6 Modular forms

In this section, we recall and develop some theory of (quasi)modular forms.

6.1 Modular forms and examples

Let \(\mathfrak {H}\) be the upper half-plane \(\{ z\in \mathbb {C}\mid \Im (z){\gt}0\} \).

Lemma 6.1

✓

The modular group \(\Gamma _1:=\mathrm{SL}_2(\mathbb {Z})\) acts on \(\mathfrak {H}\) by linear fractional transformations

\[ \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)z:=\frac{az+b}{cz+d}. \]

Let \(N\) be a positive integer.

Definition 6.2

The level \(N\) principal congruence subgroup of \(\Gamma _1\) is

\[ \Gamma (N):=\left\{ \left.\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)\in \Gamma _1\right|\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)\equiv \left(\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right)\; \mathrm{mod}\; N\right\} . \]

Definition 6.3

✓

A subgroup \(\Gamma \subset \Gamma _1\) is called a congruence subgroup if \(\Gamma (N)\subset \Gamma \) for some \(N\in \mathbb {N}\).

Definition 6.4

✓

Define the matrices

\[ S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \in \Gamma _1, T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \in \Gamma _1, \alpha = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \in \Gamma _2 \subset \Gamma _1, \beta = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} \in \Gamma _2 \subset \Gamma _1. \]

It is easily verifiable that \(\alpha = T^2\) and \(\beta = -S\alpha ^{-1}S = -ST^{-2}S\).

The following two lemmas tell us the group structure of \(\Gamma (1) = \Gamma _1\) and \(\Gamma (2)\), which we will use later on to define the theta forms.

Lemma 6.5

We have \(\Gamma (1) = \langle S, T, -I \rangle \).

Proof ▶

See [ 4 , Exercise 1.1.1 ] .

Lemma 6.6

We have \(\Gamma (2) = \langle \alpha , \beta , -I \rangle \).

Proof ▶

See [ 4 , Exercise 1.2.4 ] .

Let \(z\in \mathfrak {H}\), \(k\in \mathbb {Z}\), and \(\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)\in \mathrm{SL}_2(\mathbb {Z})\). We omit many of the proofs below when they exist in Mathlib already.

Definition 6.7

The automorphy factor of weight \(k\) is defined as

\[ j_k(z,\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)):=(cz+d)^{-k}. \]

Lemma 6.8

✓

The automorphy factor satisfies the chain rule

\[ j_k(z,\gamma _1\gamma _2)=j_k(z,\gamma _1)\, j_k(\gamma _2z,\gamma _1). \]

Definition 6.9

✓

Let \(F\) be a function on \(\mathfrak {H}\) and \(\gamma \in \mathrm{SL}_2(\mathbb {Z})\). Then the slash operator acts on \(F\) by

\[ (F|_k\gamma )(z):=j_k(z,\gamma )\, F(\gamma z). \]

Lemma 6.10

✓

The chain rule implies

\[ F|_k\gamma _1\gamma _2=(F|_k\gamma _1)|_k\gamma _2. \]

In particular, this lemma implies that if \(\Gamma = \langle M_i \rangle _{i \in \mathcal{I}}\), then the slash action \(F|\gamma \) is uniquely determined by the action of generators, i.e. \(F|M_i\) and \(F|M_i^{-1}\).

Lemma 6.11

For even \(k\), \(F|_{k}(-I) = F\).

Proof ▶

Follows from the definition of the slash operator: \((F|_{k}(-I))(z) = (-1)^{-k}F((-I)z) = F(z)\).

Definition 6.12

✓

A (holomorphic) modular form of integer weight \(k\) and congruence subgroup \(\Gamma \) is a holomorphic function \(f:\mathfrak {H}\to \mathbb {C}\) such that:

(Slash invariant) \(f|_k\gamma =f\) for all \(\gamma \in \Gamma \)
(Holomorphic at \(i\infty \)) for each \(\alpha \in \Gamma _1\; f|_k\alpha \) has the Fourier expansion \(f|_k\alpha (z)=\sum _{n=0}^\infty c_f(\alpha ,\frac{n}{n_\alpha })\, e^{2\pi i \frac{n}{n_\alpha }z}\) for some \(n_\alpha \in \mathbb {N}\) and Fourier coefficients \(c_f(\alpha ,m)\in \mathbb {C}\).

Definition 6.13

✓

Let \(M_k(\Gamma )\) be the space of modular forms of weight \(k\) and congruence subgroup \(\Gamma \).

Let us consider several examples of modular forms.

Definition 6.14

For an even integer \(k\geq 4\) we define the weight \(k\) Eisenstein series as

\begin{equation} \label{eqn:Ek-definition} E_k(z):=\frac{1}{2\zeta (k)}\sum _{(c,d)\in \mathbb {Z}^2\backslash (0,0)}(cz+d)^{-k}.\end{equation}

Lemma 6.15

For all \(k\), \(E_k\in M_k(\Gamma _1)\). Especially, we have

\begin{equation} \label{eqn:Ek-trans-S} E_k \left(-\frac{1}{z}\right) = z^k E_k(z). \end{equation}

Proof ▶

This follows from the fact that the sum converges absolutely. Now apply slash operator with \(\gamma = \left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)\) gives 10.

