof of Proposition 4.1.7.
We will proof the statement first for the cusp x 8. Fix
°
s ¡ 0 and put Ss pz q αPT8 |κpz, αq| Impz qs . We have to show that the sum defining
Ss converges for every z P F8 and that S is integrable on F8 .
Let —
A be a set of coset representatives of Γ8 zT , and put A0 A X T8 such that
T8 αPA0 Γ8 α. Following the proof of Proposition 4.1.4 using this new A0 we get
Ss pz q CΓ,k p1
¸
Impz qq Impz qs
P
|j pα, zq|k .
α A0
This time we do not have to use Lemma 4.1.6 a second time since the sum is already finite
by Lemma 4.1.8. Moreover, cα 0 for every α P A0 „ T8 , so the sum is independent of
1 pImpz qs Impz q1s q. One can easily check that this is integrable
z. Hence Ss pz q CΓ,k
on F8 with respect to ν for all s ¡ 0, and thus
¸ »
s
κpz, αq Impz q dν pz q κpz, αq Impz qs dν pz q.
F8 αPT8
F8
αPT8
»
¸
(4.1.2)
Now let psn qn be a sequence in p0, 8q that converges to 0 monotonically from above.
Then κpz, αq Impz qsn Ñ κpz, αq monotonically as n Ñ 8 for fixed z, so
»
¸
κpz, αqdν pz q lim
»
Ñ8
P
F8 α T8
¸
P
F8 α T8
n
κpz, αq Impz qsn dν pz q
(4.1.3)
by the Monotone Convergence Theorem (see Theorem 5.5 on page 139 in [Lan93]).
Combining equation (4.1.2) and equation (4.1.3) yields the claimed statement for x 8.
It remains to generalise to arbitrary cusps. Let x be a cusp of Γ and σ P SL2 pZq
such that σ 8 x. As in the proof of 4.1.4 we put Γ1 σ 1 Γσ, g 1 σ 1 gσ and
T 1 Γ1 g 1 Γ1 σ 1 T σ. Then Tx σT81 σ 1 , F 1 : σ 1 F is a fundamental domain for Γ1
and σ 1 Fx F81 . Using similar arguments as in the proof of Proposition 4.1.4 we get
»
¸
P
Fx α Tx
κpz, αqdν pz q »
¸
1 1 1
F8
α PT8
κpz, α1 qdν pz q.
Thus we have by equation (4.1.2) and (4.1.3)
»
¸
P
Fx α Tx
κpz, αqdν pz q lim
s
¸ »
×0 1 1
α PT8
1
F8
κpz, α1 q Impz qs dν pz q.
Using once more similar arguments as in the proof of Proposition 4.1.4 one can check
that
¸ »
¸ »
1
s
κpz, α q Impz q dν pz q κpσ 1 z, σ 1 ασ q Impσ 1 z qs dν pz q
1 F81
αPTx Fx
α1 PT8
¸ »
κpz, αq Impz qs |j pσ 1 , z q|2s dν pz q.
Fx
αPTx
Therefore we are done.
47
Combining Proposition 4.1.4 and Proposition 4.1.7 we see
»
¸
P
Fx α T
κpz, αqdν pz q
»
¸
P z
Fx
α T Tx
lim
s
”
κpz, αqdν pz q
¸ »
×0 αPT
x
Fx
κpz, αq Impz qs |j pσx1 , z q|2s dν pz q.
(4.1.4)
Define F 0 : xPQYt8u Fx and F 1 : F zF 0 . Then the above equality deals with F 0 ,
and it remains to consider F 1 , which is done by the following proposition.
Proposition 4.1.9. We have
»
¸
P
F1 α T
κpz, αqdν pz q
¸ »
P
F1
α T
κpz, αqdν pz q.
In Miyake’s book this statement is remarked in the middle of page 232, but not
explicitly stated as a theorem. Hence Miyake does not provide a proper proof, but only
sketches the argument in two sentences. We fill in the details at this point, starting with
a quick and obvious lemma.
Lemma 4.1.10. The set F 1 is compact in H.
”
Proof. Let D be the usual fundamental domain of SL2 pZq and write F lj 1 gj D
where g1 , . . . , gl are coset representatives of ΓzH. As remarked earlier in the proof of
Lemma 4.1.3 the set Fx is non-empty if and only if x gj 8 for some j P t1, . . . , lu.
Further, one can easily check that pgj Dq X Fx is non-empty if and only if x gj 8. Put
xj gj 8. Then
F1
¤l
j 1
gj D
¤l
z
j 1
F xj
¤l
gj pDzF8 q .
j 1
The right-hand side is compact since DzF8 is clearly compact by construction.
Proof of Proposition 4.1.9. Let g1 , . . . , gd
»
F1
¸
P
|κpz, αq|dν pzq α T
»
F1
d ¸
¸
such that T
—dj1 gj Γ. Then
|Kk pgj γz, zq||j pgj γ, zq|k Impzqk dν pzq
j 1γ Γ
d »
¸
j 1
P
PT
F1
|j pgj1, zq|k Impzqk
¸
P
|Kk pγz, gj1zq||j pγ, zq|k dν pzq.
γ Γ
Here we used Proposition 3.2.1 again. Now recall that the sum defining KkΓ pz, wq converges uniformly on any compact subset of H H as shown in Lemma 3.4.4, and note
that K : F 1 gj1 pF 1 q is compact in H H for every j since F 1 is compact by Lemma
4.1.10. Hence we find a constant Cj ¡ 0 for every j such that
sup
P
¸
P
z F1 γ Γ
|Kk pγz, gj1zq||j pγ, zq|k ¤
48
sup
pz,wqPK
|Z pΓq| KkΓpz, wq ¤ Cj .
Moreover, the continuous function |j pgj1 , z q|k Impz qk is clearly bounded on the compact
set F 1 . Finally we note that ν pF 1 q is finite as ν pF q is. Thus
»
¸
|κpz, αq|dν pzq 8
P
F1 α T
and hence we can interchange summation and integration.
Next we deduce a new trace-formula which combines
all the previous
( results. To state
it we first need to define Z pT q α P T : α a0 a0 for some a P R ,
T2
¤
P Yt8u
Tx zZ pT q,
T1
T zT 2.
x Q
a
Recall that detpαq detpg q for all α P T . Hence we either have Z pT q t detpg q idu
or Z pT q H. In particular, Z pT q is finite.
The following proposition corresponds to equation (6.4.7) on page 235 in [Miy06],
which is stated as a direct corollary of the previous results, without proof. We add a
formal proof here.
Proposition 4.1.11. We have
TrpT
ü
Sk pΓqq detpg qk1
|Z pΓq|
¸ »
α
P
T1
F
κpz, αqdν pz q
lim
s
¸ »
×0
α
P
T2
F
κpz, α, sqdν pz q
where
κpz, α, sq #
κpz, αq Impz qs |j pσx1 , z q|2s , z P Ux and αx x for some cusp x,
κpz, αq,
otherwise.
Before we start with the proof, we want to remark that the definition
of κpz, α, sq is
a
b
independent of the choice σx for cusps x of Γ. To see this let σ c d P SL2 pZq such
that σ 8 x for some cusp x of Γ. If x 8, then c 0 and a 1, so |j pσ 1 , z q| 1
is independent of σ. If x p{q P Q with p, q coprime, then a p and c q, so
|j pσ1, zq| | qz p|, which is again independent of σ.
Proof of Proposition 4.1.11. We start with equation (4.1.1) and split the integral in an
integral over F 0 and an integral over F 1 :
TrpT
ü
Sk pΓqq detpg qk1
|Z pΓq|
»
¸
P
F0 α T
κpz, αqdν pz q
»
¸
P
F1 α T
κpz, αqdν pz q .
Note that the union defining F 0 is actually a finite union since all but finitely many
neighbourhoods Fx are empty, as shown in Lemma 4.1.3. Hence we get using the equality
49
in (4.1.4)
»
¸
P
F0 α T
l
¸
κpz, αqdν pz q »
¸
P z
j 1
l »
¸
P
j 1 Fxj α T
κpz, αqdν pz q
κpz, αqdν pz q
Fxj
α T Txj
¸
lim
s
¸ »
×0 αPT
κpz, αq Impz qs |j pσxj1 , z q|2s dν pz q .
Fxj
xj
(4.1.5)
Note that for α P T zTx , z P Fx we have αx
cusp y by Lemma 4.1.2, so for any s
»
¸
P z
α T Txj
x and cannot have z P Fy for any other
κpz, αqdν pz q Fxj
»
¸
P z
Fxj
α T Txj
κpz, α, sqdν pz q.
For the second sum in (4.1.5) we have by definition
¸ »
P
α Txj
¸ »
s
1
2s
κpz, αq Impz q |j pσxj , z q| dν pz q κpz, α, sqdν pz q.
Fxj
αPTxj Fxj
Therefore
»
¸
P
F0 α T
κpz, αqdν pz q
j 1
³
l
¸
°
lim
s
×0 αPT
»
¸
lim
s
¸»
×0 αPT
F0
Fxj
κpz, α, sqdν pz q
κpz, α, sqdν pz q.
(4.1.6)
We will now consider F 1 αPT κpz, αqdν pz q. By Proposition 4.1.4 we can interchange
summation and integration, and as z P F 1 implies z R Ux for any cusp x, we can write
»
¸
P
F1 α T
κpz, αqdν pz q ¸»
P
α T
F1
κpz, α, sqdν pz q
for any s. Combining this with (4.1.6) yields
»
¸
P
F0 α T
»
κpz, αqdν pz q
¸
P
F1 α T
κpz, αqdν pz q
lim
s
¸»
×0 αPT
F
κpz, α, sqdν pz q.
To conclude the claimed formula it remains to separate some safe terms. Let α P T 1 .
We have to distinguish between two cases: Either α P Z pT q, or α R Tx for any cusp x.
In the second case we have κpz, α, sq κpz, αq for any
z P H, so we can easily separate
a
0
these terms. Suppose that α P Z pT q, then α 0 a for some a P R . Thus
»
F
|κpz, αq|dν pzq »
F
1 |a|k ν pF q 8
|Kk pz, zq||a|k Impzqk dν pzq k 4π
50
since Kk pz, z q pk 1q{p4π q Impz qk . So we can interchange integral and limit by the
Dominated Convergence Theorem giving us
»
lim
s
×0
F
κpz, α, sqdν pz q »
lim κpz, α, sqdν pz q F s
»
×0
F
κpz, αqdν pz q.
It remains to recall that Z pT q is finite as remarked earlier. Hence we have
lim
s
¸»
×0 αPT
F
κpz, α, sqdν pz q
¸ »
P
α T1
F
κpz, αqdν pz q
lim
s
×0
¸ »
P
α T2
F
κpz, α, sqdν pz q
as claimed.
We will finish this section with some group theoretic considerations, which allow us
to rearrange our trace formula in such a way that we are finally able to replace the
fundamental domain F by some appropriate quotient.
Let G be a group and H a subgroup of G. Elements g1 , g2 P G are called H-conjugate,
denoted by g1 H g2 , if there exists h P H such that g2 h1 g1 h. This gives an
equivalence relation on G. We call the corresponding equivalence class of g P G, Hconjugacy class and denote it by rg sH .
For a subset M of G which is stable under conjugation by elements in H, which means
h 1 M h M for all h P H, we define M {{H as the set of all H-conjugacy classes in M ,
so
M {{H trg sH : g P M u.
