2  Modular forms for congruence subgroups

2.1 Congruence subgroups

Let \(N\geq 1\) be an integer. In this section we will consider subgroups of \({\operatorname{SL}}_2({\mathbb{Z}})\) that are especially nice to work with. There are other subgroups that are interesting but beyond the scope of this course.

Definition 2.1 The of level \(N\) is \[\Gamma(N)=\{ \gamma\in{\operatorname{SL}}_2({\mathbb{Z}})~|~ \gamma\equiv \left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right) \pmod{N}\}.\]

Note that \(\Gamma(1)={\operatorname{SL}}_2({\mathbb{Z}})\), so we are strictly generalizing Chapter 1. Note also that \(\Gamma(N)\) can be defined as the kernel of the group homomorphism induced by the reduction map \({\mathbb{Z}}\longrightarrow{\mathbb{Z}}/N{\mathbb{Z}}\): \[\pi_N\colon {\operatorname{SL}}_2({\mathbb{Z}})\longrightarrow{\operatorname{SL}}_2({\mathbb{Z}}/N{\mathbb{Z}}).\] Therefore, \(\Gamma(N)\) is a normal subgroup of \({\operatorname{SL}}_2({\mathbb{Z}})\), of finite index.

Proposition 2.1 The map \(\pi_N\) is surjective.

Proof. Exercise.

There are too few principal congruence subgroups (only one for each \(N\geq 1\)), so it is desirable to consider more general subgroups.

Definition 2.2 A subgroup \(\Gamma\subseteq {\operatorname{SL}}_2({\mathbb{Z}})\) is a if there is some \(N\geq 1\) such that \[\Gamma(N)\subseteq \Gamma \subseteq {\operatorname{SL}}_2({\mathbb{Z}}).\] The of a congruence subgroup \(\Gamma\) is the minimum \(N\) such that \(\Gamma(N)\subseteq \Gamma\).

One can think of many different ways to construct congruence subgroups. There are two families that arise so frequently that have special notation for them:

Example 2.1 For each \(N\geq 1\), define \[\Gamma_1(N)=\{\left(\begin{smallmatrix}1&*\\0&1\end{smallmatrix}\right)\pmod N\},\] and also \[\Gamma_0(N)=\{\left(\begin{smallmatrix}*&*\\0&*\end{smallmatrix}\right)\pmod N\}.\]

Lemma 2.1 For each \(N\geq 1\) there are inclusions \(\Gamma(N) \subseteq \Gamma_1(N) \subseteq \Gamma_0(N) \subseteq {\operatorname{SL}}_2({\mathbb{Z}})\), and \[[\Gamma_1(N)\colon \Gamma(N)] = N,\quad [\Gamma_0(N)\colon \Gamma_1(N)] = N\prod_{p\mid N} \left(1-\frac 1p\right),\quad [{\operatorname{SL}}_2({\mathbb{Z}})\colon \Gamma_0(N)] = N\prod_{p\mid N} \left(1+\frac 1p\right).\]

Proof. Exercise.

The inclusions are strict except for \(N=1\) (where all groups coincide) and for \(\Gamma_0(2)=\Gamma_1(2)\).

Definition 2.3 A function \(f\colon{\mathbb{H}}\longrightarrow{\mathbb{C}}\) is of weight \(k\) with respect to \(\Gamma\) if it is meromorphic on \({\mathbb{H}}\) and it satisfies \[f|_k\gamma = f,\quad \forall \gamma\in\Gamma.\]

2.2 Cusps

Of course we will need to understand fundamental domains for the action of congruence subgroups on \({\mathbb{H}}\). Here is for example a fundamental domain for \(\Gamma_0(2)\):

Figure 2.1: A fundamental domain for \(\Gamma_0(2)\)

In this case, the fundamental domain contains two points in its closure which do not belong to \({\mathbb{H}}\): the cusp \(\infty\) as before, but also \(0\). The following result gives a construction of a fundamental domain for any congruence subgroup, using translates of the fundamental domain \(\mathcal{D}\) of \({\operatorname{SL}}_2({\mathbb{Z}})\) seen in Chapter 1.

Proposition 2.2 Let \(\Gamma\) be a congruence subgroup of \({\operatorname{SL}}_2({\mathbb{Z}})\). If there is a decomposition \[\Gamma\backslash {\operatorname{SL}}_2({\mathbb{Z}})=\bigcup_{h\in R} \Gamma h,\quad R\text{ finite,}\] then the set \(\mathcal{D}_\Gamma=\cup_{h\in R} h\mathcal{D}\) is a (possibly non-connected) fundamental domain for \(\Gamma\).