Lemma 6.16

The Eisenstein series possesses the Fourier expansion

\begin{equation} \label{eqn:Ek-Fourier}E_k(z)=1+\frac{2}{\zeta (1-k)}\sum _{n=1}^\infty \sigma _{k-1}(n)\, e^{2\pi i z}, \end{equation}

where \(\sigma _{k-1}(n)\, =\, \sum _{d|n} d^{k-1}\). In particular, we have

\begin{align} E_4(z)\, =\, & 1+240\sum _{n=1}^\infty \sigma _3(n)\, e^{2\pi i n z} \notag \\ E_6(z)\, =\, & 1-504\sum _{n=1}^\infty \sigma _5(n)\, e^{2\pi i n z}. \notag \end{align}

The infinite sum 9 does not converge absolutely for \(k=2\). On the other hand, the expression 11 converges to a holomorphic function on the upper half-plane and we will take it as a definition of \(E_2\) (See Definition 6.18 below).

The discriminant form is a unique normalized cusp form of weight 12, which can be defined as:

Definition 6.17

The discriminant form \(\Delta (z)\) is given by

\begin{equation} \label{eqn:disc-definition} \Delta (z) = e^{2 \pi i z} \prod _{n \ge 1} (1 - e^{2 \pi i n z})^{24}. \end{equation}

This product formula allows us to prove positivity of \(\Delta (it)\) for \(t {\gt} 0\) later. But we need to first check its a modular form. For this we first need some definitions/ results.

We define it as a \(q\)-series, which gives a holomorphic function on \(\mathfrak {H}\).

Definition 6.18

We set

\begin{equation} \label{eqn:E2} E_2(z):= 1-24\sum _{n=1}^\infty \sigma _1(n)\, e^{2\pi i n z}. \end{equation}

Lemma 6.19

This function is not modular, however it satisfies

\begin{equation} \label{eqn:E2-S-transform} z^{-2}\, E_2\left(-\frac{1}{z}\right) = E_2(z) -\frac{6i}{\pi }\, \frac{1}{z}. \end{equation}

Proof ▶

This is excercise 1.2.8 of [ 4 ] .

Definition 6.20

The Dedekind eta function is defined as

\[ \eta (z) = q^{1/24} \prod _{n \ge 1} (1 - q^n) \]

where \(q = e^{2\pi i z}\).

Lemma 6.21

The Dedekind eta function transforms as

\[ \eta \left(-\frac{1}{z}\right) = \sqrt{-iz} \eta (z). \]

Proof ▶

Cosider the logarithmic derivative of \(\eta \), which one can easily see is equal to \(\frac{\pi i}{12} E_2\). The result then follows from the transformation of \(E_2\).

See [ 4 , proposition 1.2.5 ] .

Lemma 6.22

\(\Delta (z) \in M_{12}(\Gamma _1)\). Especially, we have

\begin{equation} \label{eqn:disc-trans-S} \Delta \left(-\frac{1}{z}\right) = z^{12} \Delta (z). \end{equation}

Also, it vanishes at the unique cusp, i.e. it is a cusp form of level \(\Gamma _1\) and weight \(12\).

Proof ▶

The fact that it is invariant under translation is clear from the definition, so we only need to check transformation under \(S\). Now, note that \(\eta ^24 = \Delta \), and from 6.21 we have \(\eta (-1/z) = \sqrt{-iz} \eta (z)\), so \(\Delta (-1/z) = z^{12} \Delta (z)\) as required.

Using this one can now easily check that we have

Lemma 6.23

\begin{equation} \label{eqn:E2-transform-general} (cz + d)^{-2} E_2\left(\frac{az + b}{cx + d}\right) = E_2(z) - \frac{6ic}{\pi (cz + d)}, \quad \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_{2}(\mathbb {Z}). \end{equation}

Proof ▶

Modularity of \(\Delta (z)\) gives \((cz + d)^{-12}\Delta (\frac{az + b}{cz + d}) = \Delta (z)\) for \(\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right) \in \Gamma _1\), and by differentiating it we get

\begin{equation} (cz + d)^{-14} \Delta '\left(\frac{az + b}{cz + d}\right) = \Delta '(z) - \frac{6ic}{\pi (cz + d)} \Delta (z). \end{equation}

Now, divide both sides with \(\Delta (z)\) proves 16.

Lemma 6.24

We have

\begin{equation} \Delta (z) = (E_4^3-E_6^2)/1728. \end{equation}

Proof ▶

We only need to show its a cuspform, since once we have this, dividing the rhs by \(\Delta \) would give a modular form of weight \(0\) which is a constant, and so we can determine the constant easily.

To checke its a cuspform, we just look at the \(q\)-expansions of \(E_4\) and \(E_6\) and prove directly that the first term vanishes.

Corollary 6.25

\(\Delta (it) {\gt} 0\) for all \(t {\gt} 0\).

Proof ▶

By 6.17, we have

\[ \Delta (it) = e^{-2 \pi t} \prod _{n \ge 1} (1 - e^{-2 \pi n t})^{24} {\gt} 0. \]

The following nonvanishing result, which directly follows from Definition 6.17, will be used in the construction of the magic function.

Corollary 6.26

\(\Delta (z) \neq 0\) for all \(z \in \mathfrak {H}\).

Proof ▶

This follows from the product formula.

A key fact in the theory of modular forms is that the spaces \(M_k(\Gamma )\) are finite-dimensional. To prove this we will do use the following non-standard proof. First we have the following result.

