EVERHART, LANCE M., M.A. On Generators of Hilbert Modular Groups of Totally
Real Number Fields. (2016)
Directed by Dr. Sebastian Pauli. 54 pp.
In this paper we report the beginnings of the computations and tabulations of
the generators of PSL2 (OK ), where OK is the maximal order of a real eld of degree
n = [K : Q]. We discuss methods of obtaining generators in order to calculate the
values of invariants of the congruence subgroups.
ON GENERATORS OF HILBERT MODULAR GROUPS OF TOTALLY REAL
NUMBER FIELDS
by
Lance M. Everhart
A Thesis Submitted to
the Faculty of The Graduate School at
The University of North Carolina at Greensboro
in Partial Fulllment
of the Requirements for the Degree
Master of Arts
Greensboro
2016
Approved by
Committee Chair
To my family, who are always there to lift me up.
ii
APPROVAL PAGE
This thesis written by Lance M. Everhart has been approved by the following
committee of the Faculty of The Graduate School at The University of North Carolina
at Greensboro.
Committee Chair
Committee Members
Sebastian Pauli
Greg Bell
Dan Yasaki
Date of Acceptance by Committee
Date of Final Oral Examination
iii
ACKNOWLEDGMENTS
I would like to thank my advisor Dr. Sebastian Pauli, who helped me every step
of the way with my research and who is always willing to answer any question I had
with great patience.
iv
TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
CHAPTER
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
II. THE MODULAR GROUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2.1. Actions of the Special Linear Group on H . . . . . . . . . . . . .
2.2. The Modular Group . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
5
III. TOTALLY REAL NUMBER FIELDS . . . . . . . . . . . . . . . . . . . . . . . .
8
3.1. Denitions and Theorems . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2. Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3. Class Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
IV. HILBERT MODULAR GROUPS . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1.
4.2.
4.3.
4.4.
Hilbert Modular Groups . . . . . . . .
Fixed Points . . . . . . . . . . . . . . . .
Congruence Subgroups . . . . . . . . .
Cusps of the Surface Hn /PSL2 (OK )
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24
V. FUNDAMENTAL DOMAIN AND GENERATORS . . . . . . . . . . . . . . 31
5.1. Fundamental Domain and Known Generators . . .
5.2. Fundamental Domain and Generators of PSL(OK )
in the General Case . . . . . . . . . . . . . . . . . . . .
5.3. Computation . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 31
. . . . . . . 36
. . . . . . . 46
. . . . . . . 49
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
APPENDIX A. INDEX OF NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
v
LIST OF TABLES
Page
Table 1. Examples of Generators of PSL(Od ) (Part one). . . . . . . . . . . . . . . . . 50
Table 2. Examples of Generators of PSL(Od ) (Part two). . . . . . . . . . . . . . . . . 51
Table 3. Special Matrix Values Generators of PSL(Od ). . . . . . . . . . . . . . . . . . 52
vi
CHAPTER I
INTRODUCTION
Let K be a totally real algebraic number eld. The Hilbert modular group of K
is PSL2 (OK ) where OK is the ring of integers of K .
Research on the Hilbert modular groups began around 1893 with David Hilbert,
who later proposed the study of the function theory of PSL2 (OK ) to his student Ludwig Otto Blumenthal. With this, Hilbert and Blumenthal hope to create a theory of
modular functions of multiple variables that could be as important to number theory
as their classical counterparts. Blumenthal [Blu03] gave a detailed outline of the function theory used in his work. However, Blumenthal's construction of a fundamental
domain had a aw where he concluded that a fundamental domain of the action had
only one cusp as in the classical modular group. This was later corrected by Hans
Maaÿ who proved that the number of cusps are equal to the class number of K [Maa].
In this thesis, we rst give the background needed for understanding the Hilbert
modular groups, such as denitions, theorems and their applications. We then give
methods for calculating the generators of the Hibert modular groups for special cases
√
where K = Q( d) is a real quadratic eld along with methods for a general real
number eld of degree n.
1
CHAPTER II
THE MODULAR GROUP
2.1 Actions of the Special Linear Group on H
Denition 2.1. The general linear group of degree 2 over the reals, denoted GL2 (R),
is dened by
(
GL2 (R) =
Denition 2.2.
a b
!
c d
)
: ad − bc 6= 0, a, b, c, d ∈ R .
The special linear group of degree 2 over the reals, denoted SL2 (R),
is dened by
(
SL2 (R) =
a b
!
c d
)
: ad − bc = 1, a, b, c, d ∈ R .
Denition 2.3. The projective special linear group of degree 2 over the reals, denoted
PSL2 (R), is the quotient
PSL2 (R) = SL2 (R)/{E, −E}.
where E =
1 0
0 1
!
.
Let H be the complex upper half-plane,
H = {z ∈ C : =z > 0}.
2
The group SL2 (R) acts on H via the Möbius transformations
SL2 (R) × H → H
where, for any M =
a b
!
c d
∈ SL2 (R) and any z ∈ H, we have
Mz =
az + b
.
cz + d
The following lemma that we have proven shows that this action is closed in H.
Lemma 2.4. For any M ∈ SL2 (R) and any z ∈ H, M z ∈ H.
Proof. Let M ∈ SL2 (R), and let z ∈ H. Then we have
Mz =
=
(a<(z) + b) + ia=(z)
az + b
=
cz + d
(c<(z) + d) + ic=(z)
[(a<(z) + b) + ia=(z)] [(c<(z) + d) − ic=(z)]
|cz + d|2
ac<(z)2 + (bc + ad)<(z) + bd + ac=(z)2
ad=(z) − cb=(z)
=
+i
2
|cz + d|
|cz + d|2
=
ac<(z)2 + (bc + ad)<(z) + bd + ac=(z)2
=(z)
+i
.
2
|cz + d|
|cz + d|2
3
Thus, we can see that
=(M z) =
=(z)
> 0.
|cz + d|2
Therefore, M z is contained in H.
Let M =
a b
!
c d
∈ PSL2 (R), then M = −M in PSL2 (R) and, for any z ∈ H,
Mz =
−az − b
az + b
=
= (−M )z.
cz + d
−cz − d
Thus, this action descends to give an action of PSL2 (R) on H.
Now, we give some basic denitions pertaining to these group actions. (Compare
to [Fre90] and [Hir73]).
Denition 2.5.
A subgroup Γ ⊂ SL2 (R) is discrete if the intersection of Γ with any
compact subset K ⊂ SL2 (R) is a nite set.
Denition 2.6.
A subgroup Γ ⊂ SL2 (R) acts discontinuously on H if for any two
compact subsets K1 , K2 ⊂ H, the set
{M ∈ Γ : M (K1 ) ∩ K2 6= ∅}
is nite.
4
2.2 The Modular Group
In this section we give a brief explanation of the classical modular group and some
of its properties. We give more detailed results for the Hilbert modular groups later
in this paper.
Denition 2.7.
The classic modular group of Q is
(
PSL2 (Z) :=
a b
!
c d
)
: a, b, c, d ∈ Z, ad − bc = 1 /{E, −E}.
Remark. The group PSL2 (Z) is a discrete subgroup of PSL2 (R) that is a topological
group with discrete topology.
For the rest of this section we dene the special matrices of PSL2 (Z):
S=
Denition 2.8.
!
0 −1
1
0
, T =
1 1
0 1
!
.
Given an action of a group G on a topological space Y by homeo-
morphisms, a fundamental domain for this action is a set D of representatives for the
orbits.
Theorem 2.9
([Ser73], Chapter VII Ÿ1.2). Let D be the subset of H such that
1
D = z : |z| ≥ 1, |<(z)| ≤
.
2
Then D is a fundamental domain for the action of the classic modular group on H.
In other words,
5
(1) For every z ∈ H, there exists some M ∈ PSL2 (Z) such that M z ∈ D.
(2) Suppose that two distinct points z1 , z2 ∈ D are congruent modulo PSL2 (Z).
Then <(z1 ) = ± 21 and z1 = z2 ± 1, or |z1 | = 1 and z2 = − z11 .
(3) Let z ∈ D and let I(z) be the stabilizer of z in PSL2 (Z). We have I(z) = {1}
except in the following 3 cases:
(a) z = i, in which case I(z) = hSi;
(b) z = ρ = e2πi/3 , in which case I(z) = hST i;
(c) z = −ρ̄ = eπi/3 , in which case I(z) = hT Si.