Proof. If \(z\in{\mathbb{H}}\), then there exists \(g\in{\operatorname{SL}}_2({\mathbb{Z}})\) and \(z_0\in\mathcal{D}\) such that \(z = gz_0\). The coset decomposition implies that there is some \(\gamma\in R\) and some \(h\in \Gamma\) such that \(g=h\gamma\). Therefore \[z = h\gamma z_0.\] Since \(z_0'=\gamma z_0\in \gamma\mathcal{D}\subset \mathcal{D}_\Gamma\) we have written \(z=h z_0'\) with \(h\in\Gamma\).

It remains to be shown that if \(z\in\stackrel{\circ}{\mathcal{D}_\Gamma}\) and \(\gamma z \in\stackrel{\circ}{\mathcal{D}_\Gamma}\) for some \(\gamma\in\Gamma\), then \(\gamma = 1\). For that, let \(\varepsilon>0\) be small enough so that the ball \(B_\varepsilon(z)\) of radius \(\varepsilon\) around \(z\) is fully contained in \(\stackrel{\circ}{\mathcal{D}_\Gamma}\). The ball \(B_\varepsilon(z)\) intersects some translates of \(\mathcal{D}\), say: \[B_\varepsilon(z)\cap h\mathcal{D}\neq \emptyset\iff h\in R'\subseteq R.\] Consider the translated ball \(\gamma B_\varepsilon(z)=B_\varepsilon(\gamma z)\). Since \(\gamma z\) is also in the interior of \(\mathcal{D}_\Gamma\), we deduce that \(\gamma B_\varepsilon(z)\) must intersect the interior of some translate of \(\mathcal{D}\), say \(h\stackrel{\circ}{\mathcal{D}}\), for some \(h\in R\). Therefore: \[\gamma B_\varepsilon(z)\cap h\stackrel{\circ}{\mathcal{D}}\neq \emptyset \implies B_\varepsilon(z)\cap \gamma^{-1}h\stackrel{\circ}{\mathcal{D}}\neq \emptyset.\] Since we listed all the translates whose interior intersected with \(B_\varepsilon(z)\), we must have that \(\gamma^{-1}h = h_0\). But now \(\Gamma h = \Gamma \gamma^{-1}h = \Gamma h_0\), and since both \(h\) and \(h_0\) belong to \(R\), we must have \(h=h_0\). Therefore \(\gamma^{-1}=1\), or \(\gamma=1\) as we wanted.

In order to further study the cusps, we consider the \({\mathbb{P}}^1({\mathbb{Q}})={\mathbb{Q}}\cup\{\infty\}\). Note that \({\operatorname{SL}}_2({\mathbb{Z}})\) (in fact \(\operatorname{GL}_2({\mathbb{Q}})\)) acts on \({\mathbb{P}}^1({\mathbb{Q}})\) by fractional linear transformations: \[\gamma x = \frac{ax +b}{cx+d},\quad \gamma = \left(\begin{matrix}a&b\\c&d\end{matrix}\right)\in{\operatorname{SL}}_2({\mathbb{Z}}),x\in{\mathbb{P}}^1({\mathbb{Q}}).\] Here we understand that \(\gamma\infty = \frac ac\), and \(\gamma x = \infty\) if \(cx = -d\).

Proposition 2.3 The action of \({\operatorname{SL}}_2({\mathbb{Z}})\) on \({\mathbb{P}}^1({\mathbb{Q}})\) is transitive, and it induces a bijection \[{\operatorname{SL}}_2({\mathbb{Z}})/{\operatorname{SL}}_2({\mathbb{Z}})_\infty\cong {\mathbb{P}}^1({\mathbb{Q}}),\quad {\operatorname{SL}}_2({\mathbb{Z}})_\infty = \langle \pm T\rangle.\]

Proof. We will see that the orbit of \(\infty\) is all of \({\mathbb{P}}^1({\mathbb{Q}})\), where we note that \(\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right) \infty = \frac ac\). Given \(\frac ac\in{\mathbb{P}}^1({\mathbb{Q}})\) in reduced terms (that is, such that \((a,c)=1\)), then Bézout’s identity asserts the existence of integers \(b\) and \(d\) such that \(ad-bc=1\). Then the matrix \(\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\) belongs to \({\operatorname{SL}}_2({\mathbb{Z}})\) and takes \(\infty\) to \(\frac ac\).