Theorem 6.27

Let \(k \in \mathbb {Z}\) with \(k {\lt} 0\). Then \(M_k(\Gamma _1) = \{ 0\} \) and moreover \(\dim M_0(\Gamma (1)) = 1\).

Proof ▶

The proof makes use of the maximum modulus principle, as its already been formalised we skipt the details here but see the lean proof for details.

Theorem 6.28

Let \(k \in \mathbb {Z}\) with \(k \ge 0\) and even. Then \(\dim M_k(\Gamma _1) = \lfloor k / 12 \rfloor \) if \(k \equiv 2 \mod 12\) and \(\dim M_k(\Gamma _1) = \lfloor k / 12 \rfloor + 1\) if \(k \not\equiv 2 \mod 12\).

Proof ▶

First we note that for \(2 {\lt} k\) we have \(\dim (M_k(\Gamma _1)) = 1 + \dim S_k(\Gamma _1)\). This follows since we know the \(E_k\) are in \(M_k\) so by scalling appropriately, any non-cuspform \(f \in M_k\) we would have \(f - a E_k \in S_k\) for some \(a\).

Next, note that \(S_k(\Gamma _1)\) is isomorphic to \(M_{k-12}(\Gamma _1)\), since if \(f \in S_k\) then \(f/ \Delta \) is now a modular form (using the product expansion of \(\Delta \) and its non-vanishing on \(\mathfrak {H}\)) of weight \(k-12\). Note its important that \(f\) is a cuspform so that the quotient by \(\Delta \) is a modular form.

So we only need to know the dimensions of \(M_k(\Gamma _1)\) for \(0 \le k \le 12\). For \(k = 0\) we have \(\dim M_0(\Gamma _1) = 1\) by Theorem 6.27. For \(k = 4\) we have \(\dim M_4(\Gamma _1) = 1\) since if there was a cuspform \(f\) of weight \(4\) then \(f/ \Delta \) would be a modular form of negative weight, i.e. zero, so \(f=0\). Similarly for \(k=6,8,10\). For \(k=12\) we have \(\dim S_{12}(\Gamma _1) = 1\) since the discriminant form is a cusp form of weight \(12\) and any other cusp form of weight \(12\) would be a scalar multiple of \(\Delta \) (since their ratio would be a modular form of weight \(0\)). So we have \(\dim M_{12}(\Gamma _1) = 2\).

Finally we need to check that \(\dim M_2(\Gamma _1) = 0\). Firstly, there can’t be any cuspforms here by the same argument as above. So we need to check that there are no modular forms of weight \(2\). If we did have one, call it \(f\) then \(f^2\) would be a non-cuspform of weight \(4\) and so \(f^2 = a E_4\), where in fact \(a=a_0(f)^2\) (since \((f^2-a_0(f)E_4)\) is now a cuspform of weight \(4\) which means its zero). Similarly, \(f^3 = a_0(f)^3 E_6\). But now taking powers to make them weight \(12\) forms we see that \(a_0(f)^6(E_4^3 - E_6^2) = 0 = 1728 a_0(f)^6 \Delta \) but \(a_0(f) \ne 0\) (since its assumed to not be a cuspform), this would mean \(\Delta =0\) which we know can’t happen.

Theorem 6.29

Let \(\Gamma \) be a congruence subgroup. Then \(M_k(\Gamma )\) is finite-dimensional.

Proof ▶

We know that \(\dim (M_k(\Gamma _1))\) is finite dimensional from the above, now this means that there is some \(r_k\) such that any element of \(M_k(\Gamma _1)\) vanishing at infinity to degree \({\gt} r_k\) must be zero. Now take \(f \in M_k(\Gamma )\) and vanishes to degree \(n\) at infinity, then consider \(F = \prod _\gamma f\mid _k \gamma \) where the product is over a set of representatives of \(\Gamma _1 \backslash \Gamma \). Then \(F\) is a modular form of weight \(k d\) where \(d = [\Gamma _1: \Gamma ]\) and vanishes at infinity to degree at least \(n\). So if \(n {\gt} r_{kd}\) then \(F=0\) meaning the \(f=0\).

Corollary 6.30

We have

\begin{align} \dim M_2(\mathrm{SL}_{2}(\mathbb {Z})) & = 0, \label{eqn:dimM2} \\ \dim M_4(\mathrm{SL}_{2}(\mathbb {Z})) & = 1, \label{eqn:dimM4} \\ \dim M_6(\mathrm{SL}_{2}(\mathbb {Z})) & = 1, \label{eqn:dimM6} \\ \dim M_8(\mathrm{SL}_{2}(\mathbb {Z})) & = 1, \label{eqn:dimM8} \\ \dim S_4(\mathrm{SL}_{2}(\mathbb {Z})) & = 0, \label{eqn:dimS4} \\ \dim S_6(\mathrm{SL}_{2}(\mathbb {Z})) & = 0, \label{eqn:dimS6} \\ \dim S_8(\mathrm{SL}_{2}(\mathbb {Z})) & = 0. \label{eqn:dimS8} \end{align}

Another examples of modular forms we would like to consider are theta functions [ 13 , Section 3.1 ] .