Proof. !We prove statement 1 by showing that for any z ∈ H there exists M =
a b
c d
∈ PSL2 (Z) such that M z ∈ D. We know that
=M z =
=z
.
|cz + d|2
Since c and d are integers, the pairs (c, d) such that |cz + d| is less than a given
number is nite. Thus, there exists M 0 ∈ PSL2 (Z) such that =(M 0 z) is maximal.
Now choose an integer m such that T m M 0 z has real part between − 21 and 12 . Thus,
z 0 = T m M 0 z ∈ D since if |z 0 | < 1, then Sz 0 = − z10 would have an imaginary part
larger than z 0 which is impossible since we chose the (c, d) such that the imaginary
part of z 0 is maximal.
6
For statements 2 and 3, let z ∈ D and let M =
a b
!
∈ SL2 (Z) such that
c d
M z ∈ D. We can replace the pair (M, z) with (M −1 , M z), so without loss of generality, suppose that =(M z) ≥ =(z). Then |cz + d| ≤ 1, leaving only the cases
c = 0, 1, −1. If c = 0, then d must be ±1. Then M is a translation by ±b, and since
both <(M z) and <(z) are between − 21 and 12 , b must be 0 or ±1.
If c = 1, then |z +d| ≤ 1 means that d = 0 except for the case z = ρ (or −ρ̄) where
we have d = 0, ±1. The case d = 0 gives us |z| ≤ 1, thus |z| = 1. Also, this means that
b = −1, so M z = a − z1 . Thus a must be 0 unless z = ρ (or −ρ̄) in which case we have
a = 0, ±1. The case z = ρ, d = 1 gives a − b = 1 and M ρ = a − 1/(1 + ρ) = a + ρ, thus
a = 0, 1. Similarly when z = −ρ̄, d = −1. Finally, the case that c = −1 is equivalent
to the case where c = 1 since M ∈ PSL
! 2 (Z) implies that M = (−E)M with respect
1 0
to the action on H, where E =
. Therefore, statements 2 and 3 are true.
0 1
Theorem 2.10
([Ser73], Chapter VII Ÿ1.2). PSL2 (Z) is generated by S and T .
Serre gives a proof of this lemma based on the previous Theorem 2.9 for the
fundamental domain of SL2 (Z), but we will not cover the proof here since it can be
similarly proven with the same technique as the more dicult proof of Theorem 5.1,
as Z is Euclidean.
7
CHAPTER III
TOTALLY REAL NUMBER FIELDS
3.1 Denitions and Theorems
Denition 3.1. A number eld is a nite degree (and hence algebraic) eld extension
of the eld of rational numbers, Q.
Let K be a totally real number eld of degree n = [K : Q]. Then K has n dierent
embeddings into R (since K is totally real):
K → R, 1 ≤ i ≤ n
a 7→ a(i) ,
where a(i) is the ith embedding of a. We will combine these embeddings together in
a single mapping
K → Rn .
a 7→ (a(1) , a(2) , . . . , a(n) ).
We use a and the n-tuple (a(1) , a(2) , . . . , a(n) ) interchangeably.
Remark. Throughout this paper we give information for the general case of any totally
real number eld K as well as more specic information for the real quadratic eld
√
K = Q( d), where d > 0 is squarefree. From here on we consider K to be a general
real number eld unless otherwise specied.
8
A quadratic eld is an algebraic number eld of degree two. It is easily shown
that there is a bijection between the set of all quadratic elds and the set of all square
free integers d 6= 0.
√
Let d be a positive and square free integer, and let K = Q( d). The eld K
√
can be written as a vector space of degree two over Q, hence K = Q + Q d. The
discriminant of K is d if d ≡ 1 mod 4, and 4d otherwise.
Denition 3.2.
√
Let α = a + b d ∈ K , where a, b ∈ Q. The conjugate of α, denoted
α0 , is
√
α0 = a − b d
√
Let α ∈ K = Q( d). In this case, the embedding of K into R2 is given by
α 7→ (α(1) , α(2) ) = (α, α0 ).
Denition 3.3.
√
Let α ∈ K = Q( d). We dene the trace of α as
trK (α) = α + α0 ,
and the norm of α as
NK (α) = αα0 .
Denition 3.4.
The units of a ring R are all elements α ∈ K such that there exists
an element β ∈ R where αβ = βα = 1.
9
Denition 3.5.
An element α ∈ K is called an integer of K when α is the root to a
polynomial
xn + an−1 xn−1 + · · · + a1 x + a0 = 0
where a1 , . . . , an−1 are elements of Z and α satises no similar equation of degree less
than n. Equivalently, in the quadratic case, α ∈ K is an integer when its trace and
norm are elements of Z.
Denition 3.6.
An order is a subring O of K such that
(1) O spans K over Q, so that QO = K , and
(2) O is a Z-lattice in K .
Denition 3.7.
The ring of integers, also known as the maximal order, of K is the
subring of all integers of K . We denote the ring of integers of K as OK (or Od in the
quadratic case).
The ring of integers, OK , can be represented as a ring of degree n = [K : Q] over
Z, and OK takes the form Z + Zω1 + . . . + Zωn−1 where 1, ω1 , . . . , ωn−1 ∈ OK is an
integral basis of OK .
For the quadratic case, Od takes the form Z + Zω where
ω=




√
1+ d
2


 √d
if d ≡ 1
mod 4
otherwise.
10
From denition 3.2, it follows that if α = a + bω ∈ Od with a, b ∈ Z, the conjugate
α0 is
0
α =
Theorem 3.8

√
1− d

a
+
b
if d ≡ 1

2



 a − b√ d
mod 4
otherwise.
(Dirichlet Unit Theorem). Let K be a number eld with r1 real em-
beddings and 2r2 pairs of complex conjugate embeddings. The unit group of K is
generated with r1 + r2 − 1 independent generators of innite order. Moreover, where
r = r1 + r2 − 1, any order O in K contains multiplicatively independent units
1 , 2 , . . . , r of innite order such that every unit in O can be uniquely written as
m2
mr
1
ζ · m
1 · 2 · · · r
where ζ is a root of unity in O and mi ∈ Z, 1 ≤ i ≤ r. So
O× ∼
= µ(O) × Zr1 +r2 −1
where µ(O) is the nite cyclic group of roots of unity in O.
The unit group of OK , denoted OK× , takes the form
mn
1
OK× = {(−1)m0 m
1 · · · n : m0 , m1 , . . . , mn ∈ Z},
where 1 , 2 , . . . , n ∈ OK are units of OK .
11
For the quadratic case, we can give a more specic denition for the unit group
and fundamental unit (in the quadratic case there exists only one generator of the
unit group Od× ).
√
Theorem 3.9. Let K = Q( d) be a real quadratic eld and D its discriminant.
Then the unit group Od× of the maximal order Od of K is
Od× = h−1, i = {−1a · b : a ∈ Z, b ∈ Z}
where =
√
a+b D
2
and (a, b) is the smallest solution to (with respect to )
a2 − Db2 = ±4.
The element is called the fundamental unit of K .
3.2 Dedekind Domains
In this section, we give important denitions and results for Dedekind domains.
These are important because OK is a Dedekind domain, and taking advantage of these
results makes calculations much easier in later chapters.
Denition 3.10.
A Noetherian ring is a ring in which every non-empty set of ideal
has a maximal element.
A ring is Noetherian if the ascending chain condition on ideals holds, which means,
given any ideal chain:
I1 ⊆ I2 ⊆ I3 ⊆ . . .
12
there exist an N ∈ N such that, for all n ≥ N , n ∈ N,
In = IN .
There are some useful results based on Noetherian rings that we may need.
Denition 3.11.
We say that an ideal I is a primary ideal if whenever ab ∈ I , then
either a ∈ I or bm ∈ I for some m > 0.
Theorem 3.12
(Lasker-Noether Theorem). Let R be a Noetherian ring, then every
ideal of R can be expressed as a an intersection of nitely many primary ideals.
Denition 3.13.
A Dedekind domain R is an integral domain satisfying
(1) R is a Noetherian ring.
(2) R is integrally closed.
(3) Every nonzero prime ideal of R is maximal.
Proposition 3.14. Let I be a non-zero ideal in the ring of integers OK . Then I is a
free Z-module of rank n and generated by a Q-basis for K .