The stabilizer of \(\infty\), written \({\operatorname{SL}}_2({\mathbb{Z}})_\infty\), is: \[{\operatorname{SL}}_2({\mathbb{Z}})_\infty = \left\{\left(\begin{matrix}a&b\\c&d\end{matrix}\right) ~|~ \left(\begin{matrix}a&b\\c&d\end{matrix}\right) \infty = \infty\right\} = \left\{\left(\begin{matrix}a&b\\c&d\end{matrix}\right) ~|~ \frac ac = \infty\right\} = \left\{\left(\begin{matrix}a&b\\0&d\end{matrix}\right)\right\} = \langle \pm T\rangle.\]

Definition 2.4 The set of of a congruence subgroup \(\Gamma\) is the set \(\operatorname{Cusps}(\Gamma)\) of \(\Gamma\)-orbits of \({\mathbb{P}}^1({\mathbb{Q}})\). Equivalently, \[\operatorname{Cusps}(\Gamma) = \Gamma\backslash {\operatorname{SL}}_2({\mathbb{Z}}) / {\operatorname{SL}}_2({\mathbb{Z}})_\infty.\]

If \(P = [\frac ac]\) is a cusp of \(\Gamma\), set \(\Gamma_P\) for the of \(P\) in \(\Gamma\), the elements of \(\Gamma\) fixing \(P\).

Lemma 2.2 If \(\gamma_P(\infty) = P\), then \[\Gamma_P = \Gamma\cap \gamma_P{\operatorname{SL}}_2({\mathbb{Z}})_\infty \gamma_P^{-1}.\]

Proof. Let \(\gamma\in\Gamma\). Then observe that \[\begin{align*} \gamma\in\Gamma_P&\iff \gamma P = P\iff \gamma \gamma_P\infty=\gamma_P\infty\\ &\iff \gamma_P^{-1}\gamma\gamma_P\infty=\infty\\ &\iff \gamma_P^{-1}\gamma\gamma_P\in {\operatorname{SL}}_2({\mathbb{Z}})_\infty\\ &\iff \gamma\in \gamma_P{\operatorname{SL}}_2({\mathbb{Z}})_\infty \gamma_P^{-1}. \end{align*}\] This concludes the proof.

Lemma 2.3 The subgroup \(H_P = \gamma_P^{-1}\Gamma\gamma_P\cap {\operatorname{SL}}_2({\mathbb{Z}})_\infty \subseteq {\operatorname{SL}}_2({\mathbb{Z}})_\infty\) does not depend on the choice of the representative for \(P\), and has finite index in \({\operatorname{SL}}_2({\mathbb{Z}})_\infty\).

Proof. Just note that if \(\frac{a'}{c'}\) is another representative for \(P\), then \(\gamma_P\) gets modified into \(\gamma\gamma_P\) for some \(\gamma\in\Gamma\). Then \((\gamma\gamma_P)^{-1}\Gamma(\gamma\gamma_P) = \gamma_P^{-1}\Gamma\gamma_P\).

Lemma 2.4 Let \(H\) be a subgroup of finite index in \({\operatorname{SL}}_2({\mathbb{Z}})_\infty\), and let \(h\) be the index of \(\{\pm 1\}H\) in \({\operatorname{SL}}_2({\mathbb{Z}})_\infty\). Then \(H\) is one of he following: \[H = \begin{cases} \langle \left(\begin{smallmatrix}1&h\\0&1\end{smallmatrix}\right)\rangle&\\ \langle \left(\begin{smallmatrix}-1&h\\0&-1\end{smallmatrix}\right)\rangle&\\ \{\pm 1\}\times \langle \left(\begin{smallmatrix}1&h\\0&1\end{smallmatrix}\right)\rangle& \end{cases}\]

Proof. Exercise.

Definition 2.5 The integer \(h_\Gamma(P)=h\) in the above lemma is called \(P\) for \(\Gamma\). A cusp is an if \(\gamma_P^{-1}\Gamma_P\gamma_P\) is of the form \(\langle \left(\begin{smallmatrix}-1&h\\0&-1\end{smallmatrix}\right)\rangle\), and it is otherwise.

Example 2.2 In this example we show that if \(p\) is any prime, then \(\operatorname{Cusps}(\Gamma_0(p)) = \{ \infty, 0\}\).