Definition 6.31

We define three different theta functions (so called “Thetanullwerte”) as

\begin{align} \Theta _{2}(z) = \theta _{10}(z)\, =\, & \sum _{n\in \mathbb {Z}}e^{\pi i (n+\frac12)^2 z}. \notag \\ \Theta _{3}(z) = \theta _{00}(z)\, =\, & \sum _{n\in \mathbb {Z}}e^{\pi i n^2 z} \notag \\ \Theta _{4}(z) = \theta _{01}(z)\, =\, & \sum _{n\in \mathbb {Z}}(-1)^n\, e^{\pi i n^2 z} \notag \\ \end{align}

For convenience, we use the following notations for the fourth powers of the theta functions.

Definition 6.32

Define

\begin{equation} H_2 = \Theta _2^4, \quad H_3 = \Theta _3^4, \quad H_4 = \Theta _4^4. \label{eqn:H2-H3-H4} \end{equation}

Note that we only need these fourth powers to define 7.18.

The group \(\Gamma _1\) is generated by the elements \(T=\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)\), \(S=\left(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right)\), and \(-I = \left(\begin{smallmatrix} -1 & 0 \\ 0 & -1 \end{smallmatrix}\right)\) (Lemma 6.5), and the transformation of functions under \(\Gamma (2)\) is determined by that under \(\Gamma _1\) (by Lemma 6.10). The following lemma shows how the theta functions (and their powers) transform under the slash action of these matrices.

Lemma 6.33

✓

These elements act on the theta functions in the following way

\begin{align} H_2 | S & = -H_4 \label{eqn:H2-transform-S} \\ H_3 | S & = -H_3 \label{eqn:H3-transform-S} \\ H_4 | S & = -H_2 \label{eqn:H4-transform-S} \end{align}

and

\begin{align} H_2 | T & = -H_2 \label{eqn:H2-transform-T} \\ H_3 | T & = H_4 \label{eqn:H3-transform-T} \\ H_4 | T & = H_3 \label{eqn:H4-transform-T} \end{align}

Proof ▶

The last three identities easily follow from the definition. For example, 31 follows from

\begin{align} \Theta _{2}(z + 1) & = \sum _{n\in \mathbb {Z}}e^{\pi i (n+\frac12)^2 (z + 1)} = \sum _{n \in \mathbb {Z}} e^{\pi i (n + \frac{1}{2})^{2}} e^{\pi i (n + \frac{1}{2})^{2} z} \\ & = \sum _{n \in \mathbb {Z}} e^{\pi i (n^2 + n + \frac{1}{4})} e^{\pi i (n + \frac{1}{2})^{2} z} = \sum _{n \in \mathbb {Z}} (-1)^{n^2 + n}e^{\pi i / 4} e^{\pi i (n + \frac{1}{2})^{2} z} \\ & = e^{\pi i / 4} \Theta _{2}(z) \end{align}

and taking 4th power. 28 and 30 are equivalent under \(z \leftrightarrow -1/z\), so it is enough to show 28 and 29. These identities follow from the identities of the two-variable Jacobi theta function, which is defined as (be careful for the variables, where we use \(\tau \) instead of \(z\))

\begin{equation} \theta (z, \tau ) = \sum _{n \in \mathbb {Z}} e^{2 \pi i n z + \pi i n^2 \tau } \label{eqn:jacobi2} \end{equation}

and already formalized by David Loeffler. This function specialize to the theta functions as

\begin{align} \Theta _{2}(\tau ) & = e^{\pi i \tau / 4} \theta (-\tau / 2, \tau ) \label{eqn:Th2-as-jacobi2} \\ \Theta _{3}(\tau ) & = \theta (0, \tau ) \label{eqn:Th3-as-jacobi2} \\ \Theta _{4}(\tau ) & = \theta (1/2, \tau ) \label{eqn:Th4-as-jacobi2} \\ \end{align}

Possion summation formula gives

\begin{equation} \theta (z, \tau ) = \frac{1}{\sqrt{-i \tau }} e^{-\frac{\pi i z^2}{\tau }} \theta \left(\frac{z}{\tau }, -\frac{1}{\tau }\right) \label{eqn:jacobi2transform} \end{equation}

and applying the specializations above yield the identities. For example, 30 follows from

\begin{equation} \Theta _{4}(\tau ) = \theta \left(\frac{1}{2}, \tau \right) = \frac{1}{\sqrt{-i\tau }} e^{- \frac{\pi i }{4 \tau }} \theta \left(\frac{1}{2 \tau }, -\frac{1}{\tau }\right) = \frac{1}{\sqrt{-i\tau }} \Theta _{2}\left(-\frac{1}{\tau }\right) \end{equation}

and taking 4th power.

Using the above identities, we can prove that these are modular forms.

Lemma 6.34

✓

\(H_{2}\), \(H_{3}\), and \(H_{4}\) are slash invariant under \(\Gamma (2)\), i.e. for all \(\gamma \in \Gamma (2)\) and \(i \in \{ 2, 3, 4\} \), we have \(H_i|\gamma = H_i|\gamma ^{-1} = H_i\).

