The Modular Group SL_{2}(ℤ) and its congruences subgroups

Topics covered : Bump’s section 1.2 (The Modular Group) and Exercises 1.2.7 to 1.2.11

1. Notation

Let H denote the upper half place, i.e. {z = x + iy ∈ ℂ|y > 0}. Let SL_{2}(ℤ) be the

group of matrices \{( \begin{matrix}a & b \\ c & d \end{matrix} ) , ad − bc = 1, a, b, c, d ∈ ℤ\}. We use the notation ℜ and

ℑ: ℜ(z) + iℑ(z) = x + iy = z ∈ ℂ.

2. The Modular Group

2.1. Motivation. Automorphic functions for a group Γ are functions on Γ\H. We want to study functions that transform a bit differently under the action of discrete subgroups of SL_{2}(ℝ), called automorphic forms, such that the ratio of any two of them (of same weight) will yield an automorphic function.

Definition 2.1. A modular form of weight k for SL_{2}(ℤ) is a holomorphic function f : H ⟶ ℂ such that :

f(γ(z)) = (cz + d)^{k}f(z), γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ),

plus the requirement of holomorphicity at ∞ (the Fourier coefficents a_{n} = 0 ∀n < 0).

2.2. SL_{2}(ℤ)... The group G = SL_{2}(ℝ) acts on the upper half plane via fractional linear transformations :

γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) , z ↦ γ(z) = \frac{az + b}{cz + d}.

This action is not faithful since −I and I have the same effect; we thus need to mod out by \{±I\} (i.e. consider \overline{Γ} := Γ/Γ ∩ \{±1\}) if we require faithfulness. In a similar fashion, SL_{2}(ℂ) acts on ℙ^{1}(ℂ), and the subgroup of SL_{2}(ℂ) fixing H is SL_{2}(ℝ). The action on H is transitive: the subgroup of upper triangular matrices act transitively:

( \begin{matrix}\sqrt{y} & \frac{x}{\sqrt{y}} \\ 0 & \frac{1}{\sqrt{y}} \end{matrix} ) : i ↦ x + iy.

The stabilizer G_{i} of i is the special orthogonal group

S0_{2}(ℝ) := \{( \begin{matrix}a & b \\ −b & a \end{matrix} ) |a^{2} + b^{2} = 1\}.

By generalities on transitive group actions, G/G_{i} = SL_{2}(ℝ)/S0_{2}(ℝ) ≅ H.

Definition 2.2. The group Γ(N) is

\{SL_{2}(ℤ) ∋ ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) mod N\}.

For example, Γ(1) := SL_{2}(ℤ). Since Γ(N) = Ker(Γ(1) ↠ SL_{2}(ℤ/Nℤ)), Γ(N) is a normal subgroup of finite index.

Definition 2.3. A group Γ ⊂ G acts discontinuously if for all compact K_{1}, K_{2} ⊂ H, |{γ ∈ Γ|K_{2} ∩ γ(K_{1}) ≠ ∅}| < ∞.

Proposition 2.4. The group SL_{2}(ℤ) acts discontinously on H.

Proof. Let K_{1}, K_{2} be two compact subsets of H. There exists ϵ > 0 such ℑ(w) > ϵ ∀w ∈ K_{2}. Fix z = x + iy ∈ K_{1}. Since (c, d) ↦ |cz + d|^{2} is a positive definite quadratic form, ℑ(γ(z)) = \frac{y}{|cz+d|^{2}} < ϵ outside a big enough square |c|, |d| < R(z). Because K_{1} is compact, R = max_{z∈K_{1}}R(z) < ∞ and K_{2} ∩ γ(K_{1}) = ∅ unless |c|, |d| < R. This proves there are only finitely many bottom rows (c d) of matrices γ ∈ SL_{2}(ℤ) such that K_{2} ∩ γ(K_{1}) ≠ ∅. It remains to show that given c, d, there are only finitely many γ with bottom row (c d) such that K_{2} ∩ γ(K_{1}) ≠ ∅. If γ_{1} and γ_{2}, share a bottom

row, then γ_{1}^{−1} ∘ γ_{2} is of the form ( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) for n ∈ ℤ; this matrix correspond to a

translation z ↦ z + n. For fixed γ_{1}, there can be only finitely such translations such

that ( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) γ_{1}(K_{1}) ∩ K_{2} ≠ ∅.