Theorem 3.15. Let R be an integral domain, and let S be an integral domain that
contains R. Let I be an ideal of S , and suppose I contains a non-zero element α
satisfying a non-zero polynomial f (x) ∈ R[x]. Then I ∩ R 6= 0.
Proof. Because S is an integral domain, we may factor out any powers of x dividing
f (x), and can therefore assume that f (0) 6= 0, but
α | (f (α) − f (0)).
13
Thus −f (0) = f (α) − f (0) ∈ I ∩ R, as desired.
Corollary 3.16. Let I be a non-zero ideal in a ring of integers OK . Then, OK /I is
nite.
Proof. Suppose that I contains some non-zero integer m ∈ Z. Then OK /I is a
quotient of OK /(m), which is isomorphic as a Z-module to (Z/mZ)2 , hence both are
nite.
Denition 3.17.
Let I be a non zero ideal in OK . The norm of I is dened by
N (I) := |OK /I|.
Theorem 3.18. The ring of integers, OK , is a Dedekind domain. Thus, OK satises
(1) OK is a Noetherian ring.
(2) OK is integrally closed.
(3) Every nonzero prime ideal of OK is maximal.
Lemma 3.19. Let R be a Dedekind domain, and let a be a non-zero element in an
ideal I ⊆ R. Then there is an element b ∈ R such that I = (a, b). In particular, every
ideal in R can be generated by two or less elements.
3.3 Class Groups
Denition 3.20.
Let K be a number eld and R a ring such that R ⊂ K , we say
that an ideal a of K is a fractional ideal of K if there exists an element b ∈ R such
14
that
ba = {bx : x ∈ a}
is an ideal in R. To distinguish ideals and fractional ideals, we also call the ideals of
the maximal order of K integral ideals.
Let K be a number eld, let IK the group of fractional ideals, and let HK its
subgroup of principal fractional ideals. Then the quotient ClK := IK /HK is called
the class group of K . The class group is a nite abelian multiplicative group, and the
cardinality of this group is the class number of K .
Denition 3.21.
Let a ∈ IK be a fractional ideal, and let OK be the maximal order
of K . Then we dene
a−1 := {b ∈ K | ba ⊆ OK }.
So, aa−1 = OK
Denition 3.22.
Let K be a number eld. Let m0 ∈ IK be an integral ideal and m∞
a formal product of real innite places of K . Then m := m0 m∞ is called a congruence
module.
For the next denition, we say vp (α) = m if α ∈ pm (pm |α) and α 6∈ pm−1
(pm−1 6 | α) where p is a prime ideal.
Denition 3.23.
Let α(i) ∈ C be the ith embedding of α ∈ K . Multiplicative
congruences with respect to a nite place contained in the congruence module are
15
dened by
α ≡ 1 mod× pm if and only if vp (α − 1) ≥ m
(i)
and for a real innite place p∞ , which is the ith embedding of p∞ into R, by
m
(i)
α ≡ 1 mod× (p(i)
> 0.
∞ ) if and only if α
Denition 3.24.
Let I m := {a ∈ IK : gcd(a, m0 ) = 1} and Hm := {(α) ∈ HK | a ≡
1 mod× m}. Then the ray class group modulo m is dened as
Clm := I m /Hm .
16
CHAPTER IV
HILBERT MODULAR GROUPS
4.1 Hilbert Modular Groups
Extending Ÿ3.1, we dene an embedding GL2 (K) → GL(R)n by:
a b
!
c d
= M 7→ (M (1) , M (2) , . . . , M (n) ), where M (i) =
a(i) b(i)
c(i) d(i)
!
, 1 ≤ i ≤ n.
We use M and the n-tuple image of its embedding into GL2 (R)n interchangeably.
Denition 4.1.
The Hilbert modular group of K is
ΓK = PSL2 (OK ) = SL2 (OK )/{±E}.
n
The Hilbert modular
! group acts on H , the product of n copies of the upper half
a b
∈ PSL2 (OK ) and z = (z1 , . . . , zn ) ∈ Hn ,
plane. For M =
c d
M z = M (z1 , . . . , zn ) = (M
(1)
z1 , . . . , M
(n)
zn ) =
a(1) z1 + b(1)
a(n) zn + b(n)
,
.
.
.
,
c(1) z1 + d(1)
c(n) zn + d(n)
,
This gives us a well dened action of ΓK on Hn .
4.2 Fixed Points
Denition 4.2.
Let M ∈ PSL2 (OK ). We say that a point z ∈ Hn is a xed point of
M when M z = z .
17
Denition 4.3.
Let M =
a b
!
c d
∈ PSL2 (R). We say
(1) M is elliptic if |a + d| < 2.
(2) M is parabolic if |a + d| = 2.
Similarly, we say M ∈ PSL2 (OK ) is elliptic or parabolic if each embedding of M is
elliptic or parabolic, respectively.
Lemma 4.4. An element of PSL2 (OK ) has a xed point if and only if it is elliptic.
Proof. Suppose M =
a b
!
∈ PSL2 (OK ) has a xed point z ∈ Hn . From here, we
c d
will prove for only the rst embedding, each other embedding follows from this. We
have
a(1) z1 + b(1)
= z1 ,
c(1) z1 + d(1)
which implies
0 = c(1) z12 + (d(1) − a(1) )z1 − b(1) .
Using the quadratic formula, we have
z1 =
a(1) − d(1) ±
p
p
(d(1) − a(1) )2 + 4b(1) c(1)
a(1) − d(1) ± (d(1) + a(1) )2 − 4
=
.
2c(1)
2c(1)
18
Thus, applying this
! to each embedding, we can see that a matrix M ∈ PSL2 (OK )
1 0
where M 6= ±
has a xed point if and only if it is elliptic.
0 1
Denition 4.5.
For z ∈ Hn , we dene the isotropy subgroup of z in a subgroup
Γ ⊂ PSL2 (OK ) as
Γz := {M ∈ Γ : M z = z}.
Denition 4.6.
An element z ∈ Hn is called an elliptic xed point of a subgroup
Γ ⊂ PSL2 (OK ) if Γz is nontrivial.
Denition 4.7.
A group Γ is said to act discontinuously at a point z ∈ Hn if there
exist a neighborhood U of z such that M U ∩ U 6= ∅ for at most nitely many M ∈ Γ,
where M U = {M u : u ∈ U }.
Denition 4.8.
A subgroup Γ ⊂ PSL2 (OK ) is discrete if it acts discontinually on
the upper half plane.
Proposition 4.9. Let M be a matrix in PSL2 (OK ). Then the following are equivalent:
(1) M is elliptic or M = ±
1 0
0 1
!
.
(2) M is of nite order.
(3) M has a xed point in Hn .
Proposition 4.10. Let Γ ⊂ PSL2 (OK ) be a discrete subgroup. Then the stabilizer of
a point in Hn is a nite cyclic group.
The cardinality and number of these isotropy subgroups in Γ ⊂ PSL2 (OK ) (up
to conjugation) are necessary pieces of information for calculating invariants of the
19
Hilbert modular groups and the congruence subgroups. To nd these, we plan to
check in the embedding of the quotient of higher levels. We discuss this embedding
and the permutation representation of it in more detail later in the following section.
4.3 Congruence Subgroups
Denition 4.11.
Let n be an ideal of OK . The subgroup
(
ΓK (n) :=
a b
!
c d
)
∈ PSL2 (OK ) : a, d ≡ 1 mod n, b, c ≡ 0 mod n
of PSL2 (OK ) is called a principal congruence subgroup of level n of PSL2 (OK ).
Denition 4.12.
A congruence subgroup of PSL2 (OK ) is a group Γ such that it
contains a principal congruence subgroup:
ΓK (n) ≤ Γ ≤ PSL2 (OK ).
The next theorem is well known and important for making calculations on the
congruence subgroups using a homomorphism to the quotient of the congruence subgroup by the principal congruence subgroup. This will be discussed in greater detail
later in the section.