Proof ▶

By lemma 6.6 and lemma 6.10, it suffices to show that the \(H_i\) are invariant under slash actions with respect to \(\alpha \), \(\beta \), and \(-I\). Invariance under \(-I\) follows from Lemma 6.11. The rest follows from Lemma 6.10, 6.33, and the matrix identities

\begin{equation} \alpha = T^2, \quad \beta = -S\alpha ^{-1}S = -ST^{-2}S. \label{eqn:matrix} \end{equation}

For example, invariance for \(H_2\) can be proved by

\begin{align} H_2|\alpha & = H_2 |T^{2} = -H_2 |T = H_2 \\ H_2|\beta & = H_2 |(-S\alpha ^{-1}S) = H_2 | (S\alpha ^{-1}S) =-H_4 |(\alpha ^{-1}S) = -H_4 |S = H_2. \end{align}

Lemma 6.35

For all \(\gamma \in \Gamma _1\), \(H_{2}|_2 \gamma \), \(H_{3}|_2 \gamma \), and \(H_{4}|_2 \gamma \) are holomorphic at \(i\infty \).

Proof ▶

We want to show that for \(\gamma \in \Gamma _1\), \(\| H_2|_2\gamma (z)\| \) is bounded as \(z \in \mathbb {H} \to i\infty \). Firstly, by Lemma 6.33, Lemma 6.6 and induction on group elements, we notice that \(\{ \pm H_2, \pm H_3, \pm H_4\} \) is closed under action by \(\Gamma _1\). Hence, it suffices to prove that \(H_2\), \(H_3\) and \(H_4\) are bounded at \(i\infty \). Consider \(z \in \mathbb {H}\) with \(\Im (z) \geq A\). We proceed by direct algebraic manipulation:

\begin{align} \| H_2(z)\| & = \left\| \sum _{n \in \mathbb {Z}} \exp \left(\pi i \left(n + \frac{1}{2}\right)^2 z\right)\right\| ^4 \leq \left(\sum _{n \in \mathbb {Z}} \left\| \exp \left(\pi i \left(n + \frac{1}{2}\right)^2 z\right)\right)\right\| ^4 \\ & = \left(\sum _{n \in \mathbb {Z}} \left\| \exp \left(-\pi \left(n + \frac{1}{2}\right)^2 \Im (z)\right)\right)\right\| ^4 \leq \left(\sum _{n \in \mathbb {Z}} \left\| \exp \left(-\pi \left(n + \frac{1}{2}\right)^2 A\right)\right)\right\| ^4 \end{align}

Where we prove the final term is convergent by noticing that it equals \(\exp (-\pi A / 4)\theta (iA / 2, iA)\), which has been shown to converge in Mathlib. The proofs for \(H_3\) and \(H_4\) are similar (actually easier) and have been omitted.

redIt seems the MDifferentiable requirement is missing.

Lemma 6.36

\(H_{2}\), \(H_{3}\), and \(H_{4}\) belong to \(M_2(\Gamma (2))\).

Proof ▶

From lemma 6.34 and lemma 6.35, it remains ot prove that \(H_2\), \(H_3\) and \(H_4\) are holomorphic on \(\mathbb {H}\). redfill in proof.

They have Fourier expansions as follows.

Proposition 6.37

\(H_2\) admits a Fourier series of the form

\begin{equation} H_2(z) = \sum _{n \ge 1} c_{H_2}(n) e^{\pi i n z} \end{equation}

for some \(c_{H_2}(n) \in \mathbb {R}_{\ge 0}\), with \(c_{H_2}(1) = 16\) and \(c_{H_2}(n) = O(n^k)\) for some \(k \in \mathbb {N}\).

Proof ▶

We have

\begin{align} H_2(z) & = \Theta _2(z)^4 \\ & = \left(\sum _{n \in \mathbb {Z}} e^{\pi i (n + \frac{1}{2})^{2} z}\right)^{4} \\ & = \left(2\sum _{n \ge 0} e^{\pi i (n + \frac{1}{2})^{2} z}\right)^{4} \\ & = \left(2 e^{\pi i z / 4} + 2 \sum _{n \ge 1} e^{\pi i (n^2 + n + \frac{1}{4}) z}\right)^{4} \\ & = 16 e^{\pi i z}\left(1 + \sum _{n \ge 1} e^{\pi i (n^2 + n)z}\right)^{4} \\ & = 16 e^{\pi i z} + \sum _{n \ge 2} c_{H_2}(n) e^{\pi i n z} \\ & = \sum _{n \ge 1} c_{H_2}(n) e^{\pi i n z}. \end{align}

Proposition 6.38

\(H_3\) admits a Fourier series of the form

\begin{equation} H_3(z) = \sum _{n \ge 0} c_{H_3}(n) e^{\pi i n z} \end{equation}

for some \(c_{H_3}(n) \in \mathbb {R}_{\ge 0}\) with \(c_{H_3}(0) = 1\) and \(c_{H_3}(n) = O(n^k)\) for some \(k \in \mathbb {N}\). Especially, \(H_3\) is not cuspidal.

Proof ▶

We have

\begin{equation} H_3(z) = \Theta _3(z)^{4} = \left(\sum _{n \in \mathbb {Z}} e^{\pi i n^2 z}\right)^{4} = \left(1 + 2 \sum _{n \ge 1} e^{\pi i n^2 z}\right)^{4} = 1 + O(e^{\pi i z}). \end{equation}

Proposition 6.39

\(H_4\) admits a Fourier series of the form

\begin{equation} H_4(z) = \sum _{n \ge 0} c_{H_4}(n) e^{\pi i n z} \end{equation}

for some \(c_{H_4}(n) \in \mathbb {R}\) with \(c_{H_4}(0) = 1\) and \(c_{H_4}(n) = O(n^k)\) for some \(k \in \mathbb {N}\). Especially, \(H_4\) is not cuspidal.