□

Definition 2.5. A fundamental domain for Γ is an connected domain F such that • ∀z ∈ H, there exists γ ∈ Γ such that γ(z) ∈ \overline{F};

• if z_{1}, z_{2} ∈ F and γ(z_{1}) = z_{2} for some γ ∈ Γ, then z_{1} = z_{2} and γ = ±I. Proposition 2.6. The region \{z ∈ H||z| > 1, |ℜ(z)| < \frac{1}{2}\} is a fundamental domain

for Γ(1).

Proof. Pick z ∈ H, since (c, d) ↦ |cz + d|^{2} is a positive definite quadratic form, it has minimum value for integers c, d satisfying (c, d) = 1; thence ℑ(γ(z)) has a maximum value for some γ ∈ SL_{2}(ℤ). We can find n ∈ ℤ such that γ(z)+n has real part smaller or

equal to \frac{1}{2}. In fact, ( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) γ ∈ SL_{2}(ℤ) (with the same bottom row as γ is in SL_{2}(ℤ)

and produces this effect, hence |ℜ(z)| ≤ \frac{1}{2}. This implies |ℑ(γ(z))| ≥ 1, otherwise

we contradict maximality by taking γ_{1} = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) γ, |ℑ(γ_{1}(z))| = \frac{|ℑ(γ(z))|}{|γ(z)|^{2}}, thus

Property 1 is established.

Let z = x + iy ∈ F, γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) such that w = γ(z) ∈ F. If c = 0, then

γ = ± ( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) for some n ∈ ℤ, and z, γ(z) ∈ F implies that n = 0, so z = γ(z).

Assume c ≠ 0. Then a little geometric argument indicates that ℑ_{f∈F} (f) ≥ \frac{\sqrt{3}}{2}, also, |cz + d| ≥ cy. These two estimates imply that :

\frac{\sqrt{3}}{2} < ℑ(γ(z)) = \frac{y}{|cz + d|^{2}} ≥ \frac{1}{c^{2}y} < \frac{2}{c^{2}\sqrt{3}},

hence c^{2} < \frac{4}{3} and c = ±1 (0 being excluded).

Suppose c = ±1. Since γ and −γ have the same action on H, take without loss of generality c = 1. Then a little computation show that

γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & d \\ 0 & 1 \end{matrix} ) .

Let z_{1} = z + d, w_{1} = w − a. Since |ℜ(z)| < \frac{1}{2}, we have |z_{1}| ≥ |z| > 1, and similarly

|w_{1}| > 1, yet w_{1} = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) z_{1}. Since ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) is the transformation z ↦ −\frac{1}{z}

and maps the circle inside out, we get a contradiction.

□

Remark 2.7. It can be shown that every discrete subgroup Γ of SL_{2}(ℝ) has a fundamental domain.

Let S = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and T = ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ). Note that corresponding transformations

are z ↦ −\frac{1}{z} and (resp.) z ↦ z + 1.

Proposition 2.8. The group SL_{2}(ℤ) is generated by S and T.

Proof. Clearly, < S, T >⊆ SL_{2}(ℤ). We thus need to show equality. Since S^{2} = −I, it is enough to compute in the projectivized groups (modulo {±I}). Let γ ∈ Γ(1): we will decomposed γ in a product of S, T, T^{−1}. Note that we identify A and −A. Let F be the fundamental domain of Γ(1). Then

H = ∪_{γ∈Γ(1)}\overline{γ(F)},

such that the interior of the γ(F) are disjoint. Hence there are γ_{1}, …, γ_{n} ∈ Γ(1) such that γ_{1}(F) = F and γ_{n}(F), and each γ_{k}(F) is adjacent to γ_{k+1}(F). Note that the adjacent domains to F are T (F), T^{−1}(F) and S(F). Since γ_{k+1}(F) and γ_{k}(F) are adjacent, it follows that γ_{k}^{−1}γ_{k+1}(F) is adjacent to F, hence γ_{k}^{−1}γ_{k+1} is T, T^{−1} or S. Since γ = ∏ γ_{k}^{−1}γ_{k+1}, we are done. □

Remark 2.9. The same trick to determine generators can be applied for any discrete group, once the fundamental domain has been explicitly given.

2.3. ... and its congruence subgroups. We can view H ⊂ ℙ^{1}(ℂ) = ℂ ∪ ∞, hence the topological boundary of H is ℙ^{1}(ℝ) = ℝ ∪ ∞.