Note that M {{H gives a partition of M , since rg sH
„ M for all g P M .
Lemma 4.1.12. The subsets T 1 and T 2 of T are stable under conjugation by elements
in Γ.
Proof. First note that rαsΓ tαu for all α
element. Secondly, we see for any γ P Γ
P Z pT q, as they commute with any other
tα P T : γαγ 1x xu γ 1tα1 P γT γ 1 : α1x xuγ γ 1Txγ.
Here we used that γ 1 T γ T which is obvious. Therefore we have γ 1 T 2 γ T 2 and
thus also γ 1 T 1 γ T 1 .
Recall that we defined Z pαq tβ P GL2 pQq : αβ βαu in Section 2.2, and define
Γpαq tγ P Γ : γα αγ u.
Then Γpαq Z pαqX Γ. We are now able to state Theorem 6.4.8 on page 235 in [Miy06],
Tλ1 x
which will be the starting point for further considerations in the next section. For the
sake of convenience we recall the complete notation used in the theorem.
51
Theorem 4.1.13. Let k ¥ 3 be an integer, and let T ΓgΓ with Γ being a finite index
subgroup of SL2 pZq and g being an element of GL2 pQq. Then
TrpT
ü
Sk pΓqq
detpg q k 1
|Z pΓq| »
¸
α
P {{
×0
κpz, αqdν pz q
»
¸
lim
s
p qz
Γ Γ α H
T1
P {{
p qz
α T2 Γ Γ α H
κpz, α, sqdν pz q
where we use the following notation:
”
• Put Z pΓq Γ X t1u, T 2 xPQYt8u Tx zZ pT q where Tx tα P T : αx xu and
Z pT q tα P T : α is scalaru, T 1 T zT 2 and Γpαq tγ P Γ : γα αγ u.
P t1, 2u, T j {{Γ denotes the set of Γ-conjugacy classes in T j .
The limit s × 0 means s Ñ 0 monotonically from above.
• For j
•
• We have
κpz, α, sq #
κpz, αq Impz qs |j pσx1 , z q|2s , z P Ux and αx x for some cusp x,
κpz, αq,
otherwise
where κpz, αq Kk pαz, z qj pα, z qk Impz qk , σx is any element of SL2 pZq with
σx 8 x and Ux σx tz P H : Impz q ¡ δ u for some δ ¡ 1.
Proof. We want to modify the formula given by Proposition 4.1.11. Note that
T1
§
r α sΓ §
¤
P {{ P
§
tγ 1αγ u §
tγ 1αγ u.
P {{ P p qz
The last inner union is indeed disjoint since γ11 αγ1 γ21 αγ2 if and only if γ1 γ21 P Γpαq.
P {{
α T1 Γ
α T1 Γ γ Γ
α T1 Γ γ Γ α Γ
Therefore we have
¸ »
κpz, αqdν pz q ¸
¸
»
κpz, γ 1 αγ qdν pz q.
P {{ P p qz
³
³
Let α P T 1 and γ P Γ. We have F κpz, γ 1 αγ qdν pz q γF κpγ 1 z, γ 1 αγ qdν pz q. Using
Proposition 3.2.1 one can check that κpγ 1 z, γ 1 αγ q κpz, αq. Thus we see
P
α T1
»
F
F
α T1
κpz, γ 1 αγ qdν pz q »
γF
Γγ Γ α Γ F
κpγ 1 z, γ 1 αγ qdν pz q »
γF
κpz, αqdν pz q.
Let γ P Γpαq. Then κpγz, αq κpz, γ 1 αγ q κpz, αq as γ and α commute. So the
integrand κpz, αq is Γpαq-invariant. Therefore we have
¸
P p qz
»
γ Γ α Γ γF
κpz, αqdν pz q 52
»
p qz
Γ α H
κpz, αqdν pz q,
and hence
¸ »
α
P
T1
F
»
¸
κpz, αqdν pz q α
P {{
T1
p qz
Γ Γ α H
κpz, αqdν pz q.
(4.1.7)
It remains to consider the second term of the claimed formula. As before we see
¸ »
P
F
α T2
κpz, α, sqdν pz q for any s ¡ 0. Let α P T 2 , γ
cusp x, one can check that
¸
»
¸
P {{ P p qz
α T 2 Γ γ Γ α Γ γF
κpγ 1 z, γ 1 αγ, sqdν pz q
P Γ and z P γF . If pγ 1zq P Ux and pγ 1αγ qx x for some
κpγ 1 z, γ 1 αγ, sq κpz, αq Impz qs |j ppγσx q1 , z q|2s .
On the other hand these conditions imply z
definition of κpz, α, sq we also have
P
γUx
Uγx and αpγxq
γx, so by
1 , z q|2s .
κpz, α, sq κpz, αq Impz qs |j pσγx
Since pγσx q8
Therefore
γx we can choose σγx γσx and hence κpγ 1z, γ 1αγ, sq κpz, α, sq.
¸ »
α
P
F
T2
¸
κpz, α, sqdν pz q α
P {{ P p qz
T2
¸
α
¸
P {{
T2
»
Γ γ Γ α Γ γF
»
p qz
Γ Γ α H
κpz, α, sqdν pz q
κpz, α, sqdν pz q.
(4.1.8)
Using (4.1.7) and (4.1.8) with Proposition 4.1.11 gives the claimed expression.
4.2 Calculation of integrals
We aim to further simplify the trace formula obtained by Theorem 4.1.13. More precisely,
we want to compute integrals of the form
»
p qz
Γ α H
κpz, αqdν pz q and
»
p qz
Γ α H
κpz, α, sqdν pz q.
It turns out that it is convenient to use the classification of elements in GL2 pRq introduced in Section 2.2 for this purpose. Recall that Z pT q is the set of scalar elements in
T , and that the (total) set of cusps of Γ is given by Q Y t8u. We define
Te
T h1
T h2
tα P T : α ellipticu, T p tα P T : α parabolicu,
tα P T : α hyperbolic with fixed points in RzQu,
tα P T : α hyperbolic with fixed points in Q Y t8uu.
53
By Corollary 2.2.1 we know that T Z pT q Y T e Y T p Y T h1 Y T h2 , and by definition
of T 1 and T 2 we have T 1 Z pT q Y T e Y T h1 and T 2 T p Y T h2 . Clearly all these
unions are disjoint. Moreover, we note that all these sets are stable under conjugation
by elements in Γ. This is obvious for Z pT q, and also clear for T e , T p and T h1 Y T h2 since
trace and determinant are stable under conjugation. Further, it can be checked for T h1
and T h2 . Therefore we may split the two sums given in the trace formula of Theorem
4.1.13 as follows:
»
¸
P {{
α
T1
p qz
Γ Γ α H
¸
P {{
»
p qz
α T2 Γ Γ α H
κpz, αqdν pz q
¸
P p q{{Γ
α Z Γ
κpz, α, sqdν pz q
¸
P {{
¸
...
α
...
α Tp Γ
P {{Γ
Te
¸
P
{{
¸
...
α
P
T h1
{{Γ
...
...
α T h2 Γ
In the following subsections we will study all of these five terms separately, closely
following the argumentation in [Miy06], pages 236 to 240. In the process we will fill in
many (often technical) details omitted in Miyake’s book. At then of each section we
summarise our results, combining them in the end in Section 4.3 within one big trace
formula.
4.2.1 The scalar terms
Let α λ 0
0 λ
P Z pT q. Then Γpαq Γ and one can check that κpz, αq k4π1 λk . Hence
»
k 1 k
κpz, αqdν pz q λ ν pΓzHq.
p qz
4π
Γ α H
By Lemma 2.1.2 we have that ν pΓzHq dΓ π {3 whereadΓ rSL2 pZq{t1u : Γ{Z pΓqs,
and since α P T we see detpαq detpg q, so λ signpλq detpg q. Therefore
»
p qz
Γ α H
κpz, αqdν pz q k1
signpλqk detpg qk{2 dΓ .
12
a
Note that Z pT q T Xt λ0 λ0 u with λ detpg q, so if detpg q does not have a rational
square root, then Z pT q is empty. In addition, if both λ0 λ0 P T , then these elements
are clearly not Γ-conjugates of each other. Finally, one can check that the quotient
dΓ {|Z pΓq| equals 1{2 rSL2 pZq : Γs. We summarise our results for the scalar terms as:
Lemma 4.2.1. We have
»
detpg qk1 ¸
|Z pΓq| αPZ pT q{{Γ ΓpαqzH κpz, αqdν pzq
1 detpgqk{21 rSL pZq : Γs ¸ signpλ qk .
k 24
2
α
αP Z p T q
a
where Z pT q T X t λ0 λ0 u, λ detpg q and λα denotes the eigenvalue of α. In
particular, the sum is empty if detpg q does not have a rational square root.
54
4.2.2 The elliptic terms
Let α P T e . Since α is elliptic there is z0 P H such that z0 and z0 are the unique fixed
points of α. Furthermore,
there is λ P CzR such that λ and λ are the eigenvalues of α.
1 z0
Put σ 1 z0 , then σz0 0 and thus σασ 1 0 0. Hence σασ 1 is of the form 0 .
Moreover, σz0 8, so σασ 1 8 8, and therefore σασ 1is of the form 0 0 . As α
and σασ 1 have the same eigenvalues we get σασ 1 λ0 λ0 . Note that we might have
to replace λ by λ at this point. Moreover, this fixes λ since σ is unique.
Using the equality we can express α in terms of its fixed points and its eigenvalues:
λ 0
σ
α σ 1
0 λ
Now we fix z P H and put w
Moreover, we see
1
z0 z0
λz0 λz0
λλ
|z0|2pλ λq
λz0 λz0
.
σz, w1 σz. One can check that ww1 1, so w1 |ww| .
w w1
λ{λ w w1
2
z z αz z0
σz
pσασσz
1 qpσz q σz αz z z z .
0
Here we used that βa βb detpβ qpa bqj pβ, aq1 j pβ, bq1 for any β P GL2 pCq and
for any a, b P C such that j pβ, aq 0 and j pβ, bq 0.Further, one can check that
pαz z0qj pα, zq λpz z0q. Therefore
κpz, αq Since w1
|ww|
2
k1
p
2i Impz qqk rpαz z qj pα, z qsk
4π
k 1 k
λ
4π
w w1
λ{λ w w1
k
.
as noted earlier, and since detpg q detpαq λλ, we can write
κpz, αq k1 k
λ detpg qk
4π
1 | w |2
1 λ{λ |w|2
k
.
Next we note that Γpαq Γz0 by Corollary 2.2.4, and that Γz0 is a finite group by
Lemma 2.2.6. Since any non-scalar element in Γz0 has the unique fixed point z0 in H,
all but one Γz0 -orbit in H consists of exactly |Γz0 {Z pΓq| elements. Therefore we get
»
H
κpz, αqdν pz q |Γz0 {Z pΓq| »
p qz
Γ α H
κpz, αqdν pz q.
Combining these results we see
»
p qz
Γ α H
κpz, αqdν pz q 1
k1 k
λ detpg qk
4π
|Γz0 {Z pΓq|
» H
1 |σz |2
1 λ{λ |σz |2
k
dν pz q.
Obviously we want to substitute w σz. One can check that σH D where D denotes
the open unit disk in the complex plane. Moreover, one can check that the substitution
55
transforms dν pz q Impz q2 dz into dνD pwq 4p1
[Miy06] for some details on this matter.) Hence
» H
1 |σz |2
1 λ{λ |σz |2
k
|w|2q2dw.