Theorem 4.13
(The Fourth Group Isomorphism Theorem). Let G be a group and
let N be a normal subgroup in G. There is a bijection from the set of subgroups A of
G that contain N onto the set of subgroups A = A/N of G = G/N . In particular,
every subgroup G is of the form A/N for some subgroup A of G containing N . This
bijection has the following properties: For all subgroups A, B ⊆ G with N ⊆ A and
N ⊆ B,
20
(1) A ⊆ B if and only if A ⊆ B ,
(2) if A ⊆ B , then [B : A] = [B : A],
(3) hA, Bi = hA, Bi, where hA, Bi denotes the subgroup generated by A and B ,
(4) A ∩ B = A ∩ B ,
(5) A E G if and only if A E G.
Lemma 4.14. Let n be an ideal of PSL2 (OK ). The principal congruence subgroup
ΓK (n) is normal in PSL2 (OK ).
!
a b
∈ PSL2 (OK ) and let
Proof. Let
c d
!
e f
∈ ΓK (n). Then
g h
e, h ≡ 1 mod n and g, f ≡ 0 mod n.
Using these properties we get
a b
!
c d
=
e f
g h
!
d
−b
−c
a
!
dae + dbg − caf − cbh a2 f + abh − bae − b2 g
dce + d2 g − c2 f − cdh acf + adh − bce − cdg
!
.
21
We can see that
dae + dbg − caf − cbh = 1 mod n,
a2 f + abh − bae − b2 g = 0 mod n,
dce + d2 g − c2 f − cdh = 0 mod n,
acf + adh − bce − cdg = 1 mod n.
Hence, Γd (n) E PSL2 (OK ).
Using Lemma 4.14 and the Fourth Isomorphism Theorem for Groups, we can investigate the congruent subgroups Γ as nite subgroups Γ/ΓK (n).
For an integer n ∈ N, we write ΓK (n) for ΓK ((n)), where (n) = n(OK ) is the
principal ideal generated by n.
Proposition 4.15
([Fre90, Remark 3.8, Corollary 3.9]). If n is a rational integer
greater than 2, the principal congruence subgroup ΓK (n) contains no trivial element
of nite order.
Proof. Let E =
1 0
!
. Every nontrivial nite group Γ contains an element of prime
0 1
order. Suppose, to the contrary, that there exist a prime p ∈ N and a matrix
M ∈ ΓK (n) such that M 6= E and M p = E.
Consider the matrix B := M − E and let b be the ideal generated by the coecients
of B . Since M ∈ ΓK (n), b is contained in (n). Using the binomial theorem,
22
p
p
M = (B + E) =
p X
p
k=0
k
B k E p−k = E,
which gives us
p X
p
B k = 02,2
k
k=1
(4.1)
where 02,2 is the zero matrix. Consider Equation (4.1) modulo b2 , we obtain
pB ≡ 02,2 mod b2 .
Hence pb ∈ b2 and, therefore,
(4.2)
p ∈ b ⊂ (n).
We know that for a prime p the binomial coecients
p
k
, 1 ≤ k ≤ p − 1 are divisible
by p. Therefore, for k ≥ 2, pB k ≡ 02,2 mod b3 and pB k ⊆ b3 . Since p ≥ 3 due to the
number of terms in the binomial expansion, we know that the coecients of B p are in
b3 and therefore, by Equations (4.1) and (4.2), pB ≡ 02,2 mod b3 , which implies that
p ∈ b2 ⊂ (n2 ). However, this is impossible since this would mean that n2 divides the
prime p, a contradiction. Therefore, ΓK (n) contains no element of nite order other
than E .
For now, we will restrict our work on congruence subgroups to the quadratic case.
Let Γd = PSL2 (Od ) and n ∈ Z with n ≥ 3. For calculations on the congruence
23
subgroups of level n, we plan to use a permutation representation of the congruence
subgroups of level n in Γd .
The action of Γd on H2 induces a permutation on the vectors Od /(n)×Od /(n). This
permutation group is a subgroup of the symmetric group Sm2 where m = |Od /(n)|. To
create this permutation representation, we use the action of the image of each generator of Γd in Γd /Γd (n) on Od /(n)×Od /(n) as the generators of the permutation group.
4.4 Cusps of the Surface Hn /PSL2 (OK )
n
To begin, we extend the action of PSL2 (R)n to H where
H = H ∪ R ∪ {∞}.
The action is the same Möbius transformation as before with the usual behavior with
∞: For any a ∈ R,
(1) ∞ + a = ∞,
(2)
∞
a
= ∞, and
(3)
a
∞
= 0.
Denition 4.16.
The projective line of K , denoted P1 (K), is the extension of the
usual line of K by a point at innity.
Denition 4.17.
The orbits of P1 (K) under a subgroup Γ ⊂ PSL2 (K) are called the
cusps (or cusp class) of Γ respectively. We denote a representative of a cusp, σ , as
σ := (α : β)
24
(where (1 : 0) is called the cusp at innity) which we will describe in the next
proposition. We give this notation for a cusp, (α : β), with respect to its associiated
ideal class (α, β) which we discuss next. A cusp (α, β), is an element of P1 (K) and
we say (α, β) = (γ, δ) if there exists and element a ∈ K such that aα = γ and aβ = δ .
For ease of understanding, we will refer to the cusps as dened as a cusp class, the
equivalance class of cusps dened by the action of PSL2 (OK ), and a representative of
that class as a cusp from here on.
Denition 4.18.
A matrix is said to be unimodular if it is a square integer matrix
that is invertible over the integers.
Proposition 4.19
([VDG88]). The association
(α : β) 7→ class of (α, β)
creates a bijective correspondence between the set of cusp classes of ΓK and the ideal
class group ClK of K . In particular, the number of cusp classes of ΓK is equal to the
class number of K .
Proof. Let σ and τ be cusps such that (where ∼ denotes equivalence of two elements
of the same class)
τ=
aσ + b
∼ σ.
cσ + d
Then the ideal class of τ can be represented by (aα+bβ, cα+dβ) ∼ (α, β) since
a b
!
c d
is unimodular.
25
Suppose that σ = (α : β) and τ = (γ : δ) are two cusps associated with the
same ideal class. With the multiplication by a suitable element of K , we assume
that (α, β) = (γ, δ) = a. Then we know that there exist α∗ , β ∗ ∈ a−1 such that
αβ ∗ + βα∗ = 1 and, similarly, there exists γ ∗ , δ ∗ ∈ a−1 such that γδ ∗ + δγ ∗ = 1.
Dene the matrices
Mσ =
α α∗
!
, Mτ =
β β∗
γ γ∗
δ δ∗
!
.
Notice that these matrices have determinant 1 and transforms the cusp ∞ = (1 : 0)
to their respective cusps:
Mσ (1 : 0) = (α : β) and Mτ (1 : 0) = (γ : δ).
We have
Mσ Mτ−1 =
α α∗
β β∗
!
δ∗
−γ ∗
−δ
γ
!
=
αδ ∗ − α∗ δ α∗ γ − αγ ∗
βδ ∗ − β ∗ δ β ∗ γ − βγ ∗
!
.
Hence, the matrix Mσ Mτ−1 ∈ PSL2 (OK ) and
Mσ Mτ−1 τ = Mσ (1 : 0) = σ.
Hence, Mσ Mτ−1 transforms τ to σ . Therefore, every ideal class is associated to a cusp
class.
26
Denition 4.20.
The set of multipliers of OK , denoted by Λ, is the set of squares of
units of OK :
Λ = {a2 : a ∈ OK× }.
Denition 4.21.
The translation module of a subgroup Γ of PSL2 (R)n is the set of
all a ∈ Rn such that there exist a matrix M ∈ Γ where M z = z + a for all z ∈ Hn .
Denition 4.22.
A subgroup of PSL2 (R)n is said to have a cusp ∞ if its translation
module is isomorphic to Zn and its set of multipliers of OK , is isomorphic to Zn−1 .
Corollary 4.23. PSL2 (OK ) has cusp ∞.
Proof. We have that the translation module of PSL2 (OK ) is OK ∼
= Zn . Also, by
Theorem 3.9 and Theorem 3.8, we have
OK× ∼
= µ(OK ) × Z = {+1, −1} × Zn−1 .
Therefore, Λ ∼
= Zn−1 , where Λ is the set of multipliers of OK .
Proposition 4.24. The set of all cusps of ΓK is
K ∪ {∞}.
Proof. We know that ΓK has cusp ∞ by the previous proposition. We say that
PSL2 (OK ) has cusp c = (α : β) (c 6= ∞) if there exist a matrix
Ac :=
α α∗
!
