Can anyone help me to write down the derivative of $j$?

Question

Suppose that $f(z)$ is a modular form of weight $k$, therefore for any $z\in \mathbb{H}$, and any matrix in $SL_2(\mathbb{Z})$ we have: $f(\frac{az+b}{cz+d}) = (cz+d)^k f(z)$. I do not know how to use this relation to compute the derivative of $f$. Now let $G_{2k}$ be an Eisenstein series of weight $2k$ (https://en.wikipedia.org/wiki/Eisenstein_series). How can I compute derivates of $E_4$ and $E_6$ in terms of Eisenstein series? Starting from $G_{2k}(\frac{az+b}{cz+d}) = (cz+d)^{2k} G_{2k}(z)$, I can just find this immediate relation $(\frac{1}{cz+d})^2G'_{2k}(\frac{az+b}{cz+d}) = (cz+d)^{2k} G'_{2k}(z)+2kc(cz+d)^{2k-1} G_{2k}(z)$. How can I calculate the derivative of $j$-invariant?

Edit: In the third comment of this question The roots of $J' = \frac{dJ}{d\tau}$ it is written that $J'/J = -2\pi i E_6(\tau)/E_4(\tau)$. I don't know why this is true. But if we have this relation then from this point it is easy to finish. But I can not realize how to derive that relation, the link does not lead to "Ramanujan's system of ODEs". The crucial problem is that to me $j$ and $E_k$'s are defined as functions of $\tau$, $j(\tau)$ and $E_k(\tau)$. I don't know how to differentiate with respect to $\tau$.

A situation happened to me which is very similar to my old question Prove or give a counter-example about a statement on faithful simple left modules over a finite-dimensional algebra. . I was reading about modular forms. But I faced too many questions, which I can not handle them or I have some doubts about.

By $\mathbb{H}$ I mean the upper half plane. It deals with the $j$ invariant, but I do not know it well enough. It says that $j$ is a holomorphic function that gives a homeomorphism between $\mathbb{H}/PSL_2(\mathbb{Z})$ and $\mathbb{C}$.

Does the function $\tau \mapsto e^{2\pi i \tau}$ gives a holomorphic bijection between the fundamental domain and punctured disk $\mathbb{D}$? If yes, then I am curious if Is it wise to define the inverse map as $q \mapsto \frac{1}{2 \pi i}\log(q)$? Does it make sense?

Let $q=e^{2\pi i \tau}$. If we are able to define this logarithmic map then we can define $J(e^{2\pi i \tau})=J(q)=j(\frac{1}{2 \pi i}\log(q))$, which is a map from the punctured disk $\mathbb{D}$ to $\mathbb{C}$. If we are not able to define that logarithmic map, then how can we define this last bijection between punctured disk $\mathbb{D}$ and $\mathbb{C}$? Can we consider the point at infinity as the corresponding point to the center of the punctured disc? how?

These were my doubts, now this is my question. Why the set of zeros of $j'$, $j'^{-1}(0)$ is finite in the fundamental domain? The crucial problem is that to me $j$ and $E_k$'s are defined as functions of $\tau$, $j(\tau)$ and $E_k(\tau)$. I don't know how to differentiate with respect to $\tau$.

Answer 1

There is no alternative here apart from Ramanujan's system of differential equations for Eisenstein series.

Let $\tau\in\mathbb{H} $ and $q=\exp (2\pi i\tau) $ so that $dq=2\pi iq\, d\tau$ or $$\frac{d} {d\tau} =2\pi iq\frac{d} {dq} \tag{1}$$ Ramanujan defines his functions $P, Q, R$ which are equivalent to Eisenstein series as \begin{align} P(q) &=E_2(\tau)=1-24\sum_{n=1}^{\infty} \frac{nq^n} {1-q^n}\tag{2a}\\ Q(q) &=E_4(\tau)=1+240\sum_{n=1}^{\infty} \frac{n^3q^n} {1-q^n}\tag{2b}\\ R(q) &=E_6(\tau)=1-504\sum_{n=1}^{\infty} \frac{n^5q^n} {1-q^n}\tag{2c}\\ \end{align} Ramanujan used elementary methods to establish the following system of differential equations satisfied by $P, Q, R$ \begin{align} q\frac{dP(q)} {dq} &=\frac{P^2(q)-Q(q)}{12}\tag{3a}\\ q\frac{dQ(q)} {dq} &=\frac{P(q)Q(q)-R(q)}{3}\tag{3b}\\ q\frac{dR(q)} {dq} &=\frac{P(q)R(q) -Q^2(q)}{2}\tag{3c}\\ \end{align} Usually the above equations are written without the parameter $q$ so that $P(q) $ is written just as $P$. Ramanujan's proof of the above equations is given in my blog posts (see this post and the next one). Alternatively they can proved using technique of modular forms by knowing that both sides of each relation are in same space of modular forms of a specific dimension. I am not an expert in this area of modular forms so can't really give more details.

The function $J(\tau) $ is defined as $$J(\tau) =\frac{E_4^3(\tau)} {E_4^3(\tau)-E_6^2(\tau)}=\frac{Q^3}{Q^3-R^2}\tag{4}$$ Using Ramanujan's equations above one can easily prove that $$q\frac{d} {dq}\{\log(Q^3-R^2)\}=P\tag{5}$$ and hence \begin{align} q\frac{d} {dq} (\log J) &=3q\frac{d\log Q}{dq}-q\frac{d} {dq} \{\log(Q^3-R^2)\}\notag\\ &=3\cdot\frac{PQ-R} {3Q} -P\notag\\ &=-\frac{R}{Q}\notag \end{align} Now we have $$\frac{d} {d\tau} (\log J) =2\pi i q\frac{d} {dq} (\log J) =-2\pi i\frac{R} {Q} $$ ie $$\frac{J'(\tau)} {J(\tau)} =-2\pi i\cdot\frac{E_6(\tau)}{E_4(\tau)}$$ as desired.