(See Section 1.4 in
k2
p
1 | w |2 q
dν pz q 4
k dw
D 1 λ{λ |w |2
»1
» 2π
k 2
p
1 r2 q
4
dϕ
k rdr
0
0
1 λ{λ r2
»1
p1 sqk2 ds
8π
k
0
1 λ{λ s 2
k1 1
1s
1
1
4π k 1
λ{λ 1 1 λ{λ s
0
»
k 4π
1
λ
λλ
.
Therefore we have using again that detpg q λλ
»
p qz
Γ α H
κpz, αqdν pz q λk λ detpg qk
λ λ |Γz0 {Z pΓq|
λk1 detpg q1k
.
λ λ |Γz0 {Z pΓq|
Note that |Γz0 {Z pΓq| |Z pΓq| |Γz0 | |Γpαq|. We summarise our observations:
Lemma 4.2.2. For α P T e with unique fixed point z
λα
σα ασα1 0
where σα
0
λα
P H we can write
z and λα , λα are the eigenvalues of α. Using this notation we have
z
»
¸
1
detpg qk1 ¸
λαk1
1
1
|Z pΓq|
α
P {{
Te
p qz
Γ Γ α H
κpz, αqdν pz q
α
P
Te
{{Γ |Γpαq|
λα λα
.
4.2.3 The hyperbolic terms of type one
Let α P T h1 . Then there are distinct x1 , x2 P RzQ such that x1 and x2 are the unique
fixed points of α. Furthermore, α has two distinct real eigenvalues, say λ1 and λ2 . We
x2
can assume that x2 ¡ x1 without loss of generality. Put σ px2 x1 q1{2 11 x1 , then
σx1 8, σx2 0 and σ P SL2 pRq. Since σασ 1 0 0 and σασ 1 8 8 we can argue
as for elliptic α that σασ 1 λ01 λ02 . (Again, we might have to swap λ1 and λ2 at this
point.) As before we use this equation to express α by
λ1
α σ 1
0
0
σ
λ2
x x
2
1
1
56
λ2 x2 λ1 x1 x1 x2 pλ1 λ2 q
.
λ2 λ1
λ1 x2 λ2 x1
Now we fix z P H and put w σz, w1 σz. Since σ has entries in R we have w1
Exactly as in the previous subsection we find
ww
λ1 {λ2 w w
w.
z z αz x1
αz
z z x , pαz x1qj pα, zq λ2pz x1q,
1
and thus
κpz, αq k 1 k
λ2
4π
ww
λ1 {λ2 w w
k
.
Note that λ2 0 since λ1 λ2 detpαq ¡ 0. We put λ λ1 {λ2 .
Next we consider Γpαq. By Corollary 2.2.4 we have Γpαq Γx1 X Γx1 . First suppose
that this intersection is trivial, so Γpαq Z pΓq. Then ΓpαqzH H. Recall that
elements in SL2 pRq act as automorphisms on the upper half-plane, and that dν pz q is
SL2 pRq-invariant. Thus we get using the substitution w σz where σ P SL2 pRq by
construction
»
p qz
Γ α H
κpz, αqdν pz q k 1 k
λ
4π 2
» H
ww
λw w
k
dν pwq
k
» » k 1 k 8 π reiϕ reiϕ
r drdϕ
λ2
iϕ
iϕ
4π
λre re
pr sinpϕqq2 .
0
0
Taking absolut values yields
»
p qz
Γ α H
|κpz, αq|dν pzq »π
»8
k1
1
1
k
|
λ2 |
dr
2
π
0 psinpϕqq
0 r
iϕ
e
λeiϕ
eiϕ k dϕ,
eiϕ which is ³a contradiction since we know that the left-hand side is convergent, but the
8
integral 0 r1 dr does not converge, and the last integral is convergent and non-zero.
Thus we cannot have Γpαq Z pΓq, and may therefore use Lemma 2.2.6: There is u ¡ 0
such that
*
" m
u
0
1
σ pΓx1 X Γx2 qσ
t1u 0 um : m P Z .
Note that we may replace u by u1 without changing the set, and that u 1 since
Γx1 X Γx1 Z pΓq by assumption. Thus we can assume u ¡ 1. Furthermore, we note
m
that u0 u0m z u2m z for any z P C and any m P Z. Hence a fundamental domain
of the quotient pσ pΓx1 X Γx2 qσ 1 qzH is given by tw P H : 1 ¤ |w| u2 u. (The sign does
not matter since 1 acts trivially.) So we get using the substitution w σz
»
p qz
Γ α H
κpz, αqdν pz q »
k 1 k
λ
4π 2 pσΓpαqσ1 qzH
k 1 k
λ
4π 2
k 1 k
λ
4π 2
» u2 » π 1
0
1
1
dr
r
» u2
57
ww
λw w
k
reiϕ reiϕ
λreiϕ reiϕ
»π
0
dν pwq
k
eiϕ eiϕ
λeiϕ eiϕ
r drdϕ
pr sinpϕqq2
k
dϕ
psinpϕqq2 .
(4.2.1)
³ u2
For the first integral we compute 1 r1 dr 2 lnpuq, and for the second one we note
that peiϕ eiϕ q2 4psinpϕqq2 . Thus we have to consider
»π
peiϕ eiϕqk2 dϕ.
iϕ
iϕ qk
0 pλe e
Let f pϕq be the integrand, then f is π-periodic, so f pϕ
π q f pϕq for all ϕ P C, and f
is a meromorphic function on C with singularities at πn i lnpλq{2 for n P Z. (Note that
λ detpαq{λ22 ¡ 0 since α P GL2 pQq.) Suppose that λ P p0, 1q, then f is holomorphic
on the extended upper halp-plane tϕ P C : Impϕq ¡ Ru for sufficiently small R ¡ 0.
Put
AR : tϕ P C : 0 ¤ Repϕq π, Impϕq ¡ Ru ,
and let Φ denote the map ϕ ÞÑ 2iϕ, then
ΦpAR q tz
P C:
Repz q 2R, 0 ¤ Impϕq 2πiu .
Recall that the complex exponential function is bijective on ΦpAR q. We denote its inverse
defined on the punctured disc BR : exppΦpAR qq tq P C : |q | e2R u by log, and
recall that this inverse is holomorphic except for a 2πi-skip while crossing the positive
real axis. We define
logpq q
˜
.
f : BR Ñ C, q ÞÑ f
2i
Then f˜ is by construction well-defined. Moreover, f˜ is holomorphic on BR since f being
π-periodic compensates for the 2πi-skip. We claim that f˜ has a removable singularity
at 0. To see this note that
p1 e2iϕqk2 .
p1 λe2iϕqk
Hence f˜pq q q p1 q qk2 p1 λq qk and thus limqÑ0 f˜pq q 0. Therefore we can write
8̧
logpq q
˜
f
f pqq a qn, q P B
f pϕq e2iϕ
2i
n
R
n 1
for some an P C, n P N. (The constant term vanishes since limqÑ0 f˜pq q 0.) The
series converges absolutely and locally uniformly. In particular, it converges
absolutely
°8
2iϕ
uniformly on the smaller annulus BR{2 . Substituting q e gives f pϕq n1 an e2inϕ
for ϕ P AR , which correspondingly converges absolutely uniformly on AR{2 . Finally we
can compute using uniform convergence of the sum
»π
0
f pϕqdϕ It remains to consider the case λ
similar argument. Put
8̧
»π
an
n 1
¡ 1.
0
e2inϕ dϕ 0.
(Clearly λ
1 since λ1 λ2.)
AR : tϕ P C : 0 ¤ Repϕq π, Impϕq Ru ,
58
We will use a
then f is holomorphic on AR for sufficiently small R ¡ 0, and f˜ defined as before
is holomorphic on BR : exppΦpAR qq tq P C : |q | ¡ e2R u. Further, we define
fˆpq q f˜p1{q q for q P BR1 : tq P C : |q | e2R u. Then fˆ is holomorphic on BR1 , and we
claim that fˆ has a removable singularity at 0. To see this note that
1{q p1 1{q qk2
fˆpq q p1 λ{qqk
k 2
q p1pλqqqqk .
Hence limqÑ0 fˆpq q 0, and we can write
f
logp1{q q
2i
f˜p1{qq fˆpqq 8̧
an q n ,
q
n 1
for some an P C, n P N. Substituting q e2iϕ yields f pϕq converging absolutely uniformly on AR{2 , so
»π
0
f pϕqdϕ 8̧
p qz
Γ α H
»π
0
³π
0
°8
2inϕ for ϕ P AR ,
an e
n 1
e2inϕ dϕ 0.
an
n 1
Therefore we have shown in general that
»
P BR1 ,
f pϕqdϕ 0, and thus by equation (4.2.1)
»
2pk 1q lnpuq k π
λ2
f pϕqdϕ 0.
κpz, αqdν pz q π
0
Hence the terms in the trace formula of Theorem 4.1.13 corresponding to hyperbolic α
with fixed points in RzQ do not contribute anything:
Lemma 4.2.3. We have
detpg qk1
|Z pΓq|
¸
P
{{
»
p qz
α T h1 Γ Γ α H
κpz, αqdν pz q
0.
4.2.4 The hyperbolic terms of type two
Let α P T h2 . Then there are distinct x1 , x2 P Q Y t8u such that x1 and x2 are the
unique fixed points of α. Furthermore, α has two distinct real eigenvalues, say λ1 and
λ2 . If x1 , x2 8 we may choose choose σ P SL
2 pRq as in Subsection 4.2.3 (assuming
that x2 ¡ x1 ). If x1 8 we choose σ 10 1x2 , and if x2 8 we choose σ 01 x11 .
Using exactly the same arguments as in the previous subsection we get
κpz, αq where λ λ1 {λ2
k 1 k
λ2
4π
σz σz
λσz σz
k
¡ 0. By Corollary 2.2.4 we have Γpαq Z pΓq, so
»
p qz
Γ α H
κpz, α, sqdν pz q 59
»
H
κpz, α, sqdν pz q.
Let σ1 , σ2 P SL2 pZq such that σ1 8 x1 , σ2 8 x2 , and let U1 : σ1 U8 , U2 : σ2 U8
where U8 tz P H : Impz q ¡ δ u for some δ ¡ 1 as in Section 4.1. Since the only fixed
points of α are x1 and x2 we have by definition
κpz, α, sq #
κpz, αq Impz qs |j pσj1 , z q|2s
κpz, αq
, z P Uj , j P t1, 2u,
, otherwise.