We also have a nontrivial relation between these theta functions.

Lemma 6.40

These three theta functions satisfy the Jacobi identity

\begin{equation} \label{eqn:jacobi-identity} H_{2} + H_{4} = H_{3} \Leftrightarrow \Theta _{2}^4 + \Theta _{4}^4 = \Theta _{3}^4. \end{equation}

Proof ▶

Let \(f = (H_2 + H_4 - H_3)^{2}\). Obviously, \(f\) is a modular form of weight \(4\) and level \(\Gamma (2)\). However, by using the transformation rules of \(H_2, H_3, H_4\), one have

\begin{align} f|_{S} & = (-H_4 - H_2 + H_3)^{2} = f\\ f|_{T} & = (-H_2 + H_3 - H_4)^{2} = f \end{align}

so \(f\) is actually a modular form of level \(1\). By considering the limit as \(z \to i\infty \), \(f\) is a cusp form, so we get \(f = 0\) from 23.

These are also related to \(E_4\), \(E_6\), and \(\Delta \) as follows.

Lemma 6.41

We have

\begin{align} E_4 & = \frac{1}{2}(H_{2}^{2} + H_{3}^{2} + H_{4}^{2}) = H_{2}^{2} + H_{2}H_{4} + H_{4}^{2} \label{eqn:e4theta} \\ E_6 & = \frac{1}{2} (H_{2} + H_{3})(H_{3} + H_{4}) (H_{4} - H_{2}) = \frac{1}{2}(H_2 + 2H_4)(2H_2 + H_4)(H_4 - H_2) \label{eqn:e6theta} \\ \Delta & = \frac{1}{256} (H_{2}H_{3}H_{4})^2. \label{eqn:disctheta} \end{align}

Proof ▶

We can prove these similarly as Lemma 6.40. Right hand sides of 63, 64, and 65 are all modular forms of level \(\Gamma _1\) and desired weights, where 65 is a cusp form since \(H_2\) is. Now the identities follow from the dimension calculations \(\dim M_4(\Gamma _1) = \dim M_6(\Gamma _1) = \dim S_{12}(\Gamma _1) = 1\) and comparing the first nonzero \(q\)-coefficients.

The strict positivity of Jacobi theta functions might needed later.

Corollary 6.42

All three functions \(t \mapsto H_2(it), H_3(it), H_4(it)\) are positive for \(t {\gt} 0\).

Proof ▶

By Lemma 6.40 and the transformation law 28, it is enough to prove the positivity for \(\Theta _2(it)\), which is clear from its definition:

\begin{equation} \Theta _{2}(it) = \sum _{n \in \mathbb {Z}} e^{- \pi (n + \frac{1}{2})^{2} t} {\gt} 0. \end{equation}

6.2 Quasimodular forms and derivatives

Morally, quasimodular forms can be thought as modular forms with differentiations. It can be defined formally as follows: Let \(f: \mathfrak {H} \to \mathbb {C}\) be a holomorphic function, and let \(k\) and \(s \ge 0\) be integers. The function \(f\) is a quasimodular form of weight \(k\), level \(\Gamma \), and depth \(s\) if there exist holomorphic functions \(f_0, \dots , f_s : \mathfrak {H} \to \mathbb {C}\) such that

\begin{equation} \label{eqn:quasimod-def} (f|_{k}\gamma )(z) = (cz + d)^{-k} f\left(\frac{az + b}{cz + d}\right) = \sum _{j=0}^{s} f_j(z) \left(\frac{c}{cz + d}\right)^j \end{equation}

for all \(z \in \mathfrak {H}\) and \(\gamma = \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right) \in \Gamma \).

By taking \(\gamma = \left(\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right)\), one can check that we should have \(f_0 = f\). Thus, a quasimodular form of depth \(0\) is just a modular form of same weight and level. Also, it is easy to see that the space of quasimodular forms is closed under the normalized derivative.

Definition 6.43

Let \(F\) be a quasimodular form. We define the (normalized) derivative of \(F\) as

\begin{equation} \label{eqn:derivative} F' = DF := \frac{1}{2\pi i} \frac{\mathrm{d}}{\mathrm{d}z} F. \end{equation}

\(D\) is normalized as in 68 because of the following lemma.

Lemma 6.44

We have an equality of operators \(D = q \frac{\mathrm{d}}{\mathrm{d}q}\). In particular, the \(q\)-series of the derivative of a quasimodular form \(F(z) = \sum _{n \ge n_0} a_n q^n\) is \(F'(z) = \sum _{n \ge n_0} n a_n q^n\).

Proof ▶

Directly follows from the definition 6.43, where \(\frac{1}{2 \pi i}\frac{\mathrm{d}}{\mathrm{d}z}e^{2\pi i n z} = n e^{2\pi i n z}\).

The most important quasimodular form is the weight 2 Eisenstein series \(E_2\).

Definition 6.45

For \(k \in \mathbb {R}\), define the weight \(k\) Serre derivative \(\partial _{k}\) of a modular form \(F\) as

\begin{equation} \label{eqn:serre-der} \partial _{k}F := F' - \frac{k}{12} E_2 F. \end{equation}

Theorem 6.46

Let \(F\) be a modular form of weight \(k\) and level \(\Gamma \). Then, \(\partial _{k}F\) is a modular form of weight \(k + 2\) of the same level.