β β∗
27
where α∗ , β ∗ ∈ c−1 , c = (α, β), such that the group
A−1
c ΓK Ac
has cusp ∞.
Now, let c ∈ K , then we can take the matrix
0
Ac =
1
!
−1 c
which transforms c to ∞. Thus,
2
A−1
c ΓK Ac = PSL2 (OK ⊕ c ),
where
(
PSL2 (OK ⊕ a) =
a b
!
)
−1
: a, d ∈ OK , b ∈ a , c ∈ a ,
c d
has cusp ∞. Thus ΓK has cusp c. Therefore K ∪ {∞} are cusps of ΓK .
Remark. Every cusp of ΓK is a xed point of a parabolic element of ΓK . From this
we can see that there can exist no cusps outside of K ∪ {∞} since, for a cusp k , there
must exist
M=
a b
!
c d
6=
1 0
!
0 1
28
such that a + d = 2 and M k = k , which is only true for k =
a−d
2c
∈ K.
Proposition 4.25. Let m be an ideal of K . The number of cusp classes of ΓK (m) is
the ray class number hm = #Clm .
Proof. A cusp κ ∈ K corresponds (Proposition 4.19) to the ideal (κ, 1).
We rst !
show that equivalent cusps correspond to the same ideal class in Clm . Let
a b
M=
∈ ΓK (m). We have
c d
(κ, 1) = ((κ, 1) · M t ) = (aκ + b, cκ + d)
Set κ0 = M κ =
aκ+b
.
cκ+d
Since M ∈ ΓK (m) we have c ∈ m and d ≡ 1 mod m. Thus
(ck + d) ∈ Hm and therefore for the ideal classes we have
(κ, 1)Hm =
aκ + b
, 1 Hm .
cκ + d
If (κ0 , 1) and (κ, 1) are in the same ideal class in Clm then there is α ∈ K such that
(κ0 , 1) = α(κ, 1) = (ακ, α).
There are a, b ∈ (ακ, α)−1 with b ∈ m such that aα − ακb = 1. Let
29
M=
=
=
1 0
!−1
κ0 1
1
a
b
!
ακ α
0
!−1
−κ0 1
a
a
b
!
ακ α
b
−aκ0 + ακ −bκ0 + α
!
.
By construction, we have that det M = 1. Also, by Lemma 3.21, we have M ∈
SL2 (OK ). Thus a =
1
α
+ bκ ∈ OK . Since α ≡ 1 mod m and b ∈ m, we have a ≡
1 mod m and thus M ∈ ΓK (m).
30
CHAPTER V
FUNDAMENTAL DOMAIN AND GENERATORS
5.1 Fundamental Domain and Known Generators
We begin by dening the common matrices for simplicity throughout this chapter.
Our rst two methods of nding generators require that K to be a real quadratic eld.
√
For K = Q( d), we dene the matrices
S=
!
0 −1
1
0
, and U =
0
0 −1
!
,
where is the fundamental unit of Od . Also, for each a ∈ K , we dene the matrix
Ta =
1 a
!
.
0 1
Remark. These are natural relations that hold for these matrices:
S2 =
S3 =
!
−1
0
0
−1
0
1
−1 0
= −E,
!
= −S,
31
SU =
!
0 −1
1
0 −1
0
S 3 U =
0
!
0 −−1
=
0
−1
−
0
0
!
,
!
,
SU S 3 U = (SU )4 = SU SU = E.
In [Deu86], Jesse Ira Deutsch states the next theorem for the generators of Euclidean Od . Instead of a proof, he gives aa example and says that it can be generalized.
Thus, we have given our own proof for this below.
Theorem 5.1 ([Deu86]). If Od is Euclidean, then PSL2 (Od ) is generated by S, T1 , Tω ,
and U .
Proof. For a matrix of the form
1 α
0 1
!
∈ PSL2 (Od ),
where α ∈ Od , let α = α1 + α2 ω , α1 , α2 ∈ Z, then we have
Tα = T1α1 Tωα2
(1.1)
Consider the basic case,
32
a b
!
∈ PSL2 (Od ),
0 d
Then we have that ad = 1. So a = ±n and d = a−1 = ±−n . Thus, where
b−n = z1 + z2 ω , z1 , z2 ∈ Z, using (1.1),
a b
!
n n (z1 + z2 ω)
=
c d
=
!
−n
0
0
!n
1 1
−1
0 !z 1
0 1
a b
=
!
n
0
0
−n
1 ω
1 z1 + z2 ω
0
!
1
!z2
= Un T1z1 Tωz2 .
0 1
!
be an arbitrary element of PSL2 (Od ). Since
c d
Od is a Euclidean domain, there exists a Euclidean algorithm for entries c and d:
Now, for the general case, let
c = q 0 d + r0
d = q1 r0 + r1
r0 = q2 r1 + r2
..
.
.
rn−2 = qn rn−1 + rn
rn−1 = qn+1 rn ,
where q0 , . . . , qn+1 , r0 , . . . , rn ∈ Od
We can see that, multiplying by −E as needed,
a b
!
c d
=
a
b
q0 d + r0 d
!
=
b bq0 − a
d
−r0
!
1 −q0
0
1
!
!
0 −1
1
0
.
33
Also,
b bq0 − a
d
!
−r0
−bq0 + a (bq0 − a)q1 + b
=
r0
!
r1
1 q1
0
!
1
!
0 −1
1
0
.
Thus, by application of this method by induction up to index n + 1, as given in the
Euclidean algorithm, we can see
a b
!
c d
=
= ... =
−bq0 + a (bq0 − a)q1 + b
r0
α
β
0 ±rn
!
1 q1
0
r1
!
!
0 −1
1
0
1 q1
···
0
!
!
!
!
0 −1
1 −q0
0 −1
1
1
0
0
1
1
!
!
!
1 −q0
0 −1
0 −1
!
1
1
0
0
1
1
0
0
.
Thus, we can apply the basic case to the matrix
α
!
β
0 ±rn
,
and we apply (1.1) to the matrices
1 ±qi
0
1
!
.
Therefore, PSL2 (Od ) is generated by S, T1 , Tω , and U .
34
In [Vas72], Vaserstein proves that, for any Dedekind domain of arithmetic type R
with innitely many invertible elements, SL2 (R) is generated by elementary matrices
(in an algerbraic sense). In [May07], Sebastian Mayer gives the following lemma which
is a special case of Vaserstein's Theorem.
Lemma 5.2
([May07], Lemma 1.2.21, page 32). If d ≡ 1 mod 4, then PSL2 (Od ) is
generated by the set of matrices
(
1 a
0 1
!
)
: a ∈ Od
(
∪
1 0
!
)
: b ∈ Od
b 1
.
Corollary 5.3. If d ≡ 1 mod 4, then PSL2 (Od ) is generated by the matrices S , T1
and Tω .
Proof. It suces to show that all upper and lower triangular matrices of Lemma 5.2
can be obtained with these matrices.
Let a ∈ Od , denote a as a = a1 + a2 ω where a1 , a2 ∈ Z. Then
1 a
0 1
!
= T1a1 Tωa2 .
Similarly, let b ∈ Od , write b as b = b1 + b2 ω where b1 , b2 ∈ Z. Then
S 3 T1−b1 Tω−b2 S = S 3
1 −b
0
1
!
S
35
=
0
1
−1 0
!
1 −b
0
1
!
!
0 −1
1
0
=
1 0
!
b 1
.
5.2 Fundamental Domain and Generators of PSL(OK ) in the General Case
In this section, we follow an arithmetical approach to computing the fundamental
domain given by Tsueno Tamagawa. First we dene some useful notations for z =
(z1 , . . . , zn ) ∈ Cn :
The norm of z (Note that this particular denition does not behave the same as a
typical norm):
N z = z1 · . . . · zn .
The trace of z :
Sz = z1 + . . . + zn .
The absolute value of z :
|z| = (|z1 |, . . . , |zn |).
The real part of z :
<z = (<z1 , . . . , <zn ).
36
The imaginary part of z :
=z = (=z1 , . . . , =zn ).
Let V be a Euclidean vector space of dimension m, let (x, y) denote the inner
1
product for x, y ∈ V , and let kxk = (x, x) 2 denote the magnitude of x ∈ V . A lattice
ΛV is a subgroup of V such that
ΛV = Zα1 + Zα2 + . . . + Zαm
where α1 , α2 , . . . , αm is a set of linearly independent vectors. We use
p
|(αi , αj )| to
denote the determinant of ΛV , which is the volume of a fundamental parallelotope of
ΛV .