³
We split the integral H κpz, α, sqdν pz q into three parts, an integral over U1 , an integral
over U2 and an integral over H 1 : HzpU1 Y U2 q. For the first two integrals we get
substituting w σz as before
»
Uj
κpz, α, sqdν pz q k 1 k
λ2
4π
k 1 k
λ2
4π
»
σUj
»
σUj
ww
λw w
ww
λw w
k
k
Impσ 1 wqs |j pσj1 , σ 1 wq|2s dν pwq
Impwqs |j ppσσj q1 , wq|2s dν pwq.
a
0 1 a ,
{
Since σσ1 8 8 and σσ1 P SL2 pRq we have that σσ1 is of the form
Hence |j ppσσ1 q1 , wq| |a| is constant, and
σU1
pσσ1qU8 κpz, α, sqdν pz q k 1 k 2s
λ2 a
4π
z
a
(
P H:
Impz q ¡ a2 δ .
k
Therefore we get
»
U1
k 1 k 2s
λ2 a
4π
k 1 k 2s
λ2 a
4π
»π»8
0
0
{ p q
a2 δ sin ϕ
»π
»
σU1
ww
λw w
reiϕ reiϕ
λreiϕ reiϕ
eiϕ eiϕ
λeiϕ eiϕ
k
k
Impwqs dν pwq
drdϕ
pr sinpϕqqs prrsin
pϕqq2
psinpϕqqs2
»8
1
{ p q
a2 δ sin ϕ
rs 1
Similarly to the previous subsection we define
f pϕq eiϕ eiϕ
λeiϕ eiϕ
and recall that we have shown
³π
0
k
1
{ p q rs
a2 δ sin ϕ
»
U1
k2
p
eiϕ eiϕ q
p4q iϕ iϕ k ,
pλe e q
f pϕqdϕ 0. Further, we note that
»8
Hence we get
1
psinpϕqq2
κpz, α, sqdν pz q 1
dr
1
s
sinpϕq
a2 δ
k 1 k 1
λ2
4π
sδ s
60
»π
0
s
.
f pϕqdϕ 0.
dr dϕ.
P R .
Consider now the integral
over U2 . Since σσ2 8 0 and σσ2 P SL2 pRq we have that σσ2
0 b
is of the form 1{b , b P R . Thus |j ppσσ2 q1 , wq| |w{b|, and one can check that
"
sinpϕq
re : r P p0, 8q, ϕ P p0, π q,
r
σσ2U8 σU2
iϕ
¡
δ
b2
*
.
Therefore we get
»
κpz, α, sqdν pz q U2
k 1 k
λ2
4π
k 1 k 2s
λ2 b
4π
1
k 4π
1
k 4π
1
k 4π
0.
λk b2s
0
»π
0
»π
2
k
λ
2
1
sδ s
σU2
» π » b2 sinpϕq{δ 2
λk b2s
»
» 0π
0
0
ww
λw w
k
Impwqs |j ppσσ2 q1 , wq|2s dν pwq
reiϕ reiϕ
λreiϕ reiϕ
» b2 sinpϕq{δ
f pϕqpsinpϕqqs
0 1
f pϕqpsinpϕqqs
s
k
drdϕ
pr sinpϕqqsr2s prrsin
pϕqq2
rs1 dr dϕ
b2 sinpϕq
δ
s dϕ
f pϕqdϕ
It remains to compute the integral over H 1 . Note that
σH 1 HzpσU
1
Y σU2q "
b2 sinpϕq
re : r P p0, 8q, ϕ P p0, π q,
δ
iϕ
¤r¤
*
a2 δ
.
sinpϕq
Therefore we see
»
H1
κpz, αqdν pz q k 1 k
λ2
4π
k 1 k
λ2
4π
»
k 1 k
λ2
4π
σH 1
» π » a2 δ{ sinpϕq p q{
b2 sin ϕ δ
0
»π
0
f pϕq
ww
λw w
k
reiϕ reiϕ
λreiϕ reiϕ
dν pwq
k
» a2 δ{ sinpϕq
r drdϕ
pr sinpϕqq2
1
dr dϕ.
b2 sinpϕq{δ r
Note that
» a2 δ{ sinpϕq
1
dr
b2 sinpϕq{δ r
ln
a2 δ 2
b2 psinpϕqq2
Hence we have that
»
H1
κpz, αqdν pz q k1
2π
k1
2π
aδ
k
λ
ln
2
λk
»π
2
0
2
» π
b
0
ln
aδ
b
f pϕqdϕ f pϕq lnpsinpϕqqdϕ
61
lnpsinpϕqq
»π
0
.
f pϕq lnpsinpϕqqdϕ
since the first term vanishes as before. One can check that
d
dϕ
eiϕ eiϕ
λeiϕ eiϕ
k1 k 2
p
eiϕ eiϕ q
2ipk 1qpλ 1q iϕ iϕ k ipk 1qpλ2 1qf pϕq .
pλe e q
Hence we can use integration by parts to see
»
H
k »π
i λ
ipk 1qpλ 1qf pϕq
2
lnpsinpϕqqdϕ
κpz, αqdν pz q π λ1
2
1
k
λ
2
π λ1
i
eiϕ eiϕ
λeiϕ eiϕ
π
0
k1
lnpsinpϕqq
»π
0
0
eiϕ eiϕ
λeiϕ eiϕ
k1
cospϕq
dϕ .
sinpϕq
The first term vanishes, roughly since limx×0 xk1 lnpxq 0. So we are left with
»
H
k1 iϕ
k » π iϕ
1 λ
e eiϕ
e
eiϕ
2
κpz, αqdν pz q dϕ.
π λ1
λeiϕ eiϕ
eiϕ eiϕ
1
0
Let g pϕq be the integrand, then
k 2
p
1 e2iϕ q
g pϕq p1 λe2iϕqk1
1
e2iϕ ,
and g is π-periodic, and meromorphic on C with singularities at πn i lnpλq{2 for n P Z.
Hence we can argue for g as we did for the function f in the previous subsection. First
we suppose that λ P p0, 1q, then we can define a function g̃ by g̃ pq q g plogpq q{p2iqq for
q P BR : tq P C : |q | Ru which will be well-defined and holomorphic on BR for
sufficiently small R ¡ 0. Since g̃ pq q p1 q qk2 p1 λq°qk 1 p1 q q we see that g̃
n
has a removable singularity at 0, so we can°write g̃ pq q 8
n0 an q for some an P C,
2inϕ
, and since the series converges
n P N0 . Substituting q e2iϕ gives g pϕq 8
n0 an e
absolutely and locally uniformly we have
»π
0
g pϕqdϕ 8̧
»π
an
0
n 0
e2inϕ dϕ πa0 .
The constant term a0 is given by limqÑ0 g̃ pq q 1, so
have for λ P p0, 1q as detpg q detpαq λ1 λ2 that
»
p qz
Γ α H
κpz, α, sqdν pz q »
H1
κpz, αqdν pz q ³π
k
λ
2
λ1
0
g pϕqdϕ
π, and thus we
λ1k1
detpg q1k .
λ1 λ2
It remains to consider the case λ ¡ 1. (Again we have λ 1 since the given eigenvalues
are distinct.) As in the previous subsection we define ĝ g̃ p1{q q for q P BR and
sufficiently small R ¡ 0. Since
ĝ pq q p1 1{qqk2p1 1{qq pq 1qk2pq 1q
p1 λ{qqk1
pq λqk1
62
°
n
we see that ĝ has a removable singularity at 0, and thus we can°write ĝ pq q 8
n0 an q
8
for some an P C, n P N0 . Substituting q e2iϕ gives g pϕq n0 an e2inϕ , and since
the series converges absolutely and locally uniformly we have
»π
0
8̧
g pϕqdϕ »π
an
0
n 0
e2inϕ dϕ πa0 .
The constant term a0 is given by limqÑ0 ĝ pq q λk 1 , so
we have for λ ¡ 1
»
p qz
Γ α H
κpz, α, sqdν pz q k k
λ
2 λ
λ1
1
³π
0
g pϕqdϕ λk 1 π, and thus
k
k 1
1{λλ1 1 λ λ2 λ
2
1
detpg q1k .
To combine these results in a single formula we note that λ P p0, 1q if and only if
|λ1| |λ2|, and correspondingly λ ¡ 1 if and only if |λ1| ¡ |λ2|. Further, we have λ1 ¡ 0
if and only if λ2 ¡ 0 since λ1 λ2 detpαq ¡ 0. One can check that this yields
»
p qz
Γ α H
κpz, α, sqdν pz q signpλ1 qk min t|λ1 |, |λ2 |uk1
detpg q1k .
|λ2 λ1|
Note that this formula is indeed independent of the ordering of the eigenvalues of α, so
it gives the same result if we replace λ1 by λ2 and vice versa.
Lemma 4.2.4. We have
detpg qk1
×0
|Z pΓq| slim
»
¸
α
P
T h2
{{
p qz
Γ Γ α H
κpz, α, sqdν pz q
¸ signpλα,1 qk min t|λα,1 |, |λα,2 |uk1
1
|Z pΓq|
|λα,2 λα,1|
αPT {{Γ
h2
where λα,1 and λα,2 are the distinct eigenvalues of α P T h2 .
4.2.5 The parabolic terms
Let α P T p . Then there is x P Q Y t8u such that x is the unique fixed points of α.
Furthermore, α has exactly one eigenvalue
which is rational, say λ P Q . Let σ P SL2 pZq
1 , and since there is only one eigenvalue we have
such that σ 8 0 x, then σ ασ 1
σ ασ λ0 Bλ for some B P Q . (Note that B a0 since α cannot be scalar.) Further
we observe that λ2 detpαq detpg q, so λ detpg q and T does not contain any
parabolic elements if detpg q does not have a rational square root.
Put µ B {p2λq. Using that βa βb pa bqj pβ, aq1 j pβ, bq1 for any β P SL2 pRq
and for any a, b P CzR, one can check for z P H that
κpσz, αq k 1 k
λ
4π
zz
z z B {λ
63
k
k 1 k
λ
4π
Impz q
Impz q iµ
k
.
By Corollary 2.2.4 we have Γpαq Γx , and by Lemma 2.2.6
σ 1 Γ σ t1u "
x
1 hm
0 1
*
:mPZ
where h is the width of the cusp rxs for Γ. Therefore a fundamental domain of the
quotient σ 1 pΓpαqzHq pσ 1 Γx σ qzH is given by tw P H : | Repwq| ¤ h{2u. Further, we
have by definition
#
κ̃pz, α, sq, z P Ux ,
κpz, α, sq κpz, αq,
otherwise,
where
and Ux
κ̃pz, α, sq κpz, αq Impz qs |j pσ 1 , z q|2s
σU8, U8 tz P H :
»
p qz
Γ α H
κpz, α, sqdν pz q Impz q ¡ δ u for some δ
»
κ̃pz, α, sqdν pz q
p qz
Γ α Ux
¡ 1 as before. So
»
p qzpHzUx q
Γ α
κpz, αqdν pz q.
We want to reunite the two integrals on the right by introducing a limit for the second
term. One can check that
»
p qzpHzUx q
Γ α
κpz, αqdν pz q
»
lim
s
×0
p qzpHzUx q
Γ α
κ̃pz, α, sqdν pz q,
but this does not help since we are missing a limit for the first term. Therefore we have
to consider the sum of all integrals in the trace formula of Theorem 4.1.13 coming from
parabolic elements and the corresponding limit, so
lim
s
¸
»
×0 αPT p {{Γ
p qz
Γ α H
κpz, α, sqdν pz q.
Put Txp T p X Tx for all x P Q Y t8u, then we can write T p as the disjoint union of all
Txp since every α P T p fixes a unique x P Q Y t8u. We claim that no two elements in
Txp are Γ-conjugate. To see this suppose that α, β P Txp such that α γ 1 βγ for some
γ P Γ. Then β pγxq γ pαxq γx, so γx x since x is the unique fixed point of β.