Proof ▶

Let \(G = \partial _{k}F = F' - \frac{k}{12}E_2 F\). It is enough to show that \(G\) is invariant under \(|_{k+2}\gamma \) for \(\gamma \in \Gamma \). From \(F \in M_k(\Gamma )\), we have

\begin{equation} (F|_{k}\gamma )(z) := (cz + d)^{-k} F\left(\frac{az + b}{cz + d}\right) = F(z), \quad \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma . \end{equation}

By taking the derivative of the above equation, we get

\begin{align} & -kc (cz + d)^{-k - 1} F\left(\frac{az + b}{cz + d}\right) + (cz + d)^{-k} (cz + d)^{-2} \frac{\mathrm{d}F}{\mathrm{d}z}\left(\frac{az + b}{cz + d}\right) = \frac{\mathrm{d}F}{\mathrm{d}z}(z) \\ & \Leftrightarrow (cz + d)^{-k - 2} F’\left(\frac{az + b}{cz + d}\right) = F’(z) - \frac{ikc}{2\pi (cz + d)}F(z). \end{align}

Combined with 16, we get

\begin{align} ((\partial _k F)|_{k+2}\gamma )(z) & = (cz + d)^{-k-2} \left(F’\left(\frac{az + b}{cz + d}\right) - \frac{k}{12}E_2\left(\frac{az + b}{cz + d}\right)F\left(\frac{az + b}{cz + d}\right)\right) \\ & = F’(z) - \frac{ikc}{2 \pi (cz + d)} F(z) - \frac{k}{12} \left(E_2 - \frac{6ic}{\pi (cz + d)}\right) F(z) \\ & = F’(z) - \frac{k}{12} E_2(z) F(z) = (\partial _{k} F)(z) \end{align}

so \(\partial _{k}F \in M_{k+2}(\Gamma )\).

Remark 6.47

More generally, the following theorem holds: if \(F\) is a quasimodular form of weight \(k\) and depth \(s\), then \(\partial _{k-s}F\) is a quasimodular form of weight \(k + 2\) and depth \(\le s\) of the same level. We will not prove this here.

Theorem 6.48

We have

\begin{align} E_2’ & = \frac{E_2^2 - E_4}{12} \label{eqn:DE2} \\ E_4’ & = \frac{E_2 E_4 - E_6}{3} \label{eqn:DE4} \\ E_6’ & = \frac{E_2 E_6 - E_4^2}{2} \label{eqn:DE6} \end{align}

Proof ▶

In terms of Serre derivatives, these are equivalent to

\begin{align} \partial _{1}E_2 & = -\frac{1}{12} E_4 \label{eqn:SE2} \\ \partial _{4}E_4 & = -\frac{1}{3} E_6 \label{eqn:SE4} \\ \partial _{6}E_6 & = -\frac{1}{2} E_4^2 \label{eqn:SE6} \end{align}

By Theorem 6.46, all the serre derivatives are, in fact, modular. To be precise, the modularity of \(\partial _{4} E_4\) and \(\partial _6 E_6\) directly follows from Theorem 6.46, and that of \(\partial _{1}E_2\) follows from 16. Differentiating and squaring then gives us the following:

\begin{align} E_2’|_{4}\gamma & = E_2’ - \frac{ic}{\pi (cz + d)} E_2 - \frac{3c^2}{\pi ^2 (cz + d)^2} \label{eqn:DE2-transform} \\ E_2^2|_{4}\gamma & = E_2^2 - \frac{12ic}{\pi (cz + d)} E_2 - \frac{36c^2}{\pi ^2 (cz + d)^2} \label{eqn:E2sq-transform} \end{align}

Hence, 76\(-\frac{1}{12}\)83 is a modular form of weight 4. By Corollary 6.30, they should be multiples of \(E_4, E_6, E_4^2\), and the proportionality constants can be determined by observing the constant terms of \(q\)-expansions.

Corollary 6.49

\begin{equation} \label{eqn:logder-disc-E2} \Delta ' = E_2 \Delta . \end{equation}

Proof ▶

By Ramanujan’s formula 77 and 78,

\begin{equation} \Delta ' = \frac{3 E_4^2 E_4' - 2 E_6 E_6'}{1728} = \frac{1}{1728} \left(3 E_4^2 \cdot \frac{E_2 E_4 - E_6}{3} - 2 E_6 \cdot \frac{E_2 E_6 - E_4^2}{2}\right) = \frac{E_2(E_4^3 - E_6^2)}{1728} = E_2\Delta . \end{equation}

Similar argument allow us to compute (Serre) derivatives of \(H_2, H_3, H_4\).

