# Modified weight 2 Eisenstein series is a modular form for $\Gamma_0(N)$

I'm doing exercise 1.2.8(e) in Diamond & Shurman's A First Course in Modular Forms. The problem is to show that $$G_{2,N}(\tau) := G_2(\tau)-NG_2(N\tau)$$ is in $$M_2(\Gamma_0(N))$$. To show this, I need to argue that $$G_{2,N}$$ satisfies $$G_{2,N}[\gamma]_2=G_{2,N}$$ where $$f[\gamma]_2(\tau):=j(\gamma,\tau)^{-2}f(\gamma(\tau))$$ is the weight-2 operator for every $$\gamma\in\Gamma_0(N)$$, and that $$G_{2,N}$$ is holomorphic on $$\mathcal{H}$$ and holomorphic at the cusps.

So far, I proven that $$G_2(\tau)$$ satisfies the transformation formula $$G_2[\gamma]_2(\tau) = G_2(\tau)-\frac{2\pi ic}{c\tau+d},\;\;\;\;\gamma = \begin{bmatrix}a&b\\c&d\end{bmatrix}$$ and that $$\frac{\pi}{j(\gamma,\tau)^2\Im(\gamma(\tau))}=\frac{\pi}{\Im(\tau)}-\frac{2\pi ic}{c\tau+d},$$ and I've concluded that $$G_2(\tau) - \frac{\pi}{\Im(\tau)}$$ is weight-2 invariant under $$SL_2(\mathbb Z)$$.

I have no idea how to conclude that $$G_{2,N}(\tau)$$ is a modular form for $$\Gamma_0(N)$$. Specifically, how does the weight-2 operator act on $$G_{2,N}$$? Is it $$G_{2,N}[\gamma]_2(\tau) = (c\tau+d)^{-2}(G_2(\gamma(\tau))-NG_2(\gamma(N\tau)))$$ or $$G_{2,N}[\gamma]_2(\tau) = (c\tau+d)^{-2}(G_2(\gamma(\tau)) - NG_2(N\gamma(\tau))?$$ In either case, $$N\gamma(\tau)$$ and $$\gamma(N\tau)$$ are not of the form $$\alpha(\tau)$$ for some $$\alpha\in SL_2(\mathbb Z)$$, so how should $$G_2(\tau)$$ transform?

Moreover, I don't know how to verify that $$G_{2,N}$$ is holomorphic on $$\mathcal{H}$$ or at the cusps. I would greatly appreciate detailed answers to both of these.

P.S. This is not homework, I'm simply trying to better understand computational aspects of modular forms.

• This was first proved by Ramanujan using elementary techniques and the modular form in question is central to the theory of series for $1/\pi$ given by Ramanujan. +1 for bringing it up. Commented Jan 29, 2021 at 10:34

Note that $$\tau \longmapsto NG_2(N\tau)$$ is the function $$G_2[\gamma]_2$$ with $$\gamma=\begin{bmatrix}N & 0\\0&1 \end{bmatrix}$$. Moreover, if $$\gamma’=\begin{bmatrix}a &b\\c&d\end{bmatrix} \in \Gamma_0(N)$$, then $$\gamma\gamma’=\begin{bmatrix} Na & Nb \\ c & d\end{bmatrix}=\begin{bmatrix} a & Nb \\ c/N & d\end{bmatrix}\gamma:=\gamma’’\gamma,$$ and $$\gamma’’ \in SL_2(\mathbb{Z})$$, so that $$G_{2,N}[\gamma’]_2(\tau)=G_2(\tau)-\frac{2i\pi c}{c\tau+d}-(G_2[\gamma’’]_2)[\gamma]_2(\tau)=G_2(\tau)-\frac{2i\pi c}{c\tau+d}-NG_2[\gamma’’]_2(N\tau)=G_2(\tau)-\frac{2i\pi c}{c\tau+d}-NG_2(N\tau)+N\frac{2i\pi (c/N)}{(c/N)(N\tau)+d}=G_{2,N}(\tau)$$.

Edit: Now let’s address the holomorphicity issues. First, we show that $$G_2[\gamma]_2$$ is holomorphic everywhere (including at infinity) for any $$\gamma \in SL_2(\mathbb{Z})$$. Given the transformation formula, it’s enough to show that $$G_2$$ is holomorphic everywhere and at infinity.