Write an element \(\gamma\in\Gamma_0(p)\) as \(\left(\begin{smallmatrix}a&b\\pc&d\end{smallmatrix}\right)\), with \(a,b,c,d\in{\mathbb{Z}}\) satisfying \(ad-pbc =1\). The orbit of \(\infty\) is: \[\Gamma_0(p)\cdot \infty = \left\{\left(\begin{matrix}a&b\\pc&d\end{matrix}\right)\infty\right\} = \left\{\frac{a}{pc}~\colon~ a,c\in{\mathbb{Z}},\gcd(a,pc) = 1\right\} = \left\{ \frac{r}{s}~\colon~ p\mid s, \gcd(r,s)=1\right\}.\] We thus see the orbit of the cusp \(\infty\) consists of infinity together with all the rationals which when expressed in reduced terms have a denominator which is divisible by \(p\). One element which is not in this orbit is \(0=\frac{0}{1}\). Let us study then the orbit of \(0\). \[\Gamma_0(p) \cdot 0 = \{\left(\begin{matrix}a&b\\pc&d\end{matrix}\right) 0\} = \{\frac{b}{d}~\colon~ b,d\in{\mathbb{Z}}, \gcd(b,d)=1, p\nmid d\}.\]

As we have seen in the example above, each cusp may have a different width. However, if \(\Gamma\) is a normal congruence subgroup of \({\operatorname{SL}}_2({\mathbb{Z}})\), the subgroup \(H_P\) does not depend on the cusp \(P\) and hence all cusps have the same width and regularity.

Although one can have more cusps than the index of \(\Gamma\), the last result of this section says that this is basically right, once one counts in a proper way. To prove it, we will need a group-theoretic result.

Lemma 2.5 Let \(G\) be a group acting transitively on a set \(X\) and let \(H\) be a finite index subgroup of \(G\). Then for any \(x\in X\) the stabilizer of \(x\) in \(H\) has finite index in the stabilizer of \(x\) in \(G\), and the following formula holds: \[\sum_{x\in H\backslash X} [G_x\colon H_x]=[G\colon H].\]

Proof. Let \(x\in X\), and consider the inclusion map \(G_x\longrightarrow G\). By taking the quotient by \(H\) we get \(G_x\longrightarrow H\backslash G\). Suppose \(g_1, g_2\) are mapped to the same element \(Hg\) in \(H\backslash G\). This means that \(Hg_1 = Hg_2\), or that \(g_2g_1^{-1}\) belongs to \(H\). Since \(g_2g_1^{-1}\) stabilizes \(x\) as well, we deduce that \(H_x g_1 = H_x g_2\). Therefore there is an injection of \(H_x\backslash G_x\hookrightarrow H\backslash G\). Since by assumption the latter set is finite, so is the first. Note also that the image of the map is precisely \(H\backslash HG_x\), and thus we also obtain \([G_x\colon H_x] = [H G_x\colon H]\).

To prove the second assertion, fix an element \(x_0\in X\), and consider the map \[H\backslash G\longrightarrow H\backslash X,\quad Hg\mapsto Hgx_0\] which is surjective because \(G\) acts transitively on \(X\). The fibre \(T_{Hx}\) of an orbit \(Hx\) consists of the set of classes \(Hg\) such that \(Hgx_0 = Hx\). Let \(g_x\in G\) be such that \(g_x x_0 = x\). Write \(Hg = Hg'g_x\) and then we have: \[\begin{align*} T_{Hx}&\cong \{Hg'\in H\backslash G\colon Hg'g_xx_0 = Hx\} = \{Hg'\in H\backslash G\colon Hg'x=Hx\}= H\backslash(H G_x)\cong H_x\backslash G_x. \end{align*}\] This allows us to find a formula for \([G\colon H]\): \[\begin{align*} &=\# (H\backslash G) = \sum_{x\in R} \# T_{Hx} = \sum_{x\in R} [G_x\colon H_x]. \end{align*}\]

Theorem 2.1 Let \(\Gamma\) be a congruence subgroup. Then we have \[\sum_{P\in \operatorname{Cusps}(\Gamma)} h_\Gamma(P) = [{\operatorname{SL}}_2({\mathbb{Z}})\colon \{\pm 1\}\Gamma].\]