Hence γ P Γx Γpαq, and thus β γαγ 1 α. This proves the claim. Next let α P Txp
and β P Typ . If α γ 1 βγ for some γ P Γ, then β pγxq γx as before, and thus γx y
since y is the only fixed point of β. So distinct parabolic elements in T can only be
Γ-conjugate if their fixed points do not agree, but lie in the same Γ-orbit. Moreover, we
see for γ P Γ that
γ Txp γ 1
γ T p γ 1
X
γ T x γ 1
T p X Tγx Tγxp
since α is parabolic if and only if τ 1 ατ is parabolic for any invertible τ . Recall that
C pΓq denotes the set of Γ-orbits in Q Yt8u. By the above observations a complete set of
64
representatives for the set of Γ-conjugacy classes T p {{Γ is given by
we recall that C pΓq is a finite set. Hence we can write
lim
s
»
¸
×0 αPT p {{Γ
κpz, α, sqdν pz q p qz
Γ α H
¸
P p q
x C Γ
×0
α
P
Txp
¸ »
lim
P p q s×0 αPTxp
x C Γ
¸ »
lim
s
¸
p qz
Γ α Ux
p qz
Γ α H
P
α
Txp
p
P p q Tx . Further
x C Γ
κpz, α, sqdν pz q
¸ »
κ̃pz, α, sqdν pz q
”
p qzpHzUx q
Γ α
κpz, αqdν pz q .
We still want to introduce the limit s × 0 for the second term to reunite the two sums.
Fix some x P C pΓq and let σx a
P SL2pZq such that σx8 x. For α P Txp we have
λα Bα
1
σx ασx 0 λα with λα detpg q and Bα P Q as before. Put µα Bα {p2λα q
and let hx be the width of the cusp rxs for Γ. We note that a fundamental domain of
the quotient σx1 pΓpαqzpHzUx qq pσx1 Γx σx qzpHzU8 q is given by
F1 : tw
Hence
¸ »
P
α Txp
p qzpHzUx q
Γ α
P H : | Repwq| ¤ hx{2, Impwq ¤ δu.
|κ̃pz, α, sq|dν pzq
¸ »
|κpσw, αq| Impwqsdν pwq
1
αPTxp σ pΓpαqzpHzUx qq
k
» k 1 ¸
Im
w
k
Im w s dν w
λα
4π
Im
w
iµ
α
F1
αPTxp
»δ
ks2
¸ » hx {2
p q
p q
| |
1 detpgqk{2
k 4π
P
1 detpgqk{2h
¤ k 4π
x
hx {2
α Txp
p q
dx
0
p q
y
|y iµα|k dy
y ks2
k
y Pr0,δ s |y iµα |
¸
δ sup
P
α Txp
¸
k1
detpg qk{2 hx δ ks1
| µα | k .
4π
p
P
α Tx
Since g P GL2 pQq we may choose N P N such that N g has integer entries. Then N Bα
is an integer, so Bα P 1{N pZzt0uq. Clearly |µα |k 2k detpg qk{2 |Bα |k , and Bα Bβ
for distinct α, β P Txp if and only if λα λβ . Hence
¸
P
| µα | k
¤2
k 1
α Txp
k
m
¸
detpg q {
k 2
P zt u N
8.
m Z 0
Therefore the whole sum we started with is finite, and we get using Dominated Convergence Theorem
¸ »
P
α Txp
p qzpHzUx q
Γ α
κpz, αqdν pz q
lim
s
×0
65
¸ »
P
α Txp
p qzpHzUx q
Γ α
κ̃pz, α, sqdν pz q.
Hence
¸
lim
»
×0 αPT p {{Γ
p qz
s
Γ α H
κpz, α, sqdν pz q
¸
¸ »
lim
P p q s×0 αPTxp
x C Γ
p qz
Γ α H
κ̃pz, α, sqdν pz q.
Now we can compute the integral. As above we get for fixed x P C pΓq and α P Txp
»
p qz
Γ α H
κ̃pz, α, sqdν pz q Substituting y
»8
0
µαt yields
y ks2
py iµαqk dy
» 8
»
pσx1 Γx σx qzH
κpσx w, αq Impwqs dν pwq
»8
k 1 k
y k s 2
λα hx
dy.
k
4π
0 py iµα q
pµαtqks2 pµ dtq 1 » 8 pitqks2i2
pµαt iµαqk α
µ1α s 0
pit 1qk
0
s
dt.
Note that the sign of 8 is determined by the sign of µα . Next we will use the substitution
it p1 uq{u which gives
pitqks2 dt is » p1 uqks2uk idu i1 s » usp1 uqks2du
s
pit 1qk
µ1α s γ
u k s 2
u2
µα1 s γ
0
where γ denotes the transformation of the straight line from 0 to 8 by u p1 itq1 .
Thus γ p0q 1 and γ p1q 0. Since the integrand us p1 uqk2s is holomorphic for
small s ¡ 0, we can replace γ by any curve γ 1 which has the same endpoints as γ. Let
γ 1 : r ÞÑ 1 r, r P r0, 1s, then
»
»
»
s
k s 2
s
ks2
u p1 uq
du u p1 uq
du us p1 uqks2 du
1
1
i2
µα1
s
» 8
γ
γ
where pγ 1 q1 denotes the inverse path of γ 1 , so pγ 1 q1 : r
»
γ
us p1 uqks2 du »1
0
pγ q
1
ÞÑ r, r P r0, 1s. Hence
us p1 uqks2 du B ps
1, k s 1q
where B pa, bq is the beta function as introduced in Section 3.2. (Compare Section 2.1
in [BW10].) As stated before it satisfies the identity B pa, bq ΓpaqΓpbq{Γpa bq where
Γpz q denotes the gamma function. Therefore we finally have for some fixed x P C pΓq
lim
×0
s
¸ »
α
P
Txp
p qz
Γ α H
κ̃pz, α, sqdν pz q
1
¸
k1
k i
hx lim
λα
s×0
4π
µ1α
p
P
α Tx
2π
lim
s
×0
Γps
1qΓpk s 1q
Γpk q
s
1
1qΓpk s 1q ¸
2iλα
k
k{2
signpλα q detpg q
Γpk q
Bα
p
k1
Γps
hx lim
s×0
4π
hx detpg qk{2
s
¸
α
P
signpλα q
k
Txp
66
P
α Tx
iλα
Bα
1
s
.
s
Here we used that Γpnq pn 1q! for n P N where 0! 1, and that Γ is continuous
(even holomorphic) on tz P C : Repz q ¡ 0u. (This is shown in Theorem 2.1.1 on page
19 in [BW10].) Hence we have for the sum over all parabolic elements
lim
s
»
¸
×0 αPT p {{Γ
p qz
Γ α H
p qk{2
¸ hx det g
lim
2π
P p q
x C Γ
κpz, α, sqdν pz q
detpg qk{2
2π
s
×0
¸
lim
×0 αPT p {{Γ
s
¸
signpλα q
P
α Txp
signpλα q
k
k
ihα λα
Bα
iλα
Bα
1
1
s
s
where hα denotes the width of the cusp rxα s for Γ with xα being the unique fixed point
of α P T p .
As in the previous subsections we finish with a summarising lemma.
Lemma 4.2.5. For α P T p with unique fixed point x P Q Y t8u we have
λα Bα
σ 1 ασ 0
λα
where σ P SL2 pZq such that σ 8 x and λα is the only eigenvalue of α. Further, we
write hα for the width of the cusp corresponding to the fixed point x of α. We then have
¸
detpg qk1
lim
|Z pΓq| s×0 αPT p{{Γ
»
p qz
detpg qk{21
Γ α H
2π |Z pΓq|
κpz, α, sqdν pz q
lim
s
¸
×0 αPT p {{Γ
signpλα q
k
ihα λα
Bα
1
s
.
In particular, T p is empty if detpg q does not have a rational square root.
4.3 The final trace formula
Combining the formulae from Lemma 4.2.1 to Lemma 4.2.5 with Theorem 4.1.13 yields:
Theorem 4.3.1. Let T ΓgΓ with Γ being a finite index subgroup of SL2 pZq and g
being an element of GL2 pQq. Then
TrpT
ü
Sk pΓqq ts
67
te
th
tp
with
ts
te
th
1 detpgqk{21 rSL pZq : Γs
k 24
2
¸
¸
P p q
signpλα qk ,
α Z T
λkα 1
1
,
|
Γ
p
α
q|
λ
λ
α
α
e
αPT {{Γ
¸ signpλα,1 qk min t|λα,1 |, |λα,2 |uk1
|ZpΓ1 q|
,
|
λα,2 λα,1 |
αPT {{Γ
1 s
k{21
¸
ihα λα
detpg q
k
signpλα q
2π|Z pΓq| slim
×0 αPT {{Γ
Bα
h2
tp
p
where we use the following notation:
• For α P Z pT q, λα denotes the eigenvalue of α.
• For α P T e we choose λα such that σασ 1
the unique fixed point of α in H.
λα 0
0 λα
where σ
1
1
z z and z is
• For α P T h2 , λα,1 and λα,2 denote the distinct eigenvalues of α.
• For α P T p we choose λα and Bα such that σ 1 ασ λ0α Bλαα where σ P SL2 pZq
such that σ 8 x and x is the unique fixed point of α. Further, we write hα for
the width of the cusp corresponding to the fixed point x of α.
The following lemmata simplify the elliptic and parabolic terms in many situations.
They are both combined in Theorem 6.4.10 in [Miy06]. We only give a proof for the
first lemma since this is the one we will need in the next chapter.
Lemma 4.3.2. If there is ω
all α P T , then we have
te
P GL2pRq with detpωq 1 and such that ωαω1 P T
1 λkα1 λα
1 ¸
2
|Γpαq| λα λα
αPT e {{Γ
for
k 1
in the situation of Theorem 4.3.1.
This is an improvement since the expression pλkα1 λα q{pλα λα q is now independent of the choice of eigenvalue of α. Moreover, writing λα reiϕ one can easily
check that
k 1
λkα1 λα
λα λα
k 1
rk2 sinppsink pϕ1q qϕq .
Note that the right-hand side is real.
68
(4.3.1)
z . Then σασ 1 λ 0 for
Proof. Let α P T e with fixed point z P H, and put σ 11 z
0 λ
some λ P CzR. Put β ωαω 1 , then β is elliptic since α is, and β P T by assumption,
w1 w and
so β P T e . The unique fixed points of β are w ωz and w1 ωz. Clearly
µ 0
w
1
w P H since detpω q is negative. Put τ 11 w , then τ βτ 0 µ . Since β is a
conjugate of α their eigenvalues agree, so either µ λ or µ λ.
Suppose that µ λ. Then σασ 1 τ βτ 1 , so β pτ 1 σ qαpτ 1 σ q1 . Consider
τ 1 σ Let ω
a b
c d
1
ww
w w zw zw
.
0
zz
. One can check that
zw zw
ac|z|
2
adpz z q
|cz d|2
bd
pz zq.
Hence τ 1 σ has entries in R since pz z q{pw wq Impz q{ Impwq is real. Moreover,
we have detpτ 1 σ q Impz q{ Impwq ¡ 0 as z, w P H. Therefore α and β are conjugate
by an element of GL2 pRq which contradicts
Lemma 2.2.5 as β ωαω 1 . Therefore we
must have µ λ, so τ βτ 1 λ0 λ0 .
Again by Lemma 2.2.5, α and β represent different elements in T e {{Γ. Further we
have |Γpαq| |Γpβ q| since the map γ ÞÑ ωγω 1 gives a bijection Γpαq Ñ Γpβ q. Adding
the terms in the sum over T e {{Γ corresponding to α and β yields
1 λk1
|Γpαq| λ λ
k 1
1 λ
|Γpβ q| λ λ
k 1
1 λk 1 λ
|Γpαq|
.