It’s enough to show that $$G_2(\tau)$$ is holomorphic on the half plane and bounded at infinity. But $$G_2(\tau)=2\zeta(2)+2\sum_{c \geq 1}{f(c\tau)}$$ where $$f(z)=\sum_{d \in \mathbb{Z}}{\frac{1}{(z+d)^2}}$$. As $$f$$ is the sum of a (locally) normally convergent series of holomorphic functions, $$f$$ is holomorphic.

Now, if $$z$$ has imaginary part at least $$A > 0$$ and real part between $$-1$$ and $$1$$, then $$f(z)=\sum_{d \in \mathbb{Z}}{\int_{-1/2}^{1/2}{\int_0^t{\frac{-2du}{(z+d+u)^3}}\,dt}},$$ which shows that $$|f(z)| \leq 2\sum_{d \in \mathbb{Z}}{\sup_{t \in (d-1/2,d+1/2)}\, |\frac{1}{(z+t)^3}|}$$.

The term for $$d=0$$ is at most $$\frac{1}{A^3}$$, and the term for $$d \geq 1$$ is at most $$\frac{1}{(A^2+(|d|-1/2)^2)^{3/2} } \leq \frac{8}{(A^2+d^2)^{3/2}} \leq \frac{16}{A^3+|d|^3}$$ by convexity.

It follows that $$|f(z)| \leq 2A^{-3}+32\sum_{d \geq 1}{(A^3+d^3)^{-1}}$$. By separating the cases $$d \geq A$$ and $$d \leq A$$, we get finally that $$|f(z)| \leq C(A^{-3}+A^{-2})$$ where $$C$$ is a numerical constant.

Since $$f$$ is $$1$$-periodic, it follows that $$|f(z)| \leq \frac{C}{|Im(z)|^3}+\frac{C}{|Im(z)|^2}$$ and therefore the series $$\sum_{c \geq 1}{f(cz)}$$ converges normally on all $$\{Im z > A\}$$, $$A>0$$.

It follows from this and the bound on $$f$$ that that series is a holomorphic function of $$z$$ and goes to zero as $$Im(z)$$ goes to infinity, so that $$G_2(z)$$ is holomorphic on the half-plane and at the cusps.

Thus, $$G_{2,N}$$ is also holomorphic on the half-plane, $$\Gamma_0(N)$$-invariant and bounded (hence holomorphic) at infinity. Moreover, it has a well know Fourier expansion with coefficients in $$O(n^2)$$, so by Proposition 1.2.4 it is modular.

• +1 This is very helpful and does answer my first question. It remains to show that $G_{2,N}$ is holomorphic on $\mathcal{H}^*$. Do you have any suggestions for how to prove this?
– Nico
Commented Jan 28, 2021 at 22:10
• @Nico: I edited. Basically holomorphicity on the half-plane and at infinity is a consequence of the definition of $G_2$, and holomorphicity at the other cusps stems from the computation of the Fourier coefficients of $G_2$ (and show there’s a polynomial bound). Commented Jan 29, 2021 at 9:55
• Fantastic answer! I suppose the proof is not too dissimilar to the proof of normal convergence of higher weight Eisenstein series. Thank you again.
– Nico
Commented Jan 29, 2021 at 19:38

This is more of an expansion of my comment to the question.

Let $$q\in(0,1)$$ be the nome corresponding to elliptic modulus $$k\in(0,1)$$ (in the symbolism of modular forms $$q$$ takes backstage and is replaced by $$\tau$$ where $$q=\exp(2\pi i\tau)$$ and $$\tau$$ has positive imaginary part).