Denition 5.4.
We say that a vector x ∈ V is ΛV -reduced if kx + αk ≥ kxk for all
α ∈ ΛV . The set of all ΛV -reduced vectors is denoted ΠΛV .
Lemma 5.5
([Tam59], page 242). The set ΠΛV is a convex set dened by a nite
number of inequalities of inner products
±2(x, γv ) ≤ (γv , γv )
(v = 1, 2, . . . , s)
where γ1 , γ2 , . . . , γs are suitable vectors in ΛV , and ΠΛV is a fundamental domain of
the translation group {x 7→ x + α : α ∈ ΛV , x ∈ V }. Furthermore, for a subspace
W ⊂ V with dimension equal to the rank of the subset {y ∈ ΛV : N y = 1}, the
intersection ΠΛV ∩ W is compact and x ∈ V is in ΠΛV if and only if x = x1 + x2
where x1 ∈ ΠΛV ∩ W , x2 is an element of V , and x2 is orthogonal to W .
37
Let
Y = {y ∈ Rn : y1 > 0, . . . , yn > 0}.
Let Y1 = {y ∈ Y : N y = 1}, and let log y = (log y1 , . . . , log yn ) for y = (y1 , . . . , yn ).
Let ΛK = {log |u| : u ∈ K × } where K × is the group of units of K . Then ΛK is a
rank n−1 lattice in Rn , and ΛK spans the hyperplane H = {x : x ∈ K, Sx = 0} ⊂ Rn .
Let SΛK (x, y) be a positive denite inner product on Rn for x = (x1 , . . . , xn ), y =
(y1 , . . . , yn ). Taking Rn as a Euclidean space with SΛK as its inner product, a vector
x is orthogonal to H if and only if x is a scalar multiple of (1, . . . , 1) ∈ Rn .
Denition 5.6.
A vector y ∈ Y is called strongly reduced if log y is ΛK -reduced.
Let R be the set of all strongly reduced elements of Y , and let R1 = R ∩ Y1 . From
the previous lemma, we have the following:
Lemma 5.7 ([Tam59], page 242). The set R is bounded by a nite number of analytic
hypersurfaces in Y and R1 is compact. For every y ∈ Y , there exist a unique ε ∈ K ×
such that |ε|y ∈ R.
Corollary 5.8
([Tam59], page 243). If (y1 , . . . , yn ) ∈ R, then there is a constant
c > 1 that depends only on K such that for 1 ≤ i ≤ n we have
c−1 N
Denition 5.9.
√
√
n
y ≤ yi ≤ cN n y.
We call y ∈ Y reduced if
√
y=
√
√ y1 , . . . , yn is strongly reduced.
38
The set of all reduced y ∈ Rn is the set
R2 = {y 2 : y ∈ R}.
By Lemma 5.7, for y ∈ Y , there is ε ∈ K × such that ε2 y ∈ R2 . Thus, if y is reduced,
we have
ε−2 N
√
n
y ≤ y (1) , . . . , y (n) ≤ ε2 N
√
n
y.
Let a be an ideal of OK = Z + ω1 Z + . . . + ωn−1 Z. Then a is a rank n lattice of
Rn . Let
Πa = {x ∈ Rn : S(x + α)2 ≤ Sx2 for all α ∈ a}
be the set of all a-reduced vectors of Rn as dened in Denition 5.4. By Lemma 5.5,
the set Πa is a compact and convex polyhedron that is symmetric with respect to 0,
and for every x ∈ Rn , there exist an element α ∈ a such that x + α ∈ Πa .
Let A(a) be the group of all ane transformations z → ε2 z + α with ε ∈ OK× and
α ∈ a. The matrices that correspond to these transformations by the action of the
Möbius transformation are of the form
ε
β
0 ε−1
!
,
β = ε−1 α.
We can see that if a ⊂ OK then A(a) is a subgroup of ΓK . For an element z ∈ Hn ,
we say that z is a-reduced if =z = (=zn , . . . , =zn ) is reduced and <z = (<z1 , . . . , <zn )
39
is a-reduced. Take
G(a) = {z ∈ Hn : z is a-reduced},
then we see that G(a) is a fundamental domain of A(a) and is bounded by nitely
many analytic surfaces in Hn .
From here on we denote a pair of elements γ, δ ∈ K by hγ, δi to distinguish them
from the embeddings dened in previous chapters.
Denition 5.10.
Let hγ, δi be a non-zero pair of elements in K . We say that the
pairs hγ, δi and hγ ∗ , δ ∗ i are associated if there exists an element ε ∈ OK× such that
hεγ, εδi = hγ ∗ , δ ∗ i (εγ = γ ∗ and εδ = δ ∗ ).
We say that two pairs of elements are non-associated pairs if no such ε ∈ OK× exists.
Lemma 5.11
([Tam59], page 243). Let a and b be ideals of OK and let z ∈ Hn . For
any constant m > 0, there exist only a nite number of non-associated pairs hγ, δi
with γ ∈ a and δ ∈ b such that
N |γz + δ| < m,
where γz + δ = (γ (1) z1 + δ (1) , . . . , γ (n) zn + δ (n) ).
Proof. Let hγ, δi be a pair of elements where γ ∈ a and δ ∈ b. Choose ε ∈ K × such
that |εγz + εδ| is strongly reduced. Let hεγ, εδi = hγ ∗ , δ ∗ i. From Corollary 5.8, there
40
exists c > 1 where
√
√
|(γ ∗ )(1) z1 + (δ ∗ )(1) | < c m, . . . , |(γ ∗ )(n) zn + (δ ∗ )(n) | < c m.
Since z ∈ Hn , =z is positive (=z1 , . . . , =zn > 0, where z = (z1 , . . . , zn ) ∈ Hn ), then
the set {γz + δ; γ ∈ a, δ ∈ b} is a lattice in Cn and the domain dened by
√
|z1 |, |z2 |, . . . , |zn | < c m
is bounded. Thus there exist only nitely many pairs hγ, δi such that the inequality
N |γz + δ| < m is satised.
Lemma 5.12 ([Tam59], page 243). Let a be an ideal of OK . There exists a constant
c1 > 0, depending only on OK such that for every z ∈ Hn with N (=z) < c1 , there
exists a pair hγ, δi where γ ∈ a and δ ∈ a−1 such that
N |γz + δ| < 1,
where γz + δ = (γ (1) z1 + δ (1) , . . . , γ (n) zn + δ (n) ).
Let a1 , . . . , ah be a set or representatives of the ideal classes of K such that
(1) each ai is integral, and
(2) ai has minimum norm among integral ideals in its class.
We assume that a1 represents the principle class, hence a1 = OK .
Let Fi be the set of all z ∈ Hn such that
41
(1) z is a−2
reduced, and
i
(2) N |γz + δ| ≥ 1 for all pairs hγ, δi with γ ∈ ai and δ ∈ a−1
i .
Lemma 5.13
([Tam59], page 244). Let ai be the representative ideal of the ith ideal
class. There exist only a nite number of non-associated pairs hγ, δi with γ ∈ ai and
δ ∈ a−1
such that N |γτ + δ| = 1 for some τ ∈ Fi
i
From Lemma 5.13, we can see that the innite number of inequalities dening Fi
can be represented by a nite subset of inequalities among them. Thus, Fi is bounded
by a nite number of analytic surfaces in Hn .
Let ai (1 ≤ i ≤ h) be the representatives of the ideal classes of K . Let γi and
δi be integers generating ai . Then there exist αi , βi ∈ a−1
such that αi δi − βi γi = 1,
i
which we show later in this chapter in Theorem 5.17. For 2 ≤ i ≤ h we denote by Ai
the linear fractional transformation
z→
α i z + βi z
.
γi z + δi
For i = 1 we set A1 to the identity transformation I .
Theorem 5.14
([Tam59], page 245). The set
−1
F1 ∪ A−1
2 F2 ∪ . . . ∪ Ah Fh
is a fundamental domain of the group ΓK .