λλ
This proves the claimed formula for te .
Lemma 4.3.3. If there is ω
all α P T , then we have
tp
P GL2pRq with detpωq 1 and such that ωαω1 P T
1
k{21
¸
hα λα s
signpλα qk det4|pZgpqΓq| slim
×0 αPT p {{Γ
Bα s
in the situation of Theorem 4.3.1.
For a proof of this we refer to Theorem 6.4.10 on page 241 in [Miy06].
69
for
5 A trace formula for the Hecke
operators Tp acting on Sk pΓ0pN qq
In this final chapter we present an explicit formula for the trace of the Hecke operator Tp
acting on SK pΓ0 pN qq as developed by H. Hijikata in [Hij74]. Though we will not prove
this formula, we will discuss it and calculate two examples. We also mention that S. L.
Ross II slightly simplified Hijikata’s formula in [RI92] replacing some terms by tables
such that the computation of the trace of a Hecke operator ”essentially reduces [...] to
looking up values in a table” as he writes in the abstract of the corresponding paper.
Miyake presents two trace formulae in Section 6.8 of [Miy06]. The first one is Theorem
6.8.4 on pages 262 - 264 which is defined for rather general groups Γ and looks still quite
similar to Hijikata’s formula. In the second formula on page 265 Miyake specialises to
the group Γ0 pN q with N pq ν for some odd primes p, q and some ν P N0 , which results
in another ”ready to compute” formula.
We concentrate on Hijikata’s formula since out of the four mentioned formulae it is
probably the one closest to our formula as stated at the end of the previous chapter.
Moreover, it is the most original formula, too, out of the mentioned four.
5.1 Motivating observations
We start with a simple application of Theorem 4.3.1:
Corollary 5.1.1. Let k
some prime p. Then
¥ 4 be even, Γ Γ0pN q for some N P N, and T Γ
TrpT
Sk pΓqq te
ü
1 0
0 p
Γ for
th
with
¸
1 λαk1 λα
21
|Γpαq| λα λα
αPT e {{Γ
te
th
12
k 1
¸
P
{{
α T h2 Γ
where λα denotes an eigenvalue of α
values of α P T h2 .
,
min t|λα,1 |, |λα,2 |uk1
|λα,2 λα,1|
P T e, and λα,1, λα,2 denote the two distinct eigen
Proof. By Theorem 4.3.1 we have TrpT ü Sk pΓqq ts te th tp . Since detp 10 p0 q p
and p is a prime, which does not have a rational square root, we have ts tp 0 as
70
remarked in Lemma 4.2.1 and Lemma 4.2.5.
Further, we have Z pΓq 2and k even,
1 0
which gives the term th . Put ω 0 1 and let α P T , so α γ1 10 p0 γ2 for some
γ1 , γ2 P Γ. Then
ωαω 1
pωγ1ωqpω 10 p0 ωqpωγ2ωq γ11 10 p0 γ21 P T
since one can easily check that γj1 : ωγj ω P Γ, j 1, 2. Therefore we can use Lemma
4.3.2 to get te , which proves the claimed formula.
In the following we will rearrange the terms of the trace formula given by the previous
corollary. This will on the one hand lead to a nicer statement, and on the other hand
motivate the trace formula of H. Hijikata, which will be presented afterwards.
Let k ¥ 4 be even, Γ Γ0 pN q for some N P N, and T Γ 10 p0 Γ for some prime p
as in the previous theorem. Note that the case k odd is trivial, since the only modular
form of odd weight and level Γ0 pN q is the zero-function.
Let α P T , then detpαq p. Further let t be the trace of α. By definition we know
that α is elliptic if and only if t2 4p, and that α is hyperbolic if and only if t2 ¡ 4p.
Recall that α P T h2 if and only if α has two distinctfixed points
a in Q Y t8u. This is the
a
b
case if either α fixes 8, so if α is of the form 0 d , or if t2 4p is rational. In the
first case we get that ad p, and thus either a 1 and d p, or the other way round.
Hence we have t2 4p pp 1q2 4p pp 1q2 . In the second case t2 4p needs to
be a square. Therefore we have shown that for α P T
α P Te
α P T h2
| Trpαq| 2?p,
Trpαq2 4p U 2 for some U P N.
ðñ
ðñ
Clearly there are only finitely many possible values for the trace of an element in T e .
Moreover, let t0 be the largest positive integer such that t20 4p ¤ pt0 1q2 . Then one
?
can easily see that 2 p | Trpαq| ¤ t0 for any α P T h2 . Hence the trace of an element
in T h2 is bounded as well.
Next we consider eigenvalues. Let α P T with trace t P Z. The eigenvalues of α are
given by the zeros of the polynomial Φt pxq x2 tx p. Hence they are uniquely
determined by the trace of α. We denote the zeros of Φt pxq by λ1 ptq and λ2 ptq.
We may now rewrite the formula given by Corollary 5.1.1. Keeping notation as before
we get
TrpT
ü
Sk pΓqq
¸
12 tPZ,
|t| 2?p
pλ1ptqqk1 pλ2ptqqk1 ¸ 1
λ1 ptq λ2 ptq
|Γpαq|
αPT {{Γ
e
p qt
Tr α
¸
P
t Z,
t2 4p U 2
for some U N
min t|λ1 ptq|, |λ2 ptq|uk1
|λ1ptq λ2ptq|
P
1
.
h
αPT 2 {{Γ ¸
p qt
Tr α
71
(5.1.1)
Define
?
p
λ1 ptqqk1 pλ2 ptqqk1 pλ1 ptq λ2 ptqq1 , |t| 2 p,
aptq ?
min t|λ1 ptq|, |λ2 ptq|uk1 |λ1 ptq λ2 ptq|1
, |t| ¡ 2 p,
#
and
B ptq where
B1 ptq ?
#
B1 ptq , |t| 2 p,
?
B2 ptq , |t| ¡ 2 p,
¸
1
|Γpαq| ,
αPT e {{Γ
p qt
Tr α
B2 ptq ¸
P
1.
{{
p q
α T h2 Γ
Tr α t
Then we may write (5.1.1) as
TrpT
ü
Sk pΓqq 1¸
aptqB ptq
2 t
(5.1.2)
?
where the sum runs over all integers t P Z such that either |t| 2 p, or t2 4p is a
positive square. In particular, the sum is finite as mentioned earlier.
The difficult part is now the evaluation of B1 and B2 in terms of the given operator Tp ,
the given level Γ0 pN q and the current trace t. As the corresponding studies go beyond
the scope of this thesis, we only quote and explain the trace formula given in the paper
by H. Hijikata. Since it is stated in a more general context as we are working in, we will
adjust it to our situation.
5.2 Hijikata’s trace formula for Γ0pN q
Consider Theorem 0.1 on page 57 of [Hij74]. We may use the following simplifications:
• Since we only consider operators Tn for n being prime, N and n are coprime if and
only if n does not divide N .
• The second term of the formula vanishes as we assume k ¥ 4. Moreover, n is never
a square since n is prime. Hence the third term of the formula vanishes as well,
and we are left with the first one.
• We only consider Γ Γ0 pN q itself. Therefore we may take M 1, let h be the
trivial group and assume χ to be the trivial character that maps everything to 1.
• Again since n is prime, s2 4n will never be 0, so the parabolic case (p) does not
happen and might therefore be removed.
• We do not have the factors n1k{2 in the definition of apsq as we defined the
general action of Hecke operators slightly differently. Further, signpxqk 1 in the
definition of apsq since we assume k to be even.
72
• Finally, h is given by a trivial direct product, and hence we can use the slightly
simplified definition of cps, f q. (Note that χpxq χpy q 1 for any x, y as χ is
trivial.)
Theorem (Hijikata’s trace
formula for Γ0 pN q). Let k ¥ 4 be even, Γ Γ0 pN q for some
1 0
N P N, and T Γ 0 p Γ for some prime p not dividing N . Then
TrpT
ü
Sk pΓqq ¸
¹
1¸
aptq
bpt, f q
cpt, f, q q
2 t
q prime
f PN, f |U
|
t
qN
where the following notation is used:
°
• Put Dptq : t2 4p. The sum t runs over all integers t P Z such that either
Dptq is a positive square, or Dptq is negative. For every such t we choose Ut P N
and if Dptq 0 some negative squarefree integer mt such that Dptq is of one of
the following forms:
(1) Dptq Ut2
(2) Dptq Ut2 mt with mt
•
1 mod 4
(3) Dptq Ut2 4mt with mt 2, 3 mod 4
In the following we say that t is of type phq if Dptq is of the form in (1), and t is
of type peq if Dptq is of the form in (2) or (3).
Put Φt pxq x2 tx p, and let λ1 , λ2 be the solutions of Φt pxq 0. We define
#
min t|λ1 |, |λ2 |uk1 |λ1 λ2 |1 , t of type phq,
aptq , t of type peq.
λk1 1 λ2k1 pλ1 λ2 q1
• We define
bpt, f q #
ϕpUt {f q
, t of type phq,
2
2
h pDptq{f q {w pDptq{f q , t of type peq,
where ϕpnq denotes Euler’s totient function, so ϕpnq is the order of the unit group
of the ring Z{nZ.
? Further, hpdq denotes the class number of the order of the
number field Qp d q with discriminant d, and wpdq denotes 1{2 of the order of the
unit group of this order.
• Fix t P Z and f P N dividing Ut , and let q be a prime diving N . Further, let ν be
the order of q dividing N , so ν P N0 such that q ν divides N but q ν 1 does not, and
let µ be the order of q dividing f . Put
à n P Z : Φt pnq 0 mod q ν
and if Dptq{f 2
2µ
0 mod q put
!
B̃ n P Ã : Φt pnq 0 mod q ν
73
(
, 2n t mod q µ ,
2µ 1
)
.
#
Then
|A|
cpt, f, q q |A| |B |
, Dptq{f 2
, Dptq{f 2
0 mod q,
0 mod q,
where A and B are complete sets of representatives for à and B̃ mod q ν
spectively.
µ
, re-
In the following we will use Hijikata’s formula to compute some traces of Hecke operators for specific N and p. These examples give detailed explanations for all the terms
appearing in the presented formula. We start by making some general observations on
Hijikata’s trace formula which will save us some work while computing examples.
(1) Put for t P Z such that either Dptq is negative or Dptq is a positive square
Aptq : aptq
¸
P
|
f N, f Ut
bpt, f q
¹
cpt, f, q q.
q prime
qN
|
We claim Aptq Aptq. First note that Dptq Dptq, so if t P Z is a valid value
for the first sum in the trace formula then t is as well. Further, t and t are
obviously of the same type, and Ut Ut . Thus f takes the same values for t and
t, and bpt, f q bpt, f q. We claim aptq aptq. To see this let λ1, λ2 be the
solutions of Φt pxq 0, then x2 tx p px λ1 qpx λ2 q and thus
Φt pxq x2 ptqx
p px
λ1 qpx
λ2 q.
So λ1 , λ2 are the solutions of Φt pxq 0. Hence we clearly have aptq aptq
if t is of type (h). If t is of type (e) we see
pλ1qk1 pλ2qk1 p1qk1 λk11 λk21 aptq
pλ1q pλ2q
1
λ1 λ2
since k is even. It remains to show that cpt, f, q q cpt, f, q q. Fix f P N dividing
Ut and a prime q dividing N . Then Φt pnq Φt pnq, so for m P N0
Φt pnq 0 mod q m ô Φt pnq 0 mod q m .