Proof. Consider the group \(G=\operatorname{PSL}_2({\mathbb{Z}})={\operatorname{SL}}_2({\mathbb{Z}})/\{\pm 1\}\), which acts transitively on the set \(X={\mathbb{P}}^1({\mathbb{Q}})\). Let \(H\) be the image of \(\Gamma\) in \(G\). Note that \(H\backslash X=\operatorname{Cusps}(\Gamma)\). Also, \(G_\infty\) is the image in \(G\) of \({\operatorname{SL}}_2({\mathbb{Z}})_\infty\). For each \(x\in X\), let \(\gamma\in G\) be such that \(\gamma \infty = x\). Then \[G_x=\gamma G_\infty\gamma^{-1},\text{ and}\quad H_x= \gamma \left(\gamma^{-1}H\gamma\right)_\infty \gamma^{-1}.\] Therefore \[[G_x\colon H_x] = [\bar{\operatorname{SL}}_2({\mathbb{Z}})\colon \bar \Gamma_P] = h_\Gamma(P),\] where by \(\overline{(\cdot)}\) we write the image of the group inside \(G\). Then applying Lemma 2.5 to this setting gives \[\sum_{P\in \operatorname{Cusps}(\Gamma)} h_\Gamma(P) = [G\colon H] = [{\operatorname{SL}}_2({\mathbb{Z}})\colon \{\pm 1\}\Gamma].\]

2.3 Fourier expansion at infinity

Let \(\Gamma\) be a congruence subgroup of level \(N\). Note that the matrix \(\left(\begin{smallmatrix}1&N\\0&1\end{smallmatrix}\right)\) belongs to \(\Gamma(N)\), and thus there is a minimum \(h>0\) with the property that \(\left(\begin{smallmatrix}1&h\\0&1\end{smallmatrix}\right) \in\Gamma\).

Definition 2.6 The of \(\Gamma\) is the minimum \(h>0\) such that \(\left(\begin{smallmatrix}1&h\\0&1\end{smallmatrix}\right)\in\Gamma\).

The fan width of a congruence subgroup of level \(N\) is a divisor of \(N\).

Write \[q_h =q_h(z)= e^{\frac{2\pi i z}{h}},\] and note that \(z\mapsto q_h(z)\) is periodic with period \(h\). Define \(g\) by \(g = f\circ q_h^{-1}\). That is, \(g(q_h) = f(z)\). Although \(q_h\) is not invertible, the above definition makes sense, and \(g\) has a Laurent expansion.

Definition 2.7 The \(q\)-expansion of \(f\) at infinity is the Laurent expansion: \[f(z) = g(q_h)=\sum_{n=-\infty}^\infty a(n)q_h^n.\]

2.4 Expansions at cusps

Let \(s\) be a cusp, \(s\neq \infty\). Write \(s =\alpha\infty\) for some \(\alpha\in{\operatorname{SL}}_2({\mathbb{Z}})\), and consider the equation: \[f(\alpha z) = j(\alpha,z)^{k}(f|_k\alpha)(z).\] Since \(j(\alpha,z)\neq 0,\infty\) when \(z\) is near \(\infty\), the behavior of \(f(z)\) near \(s\) is related to the behavior of \((f|_k\alpha)(z)\) near \(\infty\). Assume that \(f\) is weakly modular for the congruence subgroup \(\Gamma\). Since\[(f|_k\alpha)|_k(\alpha^{-1}\gamma\alpha) = (f|_k\gamma)|_k\alpha = f|_k\alpha,\] the new function \(f|_k\alpha\) is invariant under the group \(\Gamma'=\alpha^{-1}\Gamma\alpha\). Since \(\Gamma(N)\) is normal inside \({\operatorname{SL}}_2({\mathbb{Z}})\), we deduce that \(\Gamma'\) is also a congruence subgroup of level \(N\). Hence \(f|_k\alpha\) has a Fourier expansion at infinity as in Section 2.3 in powers of \(q_N\).

Definition 2.8 The \(s\) is the expansion: \[f|_k\alpha = \sum_{n=-\infty}^\infty b(n)q_N^n.\]

2.5 Definition of modular forms

The expansions at different cusps allow us to define modular forms for arbitrary congruence subgroups.

Definition 2.9 A function \(f\colon{\mathbb{H}}\longrightarrow{\mathbb{C}}\) is a of weight \(k\) for a congruence subgroup \(\Gamma\) if:

  1. \(f\) is holomorphic on \({\mathbb{H}}\),

  2. \(f|_k\gamma = f\) for all \(\gamma\in\Gamma\), and

  3. \(f|_k\alpha\) is holomorphic at infinity for all \(\alpha\in{\operatorname{SL}}_2({\mathbb{Z}})\).