Proof. Let z ∈ Hn . By Lemma 5.11 there exists a pair hγ, δi, γ, δ ∈ OK , such that
N |γz + δ| ≤ N |γ ∗ z + δ ∗ | for all γ ∗ ∈ OK and δ ∗ ∈ OK such that hγ ∗ , δ ∗ i =
6 h0, 0i. We
42
have two cases for such pairs of integers:
Case 1:
If N |γz + δ| ≥ 1 we can nd a unit ε ∈ OK× and an integer α ∈ OK such that
ε2 z + α ∈ G(OK ).
Then z ∗ = ε2 z + α is equivalent to z with respect to ΓK and z ∗ is a point of the set
F1 .
Case 2:
If N |γz + δ| < 1 let a be the ideal generated by γ and δ and ai the
representative of its class. Then there exists an element κ ∈ a−1
such that a = κai ,
i
so κ−1 (γ, δ) = ai . So we have that κ−1 γ, κ−1 δ ∈ ai and
N |κ−1 γz + κ−1 δ| = N |κ|−1 N |γz + δ|.
With the assumption on γ and δ , we have that N |κ| ≤ 1. On the other hand, from
the assumption on ai we have that N |κ| ≥ 1. Hence N |κ| = 1.
Let M be a matrix such that
hγi , δi i = hγ, δiM = hκ−1 γ, κ−1 δi,
where γi and δi are the generators of the representative class ai . We have
N |γi M z + δi | ≤ N |γ ∗ M z + δ ∗ |
43
for every pair hγ ∗ , δ ∗ i. For every hµ, νi such that µ ∈ ai and ν ∈ a−1
i , we have
µAi M z + ν = (µ(αi M z + βi ) + ν(γi M z + δi ))(γi M z + δi )−1
= (µ∗ M z + ν ∗ )(γi M z + δi )−1 ,
where µ∗ = αi µ + γi ν ∈ OK and ν ∗ = βi µ + δi ν ∈ OK . Thus, we have
N |µAi M z + ν| ≥ 1.
(5.1)
−2
Let L be a transformation in A(a−2
i ) such that LAi M z ∈ G(ai ). Then LAi M z ∈ Fi ,
−1
−1
−2
hence A−1
i LAi M z ∈ Ai Fi . Since Ai A(ai )Ai ⊂ ΓK , we have
−2
A−1
i A(ai )Ai M ∈ ΓK .
Remark. For this previous proof of Tamagawa's theorem, we excluded the part that
shows uniqueness up to the boundaries of A−1
i Fi . The complete proof is given in
Tamagawa's work.
We now use Tamagawa's proof of Theorem 5.14 to construct a set of generators
for a generic group ΓK .
Theorem 5.15. Let h be the class number of K and n = [K : Q]. Let {γi , δi } be
a set of generators for the ideal ai , and let {γ̂i , δ̂i } be the set of generators for a−2
i
where 1 ≤ i ≤ h. Let 1 , 2 , . . . n be the generators of the unit group of OK and let
44
1, ω1 , . . . , ωn−1 be the basis of OK . Let L be the subset of OK dened as
L=
[ −1
−1
−1
A−1
i U1 Ai , . . . , Ai Un Ai , Ai Tγ̂i Ai , Ai Tδ̂i Ai .
1≤i≤h
Then the subset of ΓK ,
L ∪ {S, T1 , Tω1 , . . . , Tωn−1 , U1 , . . . , Un },
generates ΓK .
Proof. As we have seen in Theorem 5.14, for any element z ∈ H, we can be trans−1
formed to an element of F1 ∪ A−1
2 F2 ∪ . . . ∪ Ah Fh . For any z ∈ H, we take the pair
hγ, δi ∈ OK × OK such that |zγ + δ| has the smallest norm. If N |zγ + δ| ≥ 1 as in
case 1, then we can construct the transformation z 7→ ε2 z + α given in Tamigawa's
proof using the set
{T1 , Tω1 , . . . , Tωn−1 , U1 , . . . , Un } ⊂ PSL2 (OK ).
For this, we set b = ε−2 α. Using {T1 , Tω1 , . . . , Tωn−1 } we construct
{U1 , . . . , Un } we construct
!
!
ε 0
1 b
0 ε
0 1
1 b
0 1
!
, and using
!
ε 0
. Then we have the transformation
0 ε
z=
!
ε 0
0 ε
(z + b) = ε2 z + ε2 b = ε2 z + α
45
For the case!where N |zγ + δ| < 1, the matrix M from Tamagawa's proof is equal to
1 0
E=
unless z lies on the boundary of the fundamental domain, otherwise it
0 1
gives a unit k such that
hγ, δiM = hk −1 γ, k −1 δi.
Thus M can be constructed using the set {U1 , . . . , Un , S}. We use the dened
set L in our theorem to construct the transformation A−1
i LAi ∈ PSL2 (OK ) where
LAi ∈ A(a−2 ) and ai is the representative if the ideal class of (γ, δ). This gives us the
−1
transformation A−1
i LAi M z ∈ Ai Fi .
5.3 Computation
Currently for this section, we focus only on the quadratic case. We plan to extend
this application to higher degree number elds in the future. To compute the matrices
Ai , we will use an algorithm comparable to that of Algorithm 1.3.2 in [Coh00].
Algorithm 5.16
Input:
([Coh00], Ÿ1.3 page 17). Let R be a Dedekind domain.
Two coprime ideals a and b given by their HNF matrices A and B on
the integral basis of R.
Output: a ∈ a and b ∈ b such that a + b = 1.
1. Let C be the 2n × n matrix obtained by concatenating A and B , denoted (A|B).
Using HNF reduction, Compute an HNF matrix H and 2n × 2n matrix L such
that LC = (H|0).
2. Set X be the n-component column vector formed by the rst n entries of the rst
row of of the resulting matrix LC .
46
3. Let a be the element of K whose coordinate vector on the integral basis is X t A,
and set b to the value 1 − a. Output a and b.
Theorem 5.17 ([Coh00], Ÿ1.3 page 18). Let a and b be two (fractional) ideals in R,
an integer ring of a Dedekind domain K . Let a and b be two elements of K and set
o = aa + bb. There exist u ∈ ao−1 and v ∈ bo−1 such that au + bv = 1 and these
elements are found in polynomial time.
Proof. If a is equal to zero, we can take u = 0 and v = 1/b, since we have
1/b ∈ bo−1 = R/b.
Similarly, we can nd u and v where a is zero.
Assuming that both a and b are non-zero, set I = aao−1 and J = bbo−1 . By
the denition of o−1 , I and J are integral ideals and we have I + J = R. Thus by
our previous algorithm, we can nd in polynomial time e ∈ I and f ∈ J such that
e + f = 1, and u = e/a and v = f /b satisfy the desired conditions.
Algorithm 5.18
(Computation of Ai ). Let Od be the integer ring of the quadratic
eld K and a a representative of an ideal class of K .
Input:
γ and δ , the integers that generate a.
Output: α ∈ a−1 and β ∈ a−1 such that αδ − βγ = 1.
1. If δ = 0, then set (α, β) ← (0, 1/γ).
2. If γ = 0, then set (α, β) ← (1/δ, 0).
47
3. If γ and δ are both nonzero, as in Theorem 5.17:
a. a ← hδi and b ← hγi.
b. o ← δa + γb.
c. I ← δao−1 and J ← γbo−1 .
d. Pass I and J as input to Algorithm 5.16 with output e ∈ I and f ∈ J .
e. (α, β) ← (e/δ, f /γ).
4. Return α, β .
Now we have everything that we need to calculate the set of generators for all
cases of PSL2 (Od ). Thus we have the following algorithm:
Algorithm 5.19
Input:
(Computation of the Generators of PSL2 (Od )).
Od , the integer ring of the quadratic eld K .
Output: L, the list of generators of PSL2 (Od ).
1. Set to the fundamental unit of Od .
2. Let (γ1 , δ1 ), . . . , (γh , δh ) be representatives of the lowest norm of the classes of
Cl(Od ).
3. Set L ← [S, T, Tω , U ].
4. For 2 ≤ i ≤ h:
a. Get αi , βi from Algorithm 5.18 with γi and δi as input.
b. Set Ai ←
α i βi
γi δi
!
.
48
c. Set γ̂i and δ̂i to the generators of (γi , δi )−2 .
−1
d. Add the matrices A−1
i U Ai , Ai
1 γ̂i
0 1
!
Ai and A−1
i
1 δ̂i
0 1
!
Ai to the list
L.