Further we clearly have 2n t mod q m if and only if 2pnq t mod q m for
m P N0 . This proves the claim, so Aptq Aptq for all valid t P Z.
Let λ1 , λ2 be the solutions of Φt pxq x2 tx p 0. Then λ1 λ2 t and
λ1 λ2 p. Put ak ptq pλk1 1 λk2 1 qpλ1 λ2 q1 . Then
p
λ1 λ2 q λk1 2 λk2 2 λ1 λ2 λk1 3 λk2 3
ak ptq λ1 λ2
t ak1ptq p ak2ptq.
aptq (2)
74
Note that a2 ptq 1 and a3 ptq λ1 λ2 t. Hence we may use the above
recurrence formula to write down expressions for ak ptq for fixed integers k, namely
a4 ptq t2 p,
a5 ptq t3 2pt,
a6 ptq t4 3pt2
p2 ,
...
Continuing we get for example
a24 ptq
t22 21pt20
190p2 t18 969p3 t16 3060p4 t14 6188p5 t12
8008p6 t10 6435p7 t8 3003p8 t6 715p9 t4 66p10 t2 p11 .
(5.2.1)
Though this expression looks fairly messy it well be useful in the first example.
More important, the presented concept of expressing ak ptq in terms of ak1 ptq and
ak2 ptq yields that ak ptq is an integer for every t P Z since p is.
Example 5.2.1. We start with a very basic example whose result we can check afterwards by a direct computation. Let Γ be the full modular group
SL2 pZq and k 24.
1
0
We want to compute the trace of the Hecke operator T2 Γ 0 2 Γ acting on S24 pΓq. In
terms°of Hijikata’s formula we therefore have N 1 and p 2. So Dptq t2 8 and the
sum t runs over all integers |t| ¤ 3, since Dp4q 8 is not a square±and Dp5q 17
is already greater than p5 1q2 . Further, we±note that the product q is empty since
N 1 does not have any prime divisors, so q cpt, f, q q 1.
Next we consider the different possible values for t case by case. By the above observations it sufficies to consider non-negative values of t.
• Let t 0. Then Dp0q 12 4 p
?2q, so U0 1 and t is of type (e). Further, we
have Φ0 pxq x2 2, so λ1,2 i 2, and thus
ap0q °
23
λ23
1 λ2
λ1 λ2
211.
The sum f only takes the value f 1 since U0 1. To determine bp0, 1q let K
?
?
be the number field Qp 8 q Qp 2 q. We are looking for an order in K with
discriminant 8. By Theorem 2.4.3 this order is the ring of integers OK itself.
Hence we have
hp8q
hpOK q
1
bp0, 1q wp8q
|U pOK q|{2 1 1
by Proposition 2.4.4 and Table 2.4.6.
• Let t 1. Then Dp1q 12 p7?
q, so U1 1 and t is of type (e). We have
2
Φ1 pxq x x 2, so λ1,2 p1 i 7 q{2, and
ap1q 23
λ23
1 λ2
.
λ1 λ2
This is rather hard to simplify. Recall that we expressed aptq a24 ptq as a polynomial in t in equation (5.2.1). Using this with p 2 we may compute ap1q 967.
75
For the second sum we only have
? to consider f 1 as before. To determine bp1, 1q
let K be the number field Qp 7 q, then OK is an order in K with discriminant
7 and thus
hpOK q
1
hp7q
bp1, 1q wp7q
|U pOK q|{2 1 1
by Proposition 2.4.4 and Table 2.4.6.
• Let t 2. Then Dp2q 12 4 p1q, so U2 1 and t is
? still of type (e). We have
Φ2 pxq x2 2x 2, so λ1,2 1 i reiϕ with r 2 and ϕ π {4. Hence we
get using equation (4.3.1) that
ap2q r22 sinp23ϕq
sinpϕq
pπ{4q 211.
211 sin
sinpπ {4q
Clearly equation (5.2.1) would have given the same result.
As ?
before we only
? have f 1 for the second sum. To determine bp2, 1q let K be
Qp 4 q Qp 1 q, then OK is an order in K with discriminant 4 and thus
bp2, 1q hp4q
wp4q
|UhppOOKq|{q 2 12
K
by Proposition 2.4.4 and Table 2.4.6.
• Let t 3. Then Dp3q 12 , so U3 1 and t is of type (h). Further we have
Φ3 pxq x2 3x 2, so λ1 1, λ2 2 and thus
ap3q As before f
min t|λ1 |, |λ2 |uk1
|λ1 λ2|
1.
1, and hence bp3, 1q ϕp1q 1.
Combining these results we get
TrpT2
ü
1
ap0qbp0, 1q 2ap1qbp1, 1q
2
210 967 210 1
1080.
S24 pSL2 pZqqq 2ap2qbp2, 1q
2ap3qbp3, 1q
We quickly check this result: Let Ek denote the normalised Eisenstein series of weight
k, so Ek 1{2 Gk,SL2 pZq,8 , where Gk,SL2 pZq,8 is defined as in Subsection 2.1.3. One
can easily check that a basis of S24 pSL2 pZqq is given by f1 E43 ∆ and f2 ∆2 where
∆ pE43 E62 q{1782. (Note that Ek agrees with the function Gk defined in (4.1.4)
and (4.1.5) on the bottom of page 99 in [Miy06], and thus ∆ is the function defined in
(4.1.14) two pages afterwards.) One can compute
T2 pf1 q 696f1
20736000f2
76
and T2 pf1 q f1
384f2 .
Hence the linear operator T2 is given by the matrix
A
696
1
20736000 384
with respect to the basis tf1 , f2 u, and thus we have
Tr pT2
ü
S24 pSL2 pZqqq TrpAq 1080
as expected.
Finally, we also give the trace of T2 acting on Sk pSL2 pZqq for general k. Since bpt, f q
does not depend on k we can use the corresponding values computed above, so
TrpT2
ü
Sk pSL2 pZqqq 1
ap0q
2
2ap1q
ap2q
2ap3q .
Also the eigenvalues of Φt pxq do not depend on k, so we directly see
TrpT2
ü
Sk pSL2 pZqqq
p2qk{22 1
?
i 7
? ?1 i 7 k1 p1
2k1 i 7
k1
iqk1 p1 iqk1
4i
1.
The previous example was simple in two ways: First we did not have to consider terms
of the form cpt, f, q q since N 1, and secondly we did not have to work with orders
of number fields other than the ring of integers itself. In the following example we will
have to deal with these cases. However, we will not argue as detailed as before, since
the basic considerations will still be the same.
Example 5.2.2.
Let Γ Γ0 p4q. We want to compute the trace of the Hecke operator
1
0
T3 Γ 0 3 Γ acting on Sk pΓq for some even integer k ¥ 4. In terms of Hijikata’s
formula
we have N 4 and p 3, so Dptq t2 12 and one can easily check that the
°
sum t runs over all integers |t| ¤ 4. As in the first example we consider all valid and
non-negative values of t case by case:
• Let t 0. Then Dp0q 22 p
? 3q, so U0 2 and t is of type (e). Further, we have
2
Φ0 pxq x
3, so λ1,2 i 3 and thus
ap0q λk1 1 λk2 1
λ1 λ2
p3qpk2q{2.
Since U0 2 we have to consider f? 1 and f ?2 for the second sum. Let f 1,
then Dp0q{f 2 12. Let K Qp 12 q Qp 3 q. We are looking for an order
in K with discriminant
12. Recall that ∆pOq n2 p3q for an arbitrary order
?
O Z np1
?
3 q{2 Z in K as remarked in Subsection 2.4.5. Hence we want
n 2, so O Z
3 Z. Using Theorem 2.4.7 we get
hp12q hpOq 2hpOK q
rU pOK q : U pOqs
77
1
Lp3, 2q
2
.
By the observations following the mentioned theorem we have Lp3, 2q 1,
|U pOq| 2 and rU pOK q : U pOqs 3, so h±p12q 1 and wp12q 1, and thus
bp0, 1q 1. Next we consider the product q cp0, 1, q q. The only prime dividing 4
is q 2. Thus the order of q dividing N is ν 2, and the order of q dividing f is
µ 0. Hence
(
à n P Z : n2
3 0 mod 4, 2n 0 mod 1 .
The term mod 1 is redundant, and we easily see à tn P Z : n oddu. A set of
representatives of à mod 4 is given by A t1u. Since Dp0q{f 2 12 is even
we also have to consider
!
)
n P Ã : n2 3 0 mod 8 .
One can check that B̃ H, so B H and thus cp0, 1, 2q |A| |B | 2.
?
It remains to consider the case f 2. We have Dp0q{f 2 3, so K Qp 3 q
B̃
as before, but this time we directly see
bp0, 2q hp3q
wp3q
|UhppOOKq|{q 2 13
±
K
since OK has discriminant 3. Consider q cp0, 2, q q. As before q
but the order of q dividing f is now µ 1. Hence
à n P Z : n2
2 and ν 2,
(
3 0 mod 16, 2n 0 mod 2 .
Again the term 2n 0 mod 2 is redundant, and one can check à H. The set B̃
is not relevant as Dp0q{f 2 3 is not even. So cp0, 2, 2q 0.
Therefore the 0-term in the trace formula is given by
Ap0q : ap0q rbp0, 1qcp0, 1, 2q
bp0, 2qcp0, 2, 2qs 2 p3qpk2q{2 .
• Let t 1. Then Dp1q 12 p11q, so U1 1 and t is of type (e). This time we
start backwards, so by computing all relevant values cp1, f, q q. Clearly f 1 since
U1 1 and q 2. The order of q dividing N is still ν 2, and the order of q
dividing f is µ 0. Hence
à n P Z : n2 n
(
3 0 mod 4, 2n 1 mod 1 .
One can check that à H, so cp1, 1, 2q 0. Therefore the whole
trace formula vanishes.
1-term in the
• Let t 2. Then Dp2q 12 4 p2q, so U2 1 and t is of type (e). Again we start
by computing relevant values cp1, f, q q. We have f 1 since U2 1 and q 2, so
as before ν 2 and µ 0 and thus
à n P Z : n2 2n
(
3 0 mod 4, 2n 2 mod 1 .
Again one can check à H, so cp2, 1, 2q 0. Therefore the
formula vanishes as well.
78
2-term in the trace
• Let t 3. Then Dp3q 12 p3q, so U3 1 and t is of type (e). As in the previous
cases we only have f 1, q 2, and one can easily check that cp3, 1, 2q 0.
Hence the 3-term in the trace formula vanishes, too.
• Let t 4. Then Dp4q 22 , so U4 2 and t is of type (h). Further, we have
Φ4 pxq x2 4x 3, so λ1 1, λ2 3 and thus
ap4q min t|λ1 |, |λ2 |uk1
|λ1 λ2|
12 .
Since U4 2 the second sum runs over f 1, 2. Let f 1, then bp4, 1q ϕp2q 1.
Let q 2. The order of q dividing N is ν 2 as before, and the order of q dividing
f is µ 0. Hence
à n P Z : n2 4n
(
3 0 mod 4, 2n 4 mod 1 .