A function is a of weight \(k\) for a congruence subgroup \(\Gamma\) if instead of \(3\) is satisfies:

  1. \(f|_k\alpha\) vanishes at infinity for all \(\alpha\in{\operatorname{SL}}_2({\mathbb{Z}})\).

The of weight \(k\) for a congruence subgroup \(\Gamma\) is written \(M_k(\Gamma)\); the of weight \(k\) for a congruence subgroup \(\Gamma\) is written \(S_k(\Gamma)\).

Proposition 2.4 Suppose that \(f\colon{\mathbb{H}}\longrightarrow{\mathbb{C}}\) satisfies \(1\) and \(2\) above. Suppose that \(f\) is holomorphic at infinity. That is, \[f(z) = \sum_{n=0}^\infty a(n)q_N^n.\] Furthermore, suppose that there exists constants \(C>0\) and \(r>0\) such that: \[|a(n)|< Cn^r,\quad \forall n > 0.\] Then \(f\) satisfies \(3\), and thus \(f\in M_k(\Gamma)\).

Proof. Exercise.

In fact, the converse is also true: if the Fourier coefficients of \(f\) grow as \(Cn^r\) as above, then condition \(3\) in Definition 2.9 is satisfied. The proof of this fact uses Eisenstein series for congruence subgroups, and thus will be postponed until we introduce those.

Example 2.3 Let \(f\) be a weakly-modular form of weight \(k\) for the full modular subgroup. Consider the function \(g(z) = f(Nz)\). If \(\gamma\in\Gamma_0(N)\) is of the form \(\gamma=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\), then since \(N\mid c\) the matrix \[\gamma'=\left(\begin{matrix}a&bN\\c/N&d\end{matrix}\right)\] is in \({\operatorname{SL}}_2({\mathbb{Z}})\). Therefore we may compute: \[\begin{align*} g(\gamma z) &= f(N(\gamma z)) = f(\frac{Naz+bN}{cz+d}) \\ &= f(\frac{a(Nz)+bN}{c/N (Nz) + d}) = f(\gamma'(Nz))\\ &=(c/N(Nz) + d)^{k}f(Nz) = j(\gamma,z)^k g(z). \end{align*}\] Therefore the function \(g\) is weakly-modular of weight \(k\) for the congruence subgroup \(\Gamma_0(N)\). In fact, this operation defines injections \[M_k({\operatorname{SL}}_2({\mathbb{Z}}))\longrightarrow M_k(\Gamma_0(N))\] which will play an important role later in the course, in the Atkin-Lehner-Li theory of old/new forms.

We end this section by realizing that the definition of modular forms can be checked by finitely many computations. Suppose that \(\sigma=\alpha\infty\) and \(\tau=\beta\infty\) are two cusps (here \(\alpha\) and \(\beta\) are in \({\operatorname{SL}}_2({\mathbb{Z}})\)). Suppose that \(\sigma=\gamma\tau\) with \(\gamma\in\Gamma\).

Proposition 2.5 If \[f|_k\alpha = \sum_{n=-\infty}^\infty a(n)q_h^n,\] then \[f|_k\beta = \sum_{n=-\infty}^\infty b(n)q_h^n,\quad b(n) = (\pm 1)^k e^{\frac{2\pi i n j}{h}} a(n), j\in {\mathbb{Z}}.\]

Proof. By assumption \(\alpha\infty=\gamma\beta\infty\), so \(\alpha^{-1}\gamma\beta\infty=\infty\), and therefore since the only matrices that fix infinity are of the form \(\pm \left(\begin{smallmatrix}1&j\\0&1\end{smallmatrix}\right)\) we have: \[\alpha^{-1}\gamma\beta = \pm\left(\begin{matrix}1&j\\0&1\end{matrix}\right),\quad j\in{\mathbb{Z}}.\] This means that \[\beta = \pm \gamma^{-1}\alpha\left(\begin{matrix}1&j\\0&1\end{matrix}\right),\] and therefore: \[\begin{align*} f|_k\beta &= f|_k\pm I |_k \gamma^{-1} |_k\alpha |_k \left(\begin{matrix}1&j\\0&1\end{matrix}\right) = (\pm 1)^k\sum a(n)e^{\frac{2\pi i n z}{h}} |_k \left(\begin{matrix}1&j\\0&1\end{matrix}\right)\\ &=(\pm 1)^k\sum a(n) e^{\frac{2\pi i n (z+j)}{h}}. \end{align*}\]

Corollary 2.1 For each \(n\in{\mathbb{Z}}\), we have \(a(n)=0\) if and only if \(b(n)=0\). In particular, it is enough to check \(3\) or \(3'\) for one representative from each of the equivalence classes of cusps.