Let us define elliptic integrals $$K, E$$ as $$K=K(k) =\int_0^{\pi/2}\frac{dx}{\sqrt{1-k^2\sin^2x}}\tag{1}$$ and $$E=E(k) =\int_0^{\pi /2}\sqrt{1-k^2\sin^2x}\,dx\tag{2}$$ The nome $$q$$ is related to $$k$$ via $$q=\exp\left(-\pi\frac{K(k')} {K(k)} \right), k'=\sqrt{1-k^2}\tag{3}$$ Ramanujan used the function $$P(q)$$ instead of your $$G_2$$ defined by $$P(q) =1-24\sum_{n=1}^{\infty}\frac{nq^n}{1-q^n}\tag{4}$$ The following link of $$P(q)$$ and elliptic integrals defined above was well known before Ramanujan: $$P(q^2)=\left(\frac{2K}{\pi}\right)^2\left(\frac {3E}{K}+k^2-2\right) \tag{5}$$ For Ramanujan modular forms were roughly those functions $$f(q)$$ which could be expressed like $$(2K/\pi)^w A(k)$$ where $$w$$ is some positive integer or half of a positive integer and $$A(k)$$ is an algebraic function of $$k$$ ($$w$$ being weight of $$f(q)$$). The exact definition of modular forms which Ramanujan provided could not however be decoded by Berndt and his collaborators.

The term $$3E/K$$ on right of $$(5)$$ is what prevents $$P(q^2)$$ from being a modular form. Fortunately for us Ramanujan found out a way to fix this. He instead considered the function $$NP(q^{2N})-P(q^2)$$ ($$N$$ being a positive integer) and proved that it can be expressed in the form desired.

Let us start with another well known function called Dedekind eta function defined by $$\eta(q) =q^{1/24}\prod_{n=1}^{\infty} (1-q^n)\tag{6}$$ Eta function is related to $$P(q)$$ via $$P(q) =24q\frac{d}{dq}{\log \eta(q)} \tag{7}$$ The following identity $$\eta(q^2)=2^{-1/3}\sqrt {\frac{2K}{\pi}}(kk')^{1/6}\tag{8}$$ is well known and shows that $$\eta(q^2)$$ is a modular form of weight $$1/2$$.

Using $$(1),(2),(3)$$ it can be proved that $$\frac{dq} {dk} =\frac{\pi^2q}{2kk'^2K^2}\tag{9}$$ and then $$(5)$$ can be established using $$(7),(8),(9)$$.

Let $$l$$ be the elliptic modulus corresponding to nome $$q^N$$ (which corresponds to $$N\tau$$). The elliptic integrals $$K(l), K(l') =K(\sqrt {1-l^2})$$ are usually denoted by $$L, L'$$ so that $$q^N=\exp(-\pi L'/L)$$. From $$(8)$$ we get $$\eta(q^{2N})=2^{-1/3}\sqrt{\frac{2L}{\pi}}(ll')^{1/6}\tag{10}$$ From $$(7)$$ we get $$P(q^2)=12q\frac{d}{dq}\{\log\eta(q^2)\},NP(q^{2N})=12q\frac{d}{dq}\{\log\eta(q^{2N})\}$$ and thus $$F(q) =NP(q^{2N})-P(q^2)=12q\frac{d}{dq}\left(\log\frac{\eta(q^{2N})}{\eta(q^2)}\right)$$ Using $$(8),(9),(10)$$ we thus get $$F(q) = \left(\frac{2K}{\pi}\right)^2kk'^2\frac{d}{dk}\left(3\log\frac{L}{K}+\log\frac{ll'}{kk'}\right)\tag{11}$$ From the standard theory of modular equations it is known that $$l,l'$$ as well as $$K/L$$ are algebraic functions of $$k$$ and hence from the equation $$(11)$$ it follows that $$F(q)$$ is of the form $$(2K/\pi)^2A(k)$$ where $$A(k)$$ is an algebraic function of $$k$$. Thus $$F(q)$$ is a modular form of weight $$2$$.

Ramanujan calculated the function $$A(k)$$ in explicit form for many values of $$N$$. For small values of $$N$$ the values given by Ramanujan can be verified with some effort. But no one knows how Ramanujan evaluated them for large $$N$$.

• +1 I have always wondered about Ramanujan's derivation of the $\pi$-formula and its connection to modular forms. Thank you for this wonderful addendum. Do you have any (modern) reference for the full derivation?
– Nico
Commented Jan 29, 2021 at 19:32
• @Nico: you can study all the details in my blog posts starting with this one. Commented Jan 29, 2021 at 22:42
• @Nico: the series given by Chudnovsky brothers is also based on Ramanujan's theory. A proof is available in this blog post. Commented Jan 29, 2021 at 22:45
• @Nico: also see this answer. Commented Jan 30, 2021 at 9:54