One can check à tn P Z : n oddu. A set of representatives of à mod 4 is given
by t1u. Since Dp4q{f 2 is even we have to consider
B̃
!
n P Ã : n2 4n
)
3 0 mod 8 ,
and one can check B̃ Ã. Hence we have cp4, 1, 2q |A| |A| 4. Let now
f 2, then bp4, 2q ϕp1q 1. Let q 2, ν 2 as before. The order of q dividing
f is µ 1. Hence
à n P Z : n2 4n
(
3 0 mod 16, 2n 4 mod 2 .
One can check that à tn P Z : n 1, 3 mod 8u, and a set of representatives for
à mod 8 is given by A t1, 3u. Since Dp4q{f 2 is odd, we do not consider B̃, and
thus we have cp4, 2, 2q |A| 2.
Therefore the
4-term in the trace formula is given by
Ap4q : ap4q rbp4, 1qcp4, 1, 2q bp4, 2qcp4, 2, 2qs 3.
Combining all of these results we finally see
TrpT3
ü
Sk pΓ0 p4qqq 1
Ap0q
2
2Ap4q
p3qk{21 3.
As one can check in the table given on page 296 in [Miy06] the space S6 pΓ0 p4qq is 1dimensional. By the above formula the trace of T3 acting on S6 pΓ0 p4qq is 12. Hence
we have T3 pf q 12f for any f P S6 pΓ0 p4qq since the matrix representation of T3 with
respect to any basis in S6 pΓ0 p4qq is simply p12q. (This can also be verified using Sage.)
As a final example we compute the eigenvectors of the Hecke operator T2 acting on
S24 pSL2 pZqq, which we already considered in the first example. The presented method
can be generalised following the argumentation on page 266, 267 in [Miy06] to arbitrary
Tp operators acting on any space Sk pΓ0 pN qq such that p does not divide N . (For such
cases we would need a more general trace formula.)
79
Example 5.2.3. Recall that we have shown in Example 5.2.1 that
Tr pT2
ü
S24 pSL2 pZqqq 1080.
Further, we now that the space S24 pSL2 pZqq is 2-dimensional. Let µ1 , µ2 be the two
eigenvalues of T2 , then µ1 µ2 1080. So we need a second equation to determine the
eigenvalues. For this purpose we use part (2) of Lemma 4.5.7
on page 140 in [Miy06].
2
0
We have p 2 and N 1. Note that T p2, 2q SL2 pZq 0 2 , so for any cusp form f in
S24 pSL2 pZqq we have
2 0
T p2, 2qpf q f |k
2k2f.
0 2
Therefore the mentioned lemma gives
pT2q2 T4
2k1 T1
where T1 denotes the identity operator. Clearly the trace of T1 acting on S24 pSL2 pZqq is
given by 2°
since the corresponding space is 2-dimensional. Further, one can check that
TrpT m q nj1 ν m for a general linear operator T with eigenvalues ν1 , . . . , νn . Hence
Tr
pT2q2
ü
S24 pSL2 pZqq
µ21
µ22 .
Therefore it remains to compute the trace of T4 acting on SL2 pZq. Unfortunately, the
trace formula presented at the beginning of this section is only valid for Tp operators
with p being a prime. Using the more general formula given in Hijikata’s paper [Hij74]
one can compute
Tr pT4 ü S24 pSL2 pZqqq 25326656.
We are not giving any details here as the purpose of this example is the computation of
the eigenvalues of T2 . Using that µ2 1080 µ1 we see
25326656
223 2 Tr
T4
223 T1
ü
S24 pSL2 pZqq
Tr pT2q2 S24pSL2pZqq
µ21 p1080 µ1q2,
ü
so 0 µ21 1080µ1 20468736. We note that the same equation holds if we replace µ1
by µ2 . Therefore the eigenvalues µ1 , µ2 are the roots of Ψpxq x2 1080x 20468736.
These are given by
µ1,2
540 ?
5402
?
20468736 540 12 144169.
We may check this result using the matrix representation of T2 developed at the end of
Example 5.2.1.
80
6 Summary and outlook
In this thesis we developed a trace formula for Hecke operators for modular groups
following Section 6.1 to 6.4 of [Miy06]. We began by showing
of holo³ that 2the space
k
morphic functions on H being integrable in the sense that H |f pz q| Impz q dν pz q 8
is a reproducing kernel Hilbert space with kernel
Kk
k1
4π
zw
2i
k
.
Afterwards we introduced a similar space for Γ-invariant functions with Γ being a modular group. We denoted this space by Hk2 pΓq and proved that it, too, is a reproducing
kernel Hilbert space with kernel
KkΓ pz, wq |Z pΓq|1
¸
P pKk p, wq|k γ qpz q.
γ Γ
This is Theorem 3.4.5. The reason to consider these spaces is given by Theorem 3.3.3,
which states that Hk2 pΓq and Sk pΓq agree as Hilbert spaces. Hence the space of cusp
forms of weight k and level Γ is a reproducing kernel Hilbert space with kernel KkΓ . We
used this fact in Section 3.5 to write down a first trace formula: For k ¥ 3 the trace of
an Hecke operator T ΓgΓ acting on Sk pΓq is given by
TrpT
ü
Sk pΓqq detpg qk1
|Z pΓq|
»
¸
z
P
Γ Hα T
Kk pαz, z qj pα, z qk Impz qk dν pz q.
Chapter 4 is concerned with the simplification of this formula closely following Section
6.4 of [Miy06]. We started by interchanging summation and integration. This turned out
to be quite challenging near cusps where we had to introduce an extra term to guarantee
convergence. Moreover, we had to work with a fixed fundamental domain for Γ which
was replaced by some appropriate quotient at the end of the section using the notation
of conjugacy classes. Finally, we got
TrpT
ü
Sk pΓqq
detpg q k 1
|Z pΓq| »
¸
p qz
P {{
α T1 Γ Γ α H
¸
lim
s
×0
α
P {{
T2
κpz, αqdν pz q
»
p qz
Γ Γ α H
κpz, α, sqdν pz q .
The notation used in this statement is explained in Theorem 4.1.13.
81
In the following section we calculated the integrals appearing in the above formula
depending on the type of α P T , so whether α is scalar, elliptic, parabolic or hyperbolic.
The corresponding calculations were very technical, though most of them yielded fairly
nice results. In particular, it turned out that a large subset of the α’s in T , namely the
hyperbolic elements with fixed points in RzQ, do not contribute anything to the trace.
Finally, we summarised our results in a simplified trace formula (Theorem 4.3.1) at the
end of the chapter.
Eventually we presented Hijikata’s trace formula in Chapter 5 motivating it with the
formula presented at the end of the previous chapter and explaining which terms would
have to be studied further to get Hijikata’s formula. In the end we computed some
explicit traces of Hecke operators using the trace formula of Hijikata.
It remains to comment on possible further studies. The most intuitive extension
would be to close the gap between the trace formula presented at the end of Chapter
4 and Hijikata’s formula. Therefore one might want to follow Section 6.5 to 6.8 in
[Miy06]. Further, one could generalise the concepts introduced in this work to more
general groups. Miyake shows in his book that everything works exactly the same if we
use Fuchsian groups of the first kind (see Section 1.5 in [Miy06]) possibly in combination
with characters of these groups of finite order instead of modular groups. Moreover, one
could consider trace formulae for generalised spaces of modular forms. An example of
such are spaces of Siegel modular forms which are functions holomorphic on the space
of symmetric n n matrices with positive definite imaginary part that are invariant
under the action of some symplectic group. These symplectic groups generalise modular
groups.
It would also be interesting to study applications of trace formulae. For example one
could use the formula to compute eigenvalues of Tp operators as presented on pages 266,
267 in Miyake’s book, or to compute dimensions of spaces of cusp forms. The latter
can be done by computing the trace of the trivial operator acting on the corresponding
space of cusp forms. However, in both cases it would be useful to either have a look
at implementations of trace formulae for computer algebra systems (see for example
[Ste12]), or to consider implementing a trace formula on ones own. In this context one
might also compare different approaches to compute traces (or eigenvectors) of Hecke
operators with respect to their runtimes.
82
Bibliography
[Aro50] N. Aronszajn. Theory of reproducing kernels. Transactions of the American
Mathematical Society, 68(3):337–404, 1950.
[BW10] Richard Beals and Roderick Wong. Special functions: A Graduate Text. Cambridge University Press, Cambridge; New York, 2010.
[Con97] John B. Conway. A Course in Functional Analysis. Graduate Texts in Mathematics. Springer-Verlag, New York, 2 edition, 1997.
[DS05] Fred Diamond and Jerry Michael Shurman. A First Course in Modular Forms.
Springer Verlag, New York, 2005.
[Eic55] M. Eichler. Über die Darstellbarkeit von Modulformen durch Thetareihen.
Journal für die reine und angewandte Mathematik, 195:156–171, 1955.
[Eic56] M. Eichler. Zur Zahlentheorie der Quaternionen-Algebren. Journal für die reine
und angewandte Mathematik, 195:127–151, 1956.
[Eic57] M. Eichler. Eine Verallgemeinerung der Abelschen Integrale. Mathematische
Zeitschrift, 67(1):267–298, 1957.
[Hij74] Hiroaki Hijikata. Explicit formula of the traces of Hecke operators for Γ0 pnq.
Journal of the Mathematical Society of Japan, 26(1):56–82, 1974.
[Kra01] Steven G. Krantz. Function Theory of Several Complex Variables. American
Mathematical Society, Providence, RI, 2001.
[Lan93] Serge Lang. Real and Functional Analysis. Springer-Verlag, New York, 1993.
[Lan02] Serge Lang. Algebra. Graduate Texts in Mathematics. Springer-Verlag, New
York, 3rd edition, 2002.
[Miy06] Toshitsune Miyake. Modular Forms. Springer Monographs in Mathematics.
Springer-Verlag, 2nd edition, 2006.
[MV07] Hugh L. Montgomery and Robert C. Vaughan. Multiplicative Number Theory
I : Classical Theory. Cambridge University Press, Cambridge, UK ; New York,
2007.
[Ono90] Takashi Ono. An Introduction to Algebraic Number Theory. University Series
in Mathematics. Plenum Press, 2nd edition, 1990.
83
[RI92]
Shepley L. Ross II. A simplifed trace formula for Hecke operators for Γ0 pnq.
Transactions of the American Mathematical Society, 331(1):425–447, 1992.
[Sch05] Volker Scheidemann. Introduction to Complex Analysis in Several Variables.
Birkhäuser Verlag, Basel ; Boston, 2005.
[Sel56]
A. Selberg. Harmonic analysis and discontinuous groups in weakly symmetric
Riemannian spaces with applications to Dirichlet series. Journal of the Indian
Mathematical Society, 20(956):47–87, 1956.
[Shi63] Hideo Shimizu. On traces of Hecke operators. Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, astronomy, physics, chemistry,
10(1):1–19, 1963.
[SS05]
Elias M. Stein and Rami Shakarchi. Real analysis: Measure Theory, Integration,
and Hilbert Spaces. Princeton University Press, Princeton, N.J., 2005.
[ST02]
Ian Stewart and David Orme Tall. Algebraic Number Theory and Fermat’s Last
Theorem. AK Peters, Natick, Mass., 2002.
[Ste08] Peter Stevenhagen. The arithmetic of number rings. Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, 44:209–266, 2008.
[Ste12] William Stein. Implementations of Hijikata’s trace formula. http://wstein.
org/Tables/hijikata.html, September 2012.
84