2.6 Valence formula for congruence subgroups

Let \(\Gamma\) be a congruence subgroup of level \(N\). In order to state the next result we need to define the order of a weakly-modular function at a cusps \(P\in\operatorname{Cusps}(\Gamma)\).

Definition 2.10 Let \(f\) be a weakly-modular form of weight \(k\) for \(\Gamma\), and let \(P\) be a cusp of \(\Gamma\) of width \(h_\Gamma(P)\). Since \(f(z+N)=f(z)\), we can write \(f\) as a Laurent series in \(q_N=e^{\frac{2\pi i z}{N}}\), say \[f(q_N) = \sum_{n\geq n_0} a_n q_N^n,\quad a_{n_0}\neq 0.\] The of \(f\) at \(P\) is \(v_P(f) = \frac{h_\Gamma(P)}{N}n_0\).

Here is a generalization of Theorem 1.9 to arbitrary congruence subgroups.

Theorem 2.2 Let \(\Gamma\) be a congruence subgroup, and let \(k\) be an integer. Let \(f\) be a non-zero meromorphic function on \({\mathbb{H}}\cup\{\infty\}\), which is weakly-modular of weight \(k\) for \(\Gamma\). Then we have \[\sum_{z\in\Gamma\backslash{\mathbb{H}}} \frac{v_z(f)}{\#\overline\Gamma_z} +\sum_{P\in\operatorname{Cusps}(\Gamma)} v_P(f)=\frac{k}{12}[\operatorname{PSL}_2({\mathbb{Z}})\colon\overline\Gamma].\]