5. Return L.
5.4 Examples
From Corollary 5.3 we see that the generators given in Theorem 5.1 and Theorem 5.15 are not necessarily a minimal set of generators for PSL2 (Od ), but a working
one nonetheless.
Using Theorem 5.15, Theorem 5.1, and Corollary 5.3 we give data of Γd up to
d = 59 in Tables 1, 2, and 3. Table 1 contains the ideal class representatives of lowest
norm and the fundamental unit of these quadratic elds. In Table 2, we list the set
of generators for Γd and the methos used to aquire them. Also, for elds with class
number 2 or higher, we give the values of the special matrices Ai , Tγˆi , and Tδˆi for
2 ≤ i ≤ h where needed. Note that for some sets provided, like that of O10 , the ring
of integers has a class number of two and there seems to be one generator missing
according to Theorem 5.15 in Table 2. This is due to the fact that, for the ideal a2 ,
a−2
2 is generated by a single element.
49
Table 1. Examples of Generators of PSL(Od ) (Part one).
Field
√
Q(√2)
Q(√3)
Q(√5)
Q(√6)
Q(√ 7)
Q(√10)
Q(√11)
Q(√13)
Q(√14)
Q(√15)
Q(√17)
Q(√19)
Q(√21)
Q(√22)
Q(√23)
Q(√26)
Q(√29)
Q(√30)
Q(√31)
Q(√33)
Q(√34)
Q(√35)
Q(√37)
Q(√38)
Q(√39)
Q(√41)
Q(√42)
Q(√43)
Q(√46)
Q(√47)
Q(√51)
Q(√53)
Q(√55)
Q(√57)
Q(√58)
Q( 59)
Ideal Class Representatives
(1)
(1)
(1)
(1)
(1)
(1), (2, ω)
(1)
(1)
(1)
(1), (2, 1 − ω)
(1)
(1)
(1)
(1)
(1)
(1), (2, ω)
(1)
(1), (2, ω)
(1)
(1)
(1), (3, 1 + ω)
(1), (2, 1 + ω)
(1)
(1)
(1), (2, 1 + ω)
(1)
(1), (2, ω)
(1)
(1)
(1)
(1), (3, ω)
(1)
(1), (2, 1 + ω)
(1)
(1), (2, ω)
(1)
Fundamental Unit
=1−ω
=2+ω
=ω
= −5 − 2ω
= −8 + 3ω
=3−ω
= 10 − 3ω
=1+ω
= 15 − 4ω
=4−ω
= 5 − 2ω
= 170 − 39ω
=2+ω
= −197 − 42ω
= −24 + 5ω
=5−ω
=2+ω
= −11 + 2ω
= −1520 + 273ω
= −19 − 8ω
= −35 − 6ω
=6+ω
= 7 − 2ω
= 37 − 6ω
= −25 + 4ω
= −27 − 10ω
= 13 + 2ω
= 3482 + 531ω
= 24335 − 3588ω
= 48 + 7ω
= −50 − 7ω
=3+ω
= −89 + 12ω
= 131 + 40ω
= −99 + 13ω
= 530 − 69ω
50
Table 2. Examples of Generators of PSL(Od ) (Part two).
Field
√
Q(√2)
Q(√3)
Q(√5)
Q(√6)
Q(√ 7)
Q(√10)
Q(√11)
Q(√13)
Q(√14)
Q(√15)
Q(√17)
Q(√19)
Q(√21)
Q(√22)
Q(√23)
Q(√26)
Q(√29)
Q(√30)
Q(√31)
Q(√33)
Q( 34)
√
Q(√35)
Q(√37)
Q(√38)
Q(√39)
Q(√41)
Q(√42)
Q(√43)
Q(√46)
Q(√47)
Q(√51)
Q(√53)
Q(√55)
Q(√57)
Q(√58)
Q( 59)
Method
5.1
5.1
5.3
5.1
5.1
5.15
5.1
5.3
5.15
5.15
5.3
5.1
5.3
5.15
5.15
5.15
5.3
5.15
5.15
5.3
5.15
5.15
5.3
5.15
5.15
5.3
5.15
5.15
5.15
5.15
5.15
5.3
5.15
5.3
5.15
5.15
Generators of Γd
S, T1 , Tω , U
S, T1 , Tω , U
S, T1 , Tω
S, T1 , Tω , U
S, T1 , Tω , U
−1
S, T1 , Tω , U , A−1
2 U A2 , A2 Tγˆ2 A2
S, T1 , Tω , U
S, T1 , Tω
S, T1 , Tω , U
−1
S, T1 , Tω , U , A−1
2 U A2 , A2 Tγˆ2 A2
S, T1 , Tω
S, T1 , Tω , U
S, T1 , Tω
S, T1 , Tω , U
S, T1 , Tω , U
−1
S, T1 , Tω , U , A−1
2 U A2 , A2 Tγˆ2 A2
S, T1 , Tω
−1
S, T1 , Tω , U , A−1
2 U A2 , A2 Tγˆ2 A2
S, T1 , Tω , U
S, T1 , Tω
−1
−1
S, T1 , Tω , U , A2 U A2 , A−1
2 Tγˆ2 A2 , A2 Tδˆ2 A2
−1
−1
S, T1 , Tω , U , A2 U A2 , A2 Tγˆ2 A2
S, T1 , Tω
S, T1 , Tω , U
−1
S, T1 , Tω , U , A−1
2 U A2 , A2 Tγˆ2 A2
S, T1 , Tω
−1
S, T1 , Tω , U , A−1
2 U A2 , A2 Tγˆ2 A2
S, T1 , Tω , U
S, T1 , Tω , U
S, T1 , Tω , U
−1
S, T1 , Tω , U , A−1
2 U A2 , A2 Tγˆ2 A2
S, T1 , Tω
−1
S, T1 , Tω , U , A−1
2 U A2 , A2 Tγˆ2 A2
S, T1 , Tω
−1
S, T1 , Tω , U , A−1
2 U A2 , A2 Tγˆ2 A2
S, T1 , Tω , U
51
Table 3. Special Matrix Values Generators of PSL(Od ).
Field
A2
√
Q( 10)
√
Q( 15)
3ω
1+ω
√
Q( 26)
√
Q( 30)
√
Q( 34)
√
Q( 39)
− 21 ω
2
−6 − 21 ω
ω
2
−7 − 21 ω
ω
2
8ω
1+ω
9ω
1+ω
√
Q( 42)
4
3
− 37 ω
3
33
2
− 21 ω
2
37
2
− 21 ω
2
−10 − 12 ω
ω
2
√
Q( 51)
√
Q( 58)
13
2
− 77
− 13 ω
3
1+ω
√
Q( 35)
√
Q( 55)
−2 − 21 ω
ω
2
5
ω
13ω
1+ω
1
ω
3
3
53
2
− 21 ω
2
−14 − 12 ω
ω
2
Tγˆ2
Tδˆ2
1
1 2
0 1
−
1
1 2
0 1
−
1
1 2
0 1
−
1
1 2
0 1
−
1 1
0 1
1
0
59
9
+ 55
ω
9
1
1
1 2
0 1
−
1
1 2
0 1
−
1
1 2
0 1
−
1
1 3
0 1
−
1
1 2
0 1
−
1
1 2
0 1
−
52
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Otto Blumenthal, Über Modulfunktionen von Mehreren Veränderlichen,
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Henri Cohen, Advanced Topics in Computational Number Theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR
1728313
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Jesse Ira Deutsch, Identities on Modular Forms in Several Variables Derivable from Hecke Transformations, Ph.D. thesis, 1986, Thesis (Ph.D.)
Brown University, p. 63. MR 2634902
[Fre90]
Eberhard Freitag, Hilbert Modular Forms, Springer-Verlag, Berlin, 1990.
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L. N. Vaser²ten, The Group SL2 Over Dedekind Rings of Arithmetic Type,
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53
APPENDIX A
INDEX OF NOTATION
α0
the conjugate of α ∈ K
the fundamental unit of Od
i
the ith fundamental unit of OK
H
the upper half plane
D
the discriminant
d
a positive, square-free integer
h
the class number of Od
K
√
the real quadratic eld Q( d)
NK (α)
the norm of α ∈ K
Sz
the trace of the vector z
SΛ (x, y) the inner product of vectors x and y in Λ
SL2 (R)
the special linear group of degree 2 of R
trK (α)
the trace of α ∈ K
54