Definition 2.10. Let Γ act discontinously on H. The quotient space Γ\H is composed of the orbits under the action of Γ.

The quotient topology on Γ\H is given by:

U ⊂ Γ\H is open iff π^{−1}(U) is open, for π : H ⟶ Γ\H.

Fact 2.11. The space Γ\H is Hausdorff (this is assured by the discontinuity of the group action on H).

The cusps, intuitively speaking, are the points at which the fundamental domain of the group Γ touch the boundary of H.

Note that SL_{2}(ℤ) acts transitively on ℙ^{1}(ℚ), hence a subgroup Γ ⊆ SL_{2}(ℤ) of finite index can only have finitely many orbits. In particular, SL_{2}(ℤ) has only one orbit (∞).

Definition 2.12. An orbit of Γ in ℙ^{1}(ℚ) is called a cusp .

The cusps are used to compactify Γ\H (not the fundamental domain!!) by adding one point for every cusp, in order to obtain a compact Riemann surface at the end of the day.

In particular, adding ∞ to SL_{2}(ℤ)\H will yield a compact Riemann surface of genus 0.

Remark 2.13. When is Γ\H^{∗} a compact Riemann surface ? A Fuchsian group of the 1st kind is a discrete subgroup Γ ⊂ SL_{2}(ℝ) (or of SL_{2}(ℝ)/±1) such that Γ\H^{∗} is compact.

We have the following :

Proposition 2.14. [1, Proposition 1.32] If Γ\H^{∗} is compact, then the number of Γ-inequivalent cusps is finite.

Two subgroups Γ and Γ′ of a group G are said to be commensurable if Γ ∩ Γ′ is of finite index in Γ and in Γ′.

Proposition 2.15. [1, Proposition 1.31] If Γ, Γ′ are commensurable, then Γ\H^{∗} is compact iff Γ′\H^{∗}

If Γ is a discrete subgroup of SL_{2}(ℝ) commensurable with SL_{2}(ℤ), then Γ\H^{∗} is compact.

A possible source of confusion is that adding cusps gives a compact Riemann surface only if Γ\H is of finite volume under the SL_{2}(ℝ)-invariant measure \frac{1}{|y^{2}|}dxdy, and it is clearly possible to take Γ small enough so the volume of Γ\H is infinite. Thus we topologize H^{∗} := H ∪ ℙ^{1}(ℚ).

If a ∈ H, just pick a usual neighborhood (in the Euclidean topology). If a = ∞, take for a neighborhood basis ∞ ∪ {z|ℑ(z) > c} for 0 ≤ c ∈ ℝ. If a ∈ ℚ, take for a neighborhood basis a∪ the interior of a tangent circle to the real line at a. Note that a fractional linear transformation mapping a to ∞ will map the circle to a horizontal line, and vice-versa.

We give Γ\H^{∗} the quotient topology, and we obtain a manifold. Fact 2.16. The manifold Γ\H^{∗} is Hausdorff and locally compact. We proceed to define a complex structure on it (we shall use the nomenclature of

elliptic, hyperbolic and parabolic elements and elliptic fixed point: look up Exercises 1.2.7 and 1.2.8 for more information) by giving charts around each point.

• If a ∈ H is not elliptic, the usual chart on the upper half place will do.

• If a is elliptic, then we use the transform ϕ = \frac{z−a}{z−\overline{a}} to map H to the unit disc

𝔻.

Fact 2.17. The group Γ(N), for N > 1, does not contain elliptic elements. By Schwarz’s Lemma, the stabilizer of a is mapped to the cyclic group generated by a rotation of angle \frac{2π}{n} by conjugating with ϕ. Let w be the coordinate function on 𝔻, then the map z ↦ w^{n} maps a neighborhood of a ∈ Γ\H^{∗} homeomorphically to a neighborhood of 0 in ℂ.

• If a is a cusp, we can pick ρ ∈ SL_{2}(ℤ) to send a to ∞. Let \overline{Γ_{a}} be the stabilizer of a ∈ \overline{Γ}. Then \overline{ρΓρ^{−1}} is a subgroup of finite index in \overline{Γ(1)}¡ and the stabilizer of ∞ is \overline{ρΓ_{a}ρ^{−1}}. Hence this is a subgroup of finite index in the stabilizer of infinity in SL_{2}(ℤ), which is generated by z ↦ z + 1, hence \overline{ρΓ_{a}ρ^{−1}} is generated by z ↦ z + n. The map z ↦ e^{2πiρ(z)/n} maps a neighborhood of a ∈ Γ\H^{∗} homeomorphically onto a neighborhood of 0 in ℂ.