Proof. Write \(d_\Gamma=[\operatorname{PSL}_2({\mathbb{Z}})\colon\overline\Gamma]\), let \(R\) be a set of coset representatives for \(\overline\Gamma\backslash\operatorname{PSL}_2({\mathbb{Z}})\), and define \(F=\prod_{\gamma\in R} f|_k\gamma\). Note that \(F\) is weakly-modular of weight \(kd_\Gamma\) for \({\operatorname{SL}}_2({\mathbb{Z}})\), and it is meromorphic at \(\infty\). By Theorem 1.9 we have \[v_\infty(F)+\frac 12 v_i(F) + \frac 13 v_\rho(F) +\sum_{w\in W} v_w(F) = \frac{k}{12} d_\Gamma.\] Another way to write the above is: \[v_\infty(F) + \sum_{z\in\operatorname{PSL}_2({\mathbb{Z}})\backslash{\mathbb{H}}} \frac{v_z(F)}{\#\operatorname{PSL}_2({\mathbb{Z}})_z} = \frac{k}{12} d_\Gamma.\] We may now compute: \[\begin{align*} v_z(F) &= \sum_{\gamma\in\overline\Gamma\backslash\operatorname{PSL}_2({\mathbb{Z}})} v_z(f|_k\gamma) = \sum_{\gamma\in\overline\Gamma\backslash\operatorname{PSL}_2({\mathbb{Z}})} v_{\gamma z}(f)=\sum_{w\in\overline\Gamma\backslash \operatorname{PSL}_2({\mathbb{Z}})z} [\operatorname{PSL}_2({\mathbb{Z}})_w\colon\overline\Gamma_w] v_w(f). \end{align*}\] The last equality follows by grouping all elements \(\gamma\) such that \(\gamma z=w\), for each possible \(w\). Now, since \(\operatorname{PSL}_2({\mathbb{Z}})_w\) is finite and independent of \(w\in \overline\Gamma\backslash\operatorname{PSL}_2({\mathbb{Z}})z\), we get \([\operatorname{PSL}_2({\mathbb{Z}})_w\colon\overline\Gamma_w]=\frac{\# \operatorname{PSL}_2({\mathbb{Z}})_z}{\#\overline\Gamma_w}\). Dividing by \(\#\operatorname{PSL}_2({\mathbb{Z}})_z\) we obtain \[\frac{v_z(F)}{\#\operatorname{PSL}_2({\mathbb{Z}})_z} = \sum_{w\in \overline\Gamma\backslash \operatorname{PSL}_2({\mathbb{Z}})z} \frac{v_w(f)}{\#\overline\Gamma_w}.\] By summing over a set of representatives for \(\operatorname{PSL}_2({\mathbb{Z}})\backslash {\mathbb{H}}\) we finally obtain \[\begin{align*} \sum_{z\in\operatorname{PSL}_2({\mathbb{Z}})\backslash{\mathbb{H}}} \frac{v_z(F)}{\#\operatorname{PSL}_2({\mathbb{Z}})_z} &=\sum_{z\in\operatorname{PSL}_2({\mathbb{Z}})\backslash {\mathbb{H}}}\sum_{w\in\overline\Gamma\backslash \operatorname{PSL}_2({\mathbb{Z}})z} \frac{v_w(f)}{\#\overline\Gamma_w}=\sum_{w\in\overline\Gamma\backslash{\mathbb{H}}} \frac{v_w(f)}{\#\overline\Gamma_w}. \end{align*}\] In order to conclude the proof it remains to be shown that \(v_\infty(F) = \sum_{P\in\operatorname{Cusps}(\Gamma)} v_P(f)\). We first prove it assuming that \(\overline\Gamma\) is normal in \(\operatorname{PSL}_2({\mathbb{Z}})\). In this case, we have \[\begin{align*} d_\Gamma v_\infty(F) &= \sum_{P\in\operatorname{Cusps}(\Gamma)} h_\Gamma(P)v_\infty(F)\\ &=\sum_{P\in\operatorname{Cusps}(\Gamma)} v_P(F)\\ &=\sum_{P\in\operatorname{Cusps}(\Gamma)} \sum_{\gamma\in R} v_{\gamma P}(f)\\ &=\sum_{P\in\operatorname{Cusps}(\Gamma)} \sum_{P'\in\operatorname{Cusps}(\Gamma)} \#\{\gamma\in R ~|~ \gamma P = P'\} v_{P'}(f)\\ &=\sum_{P'\in\operatorname{Cusps}(\Gamma)} \sum_{P\in\operatorname{Cusps}(\Gamma)} \#\{\gamma\in R ~|~ \gamma P = P'\} v_{P'}(f)\\ &=d_\Gamma\sum_{P'\in\operatorname{Cusps}(\Gamma)} v_{P'}(f). \end{align*}\] Note that any congruence subgroup \(\Gamma\) contains a subgroup (for instance \(\Gamma(N)\)) which is normal in \({\operatorname{SL}}_2({\mathbb{Z}})\), and such that it is of finite index. Therefore it is enough to show that, if \(\Gamma'\subset \Gamma\) have finite index and \(g\) is weakly modular of weight \(k\) for \(\Gamma\), then \[\sum_{P'\in \operatorname{Cusps}(\Gamma')} v_{P'}(f) = \frac{d_{\Gamma'}}{d_\Gamma} \sum_{P'\in \operatorname{Cusps}(\Gamma')} v_{P'}(f).\] Let \(P\in\operatorname{Cusps}(\Gamma)\) and \(P'\in\operatorname{Cusps}(\Gamma')\) be such that \([P]=[P']\) inside \(\operatorname{Cusps}(\Gamma)\). Pick \(\sigma\in{\operatorname{SL}}_2({\mathbb{Z}})\) such that \(\sigma\infty = [P]\) in \(\operatorname{Cusps}(\Gamma)\). Write also \(n_0 = v_P^\Gamma(g)\), and \(m = \frac{h_{\Gamma'}}{h_\Gamma}\). Then: \[(g|_k\sigma)(z) = \sum_{n\geq n_0} a_n(g) e^{\frac{2\pi i n z}{h_\Gamma}} = \sum_{n\geq n_0} a_n(g)e^{\frac{2\pi i nmz}{h_{\Gamma'}}} = \sum_{n\geq mn_0} a_n(g)e^{\frac{2\pi i nz}{h_{\Gamma'}}}.\] Hence we have \(v_{P'}(g) = m v_P(g)\), as we wanted. This concludes the proof of the valence formula.

As for level \(1\), the valence formula gives a criterion for equality of modular forms:

Corollary 2.2 Let \(f\) and \(g\) be two modular forms in \(M_k(\Gamma)\), whose \(q\)-expansions (at one cusp of \(\Gamma\)) coincide up to the term \(\frac{k}{12}[\operatorname{PSL}_2({\mathbb{Z}})\colon\overline\Gamma]\). Then \(f\) and \(g\) are equal.

There are dimension formulas for congruence subgroups (see (Diamond and Shurman 2005, vol. 228, chap. 3)) but we will not see them in this course.