Proposition 6.50

We have

\begin{align} H_2’ & = \frac{1}{6} (H_{2}^{2} + 2 H_{2} H_{4} + E_2 H_2) \label{eqn:H2-der}\\ H_3’ & = \frac{1}{6} (H_{2}^{2} - H_{4}^{2} + E_2 H_3) \label{eqn:H3-der}\\ H_4’ & = -\frac{1}{6} (2H_{2} H_{4} + H_{4}^{2} - E_2 H_4) \label{eqn:H4-der} \end{align}

or equivalently,

\begin{align} \partial _{2} H_{2} & = \frac{1}{6} (H_{2}^{2} + 2 H_{2} H_{4}) \label{eqn:H2-serre-der} \\ \partial _{2} H_{3} & = \frac{1}{6} (H_{2}^{2} - H_{4}^{2}) \label{eqn:H3-serre-der} \\ \partial _{2} H_{4} & = -\frac{1}{6} (2H_{2} H_{4} + H_{4}^{2}) \label{eqn:H4-serre-der} \end{align}

Proof ▶

Equivalences are obvious from the definition of the Serre derivative. Define \(f_{2}, f_{3}, f_{4}\) be the differences of the left and right hand sides of 89, 90, 91.

\begin{align} f_{2} & := \partial _{2} H_{2} - \frac{1}{6} H_{2}(H_{2} + 2H_{4}) \\ f_{3} & := \partial _{2} H_{3} - \frac{1}{6} (H_{2}^2 - H_{4}^2) \\ f_{4} & := \partial _{2} H_{4} + \frac{1}{6} H_{4}(2H_{2} + H_{4}). \end{align}

Then these are a priori modular forms of weight \(4\) and level \(\Gamma (2)\), and our goal is to prove that they are actually zeros. By Jacobi’s identity 60, we have \(f_{2} + f_{4} = f_{3}\). Also, the transformation rules of \(H_2, H_3, H_4\) give

\begin{align} f_{2}|_{S} & = -f_{4} \\ f_{2}|_{T} & = -f_{2} \\ f_{4}|_{S} & = -f_{2} \\ f_{4}|_{T} & = f_{3} = f_{2} + f_{4}. \end{align}

Now, define

\begin{align} g & := (2 H_2 + H_4) f_2 + (H_2 + 2 H_4) f_4 \\ h & := f_{2}^{2} + f_{2}f_{4} + f_{4}^{2}. \end{align}

Then one can check that both \(g\) and \(h\) are invariant under the actions of \(S\) and \(T\), hence they are modular forms of level \(1\). Also, by analyzing the limit of \(g\) and \(h\) as \(z \to i \infty \), one can see that \(g\) and \(h\) are cusp forms, hence \(g = h = 0\) by 24 and 25. This implies

\begin{align} 3 E_4 f_2^{2} & = 3 (H_2^2 + H_2 H_4 + H_4^2) f_2^{2} = ((2 H_2 + H_4)^{2} - (2H_2 + H_4)(H_2 + 2H_4) + (H_2 + 2H_4)^{2}) f_2^{2}\\ & = (2 H_2 + H_4)^{2} (f_2^2 + f_2 f_4 + f_4^2) = 0 \end{align}

and by considering \(q\)-series (\(E_4\) has an invertible \(q\)-series), we get \(f_2 = 0\).

Theorem 6.51

The Serre derivative satisfies the following product rule: for any quasimodular forms \(F\) and \(G\),

\begin{equation} \partial _{w_1 + w_2} (FG) = (\partial _{w_1}F)G + F (\partial _{w_2}G). \end{equation}

103

Proof ▶

It follows from the definition:

\begin{align} \partial _{w_1 + w_2} (FG) & = (FG)’ - \frac{w_1 + w_2}{12} E_2 (FG) \\ & = F’G + FG’ - \frac{w_1 + w_2}{12} E_2(FG) \\ & = \left(F’ - \frac{w_1}{12}E_2 F\right)G + F \left(G’ - \frac{w_2}{12}E_2 G\right) \\ & = (\partial _{w_1}F)G + F(\partial _{w_2}G). \end{align}

We also have the following useful theorem for proving positivity of quasimodular forms on the imaginary axis, which is [ 7 , Proposition 3.5, Corollary 3.6 ] .

Theorem 6.52

Let \(F\) be a holomorphic quasimodular cusp form with real Fourier coefficients. Assume that there exists \(k\) such that \((\partial _{k}F)(it) {\gt} 0\) for all \(t {\gt} 0\). If the first Fourier coefficient of \(F\) is positive, then \(F(it) {\gt} 0\) for all \(t {\gt} 0\).

Proof ▶

By 84, we have

\begin{align} \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{F(it)}{\Delta (it)^{\frac{k}{12}}}\right) & = (-2 \pi ) \frac{F'(it) \Delta (it)^{\frac{k}{12}} - F(it) \frac{k}{12} E_{2}(it) \Delta (it)^{\frac{k}{12}}}{\Delta (it)^{\frac{k}{6}}} \\ & = (-2 \pi ) \frac{(\partial _{k} F)(it)}{\Delta (it)^{\frac{k}{12}}} {\lt} 0, \end{align}

hence

\[ t \mapsto \frac{F(it)}{\Delta (it)^{\frac{k}{12}}} \]

is monotone decreasing. Because of the assumption on the positivity of the first nonzero Fourier coefficient of \(F\), \(F(it) {\gt} 0\) for sufficiently large \(t\) since

\[ F = \sum _{n \geq n_{0}} a_{n} q^{n} \Rightarrow e^{2 \pi n_{0} t} F(it) = a_{n_{0}} + e^{-2 \pi t}\sum _{n\geq n_{0} + 1} a_{n} e^{-2 \pi (n - n_{0} - 1)t} \]

and \(\lim _{t \to \infty } e^{2 \pi n_{0}t} F(it) = a_{n_0} {\gt} 0\), hence the result follows.