This completes the description of the complex structure. The manifold Γ\H^{∗} is thus a compact Riemann surface.

3. Exercises

Exercise 1.2.7

Solution : Put γ in Jordan canonical form: the possible matrices are

( \begin{matrix}γ & 1 \\ 0 & γ \end{matrix} ) or ( \begin{matrix}γ & 0 \\ 0 & μ \end{matrix} ) , γ ≠ μ.

In the first case, if γ^{2} = 1, then γ = ±1, and the trace is ±2. In the second case, γμ = 1. If ±1 ≠ γ ∈ ℝ, then γ + \frac{1}{γ} > 2 by the arithmetic-geometric inequality. If γ ∈ ℂ\ℝ, then μ = \overline{γ} (both being roots of the same quadratic equation) and |γ + \overline{γ}| = 2Re(γ) < 2.

Note that any γ has two fixed points (with multiplicity). We will cover all cases in two strokes :

• For every z ∈ H, take τ ∈ SL_{2}(ℝ) such that τ(i) = z. Then

τ · SO_{2}(ℝ)τ^{−1} = {α ∈ SL_{2}(ℝ)|α(z) = z}.

All elements in S0_{2}(ℝ) = \{( \begin{matrix}a & b \\ −b & a \end{matrix} ) |a^{2} + b^{2} = 1\} have eigenvalues of

norm 1, hence an element with one fixed point in H must of trace ±2 or of trace small than 2.

• For r ∈ ℝ ∪ ∞, look at :

F (r) = {α ∈ SL_{2}(ℝ)|α(r) = r},
P (r) = {α ∈ F (r)|α parabolic or ± 1}.

Since SL_{2}(ℝ) acts transitively on ℝ ∪ ∞, there exists σ ∈ SL_{2}(ℝ) so that σ(∞) = r. But

F (∞) = \{( \begin{matrix}a & b \\ 0 & a^{−1} \end{matrix} ) |a ∈ ℝ^{×}, b ∈ ℝ\},

P (∞) = \{± ( \begin{matrix}1 & h \\ 0 & 1 \end{matrix} ) |h ∈ ℝ\},

thence if σ ≠ ±1 has a fixed point in ℝ, its trace is greater or equal than 2.

Note that the only γ ∈ SL_{2}(ℝ) of trace 2 fixing a point in H are ±I, so the trichotomy is established.

Exercise 1.2.8 a) Note that

( \begin{matrix}γ & 1 \\ 0 & γ \end{matrix} )^{n} = ( \begin{matrix}γ^{n} & nγ^{n−1} \\ 0 & γ^{n} \end{matrix} ) ,

( \begin{matrix}γ & 0 \\ 0 & μ \end{matrix} )^{n} = ( \begin{matrix}γ^{n} & 0 \\ 0 & μ^{n} \end{matrix} ) .

If there is a n ∈ ℕ such that γ^{n} = μ^{n} = 1, then |γ + μ| < 2, hence the element γ ∈ SL_{2}(ℝ) is elliptic. If γ is elliptic, then γ^{n} is elliptic or ±1; but {γ ∈ Γ|γz = z} is finite : take g ∈ SL_{2}(ℝ) such that g · i = z. Then g^{−1}γg · i = i hence {γ ∈ Γ : γz = z} = gSO_{2}(ℝ)g^{−1} ∩ Γ. Since S0_{2}(ℝ) is compact and Γ is discrete (Γ acting discontinuously), the intersection is finite.

b) Pick σ elliptic in SL_{2}(ℤ), tr(σ) < 2 hence tr(σ) ∈ {0, ±1}. so the characteristic polynomial is either x^{2} + 1 or x^{2} ± x + 1, so σ^{4} = 1 or σ^{6} = 1 (i.e. σ^{3} = ±1, but if (σ)^{3} = 1 if σ^{3} = −1 ) hence we need only look at σ^{4} = 1 or σ^{3} = 1. In \overline{F}, only i and the non-real third root of unity fit either one of these. The orbits are clearly disjoint (F is a fundamental domain).

Exercise 1.2.9

If Γ has no parabolic element, then H = H^{∗} and Γ\H is compact .

Suppose Γ\H is compact. Let π : H ⟶ Γ\H. Suppose ∞ is a cusp. Take an infinite sequence of points {z_{n}} ⊂ H such that ℑ(z_{n}) ⟶ ∞.

We find the proofs of the following lemmas in [1].

Lemma 3.1. For every cusp of Γ, there is a neighborhood U of s in H^{∗} such that

Γ_{s} = {σ ∈ Γ|σ(U) ∩ U ≠ ∅}.

Lemma 3.2. For every cusp s of Γ, for every compact K of H, there is a neighborhood U of s such that U ∩ γ(K) = ∅∀γ ∈ Γ.

Then the first lemma tells us that there is a neighborhood U = {z ∈ H^{8}|ℑ(z) > c} of ∞ such that Γ_{∞} = {γ ∈ Γ|γ(U) ∩ U ≠ ∅}. Then z_{n} ∈ U for n >> 0. Since no elements of Γ_{∞} modifies ℑ(z), if two points have distinct and sufficiently large imaginary parts, then they are not Γ-equivalent. Therefore {π(z_{n})} contains a sequence of infinitely many distinct points of Γ\H. If Γ\H is compact, there is a w ∈ H such that π(w) is a limit point of {π(z_{n})}. Let K be a compact neighborhood of w. By the second lemma, there is a neighborhood V of ∞ such that K ∩ΓV = ∅. Since π(z_{n}) ∈ π(K)∩π(V) for n >> 0, we get a contradiction.

Exercise 1.2.10

If γ(a) = a, γ ∈ SL_{2}(ℝ), then a is the unique solution of a quadratic equation with rational coefficients, hence a ∈ ℚ. The other direction follows from the definition.

Exercise 1.2.11

Proposition 3.3. An ideal class in K = ℚ(ϵ) uniquely determines a conjugacy class in GL_{2}(ℤ) of matrices with eigenvalue γ (and determinant 1).

Proof.

Lemma 3.4. Let I be an ideal, (a_{1}, a_{2}) a ℤ-basis of I. Let γ be a unit of norm 1. Then there exists A ∈ SL_{2}(ℤ) such that

A ∘ (a_{1}, a_{2})^{t} = γ(a_{1}, a_{2})^{t}.

Proof. If γ is a unit, then (γa_{1}, γa_{2}) is a ℤ-basis of I, hence there is a matrix A ∈ GL_{2}(ℤ) taking (a_{1}, a_{2})^{t} to (γa_{1}, γa_{2})^{t}. But γ satisfies a quadratic equation, hence \overline{γ} is also an eigenvalue, so γ\overline{γ} = 1 = det(A). □

Lemma 3.5. Take another base (w_{1}, w_{2}) of I. Then there is a matrix B ∈ SL_{2}(ℤ) conjugate by g ∈ GL_{2}(ℤ) to A.

Proof. If a_{1}, a_{2} and w_{1}, w_{2} are bases, then there is a matrix C ∈ GL_{2}(ℤ) such that C ∘ (w_{1}, w_{2})^{t} = (a_{1}, a_{2})^{t}. Put B = C^{−1}AC; thus B ∘ (w_{1}, w_{2})^{t} = C^{−1}AC(w_{1}, w_{2})^{t} = C^{−1}A ∘ (a_{1}, a_{2})^{t} = C^{−1} ∘ γ(a_{1}, a_{2})^{t} = γC^{−1}(a_{1}, a_{2})^{t} = γ(w_{1}, w_{2})^{t} . □

If w_{1}, w_{2} is a basis of I, and J is in the same class as I, there is a constant k such that kw_{1}, kw_{2} is a basis of J, hence by the first lemma, (kw_{1}, kw_{2}) is an eigenvector of A. We may just take w_{1}, w_{2} to represent the whole ideal class, and all is left to prove is the converse of the second lemma : if B = D^{−1}AD, D ∈ GL_{2}(ℤ), then B and A correspond to the same ideal class. But B = C^{−1}AC with eigenvalue γ is exactly C^{−1}(a_{1}, a_{2})^{t} and C^{−1}(a_{1}, a_{2})^{t} also has entries which are a basis of I.

□

Remark 3.6. If we take SL_{2}(ℤ)-conjugacy classes instead of GL_{2}(ℤ)-conjugacy classes, we obtain the narrow class number.

Bibliography

[1] Shimura, Goro, Introduction to the arithmetic theory of automorphic functions. Kano Memorial Lectures, No. 1. Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. xiv+267 pp.