j-invariant duplication, triplication and quintuplication formulae... how?

Question

I am interested in finding the derivation of the duplication, triplication and quintuplication formulae for Klein’s j-invariant, which are equations (13) – (24) of the corresponding page (Klein’s j-invariant) of MathWorld:

http://mathworld.wolfram.com/KleinsAbsoluteInvariant.html

Yes, I've asked the MathWorld team (without luck).

I'm guessing this is a easy, standard result for those who are in the know, apologies for that.

I have looked at Apostol, ‘Modular Functions and Dirichlet Series in Number Theory’ (the first reference on the MathWorld page) and Thm 4.11 on page 89 comes close but doesn't seem to do it for me.

I would be very grateful if someone could give me the reference for the derivation, in the least general case (as I am a bit of a novice in this area).

Best, m

PS The duplication formula is $J(\tau)=f(t)$ and $J(2\tau)=f(1/t)$ with $t=\frac{1}{64}\left[\frac{\eta(\tau)}{\eta(2\tau)}\right]^{24}$,
$f(u)=\frac{(u+4)^3}{27u^2}$ and $\eta(z)$ the Dedekind eta.

Answer 1

One can explain this conceptually in terms of "modular equations".

It's been known since the late 19th century that for any integer $N$, there is some polynomial $\Phi_n(X, Y)$ with the property that $\Phi_n(J(\tau), J(N\tau)) = 0$ for all $\tau$ in the upper half-plane. This is the "classical modular equation". For a few small values of $N$, the plane curve defined by $\Phi_n(X, Y) = 0$ is a rational curve, so we can find rational functions $a(u), b(u)$ of an auxilliary variable $u$ such that for every $\tau$, there is a $u$ such that $J(\tau) = a(u)$ and $J(N\tau) = bu$. Moreover, the map $\tau \mapsto -1/N\tau$ gives an involution on the curve, and you can choose your parametrization so this corresponds to $u \mapsto 1/u$, which tells you that $a(u) = b(1/u)$. This is what is going on in the formulae quoted on MathWorld.

What MathWorld doesn't tell you is that the curve defined by $\Phi_n(X, Y) = 0$ is actually a rather important object. It's singular in general, but its normalisation is a smooth curve over $\mathbb{C}$ which turns out to be isomorphic to the quotient of the upper half-plane by the group of matrices $$\Gamma_0(N) = \left\{\begin{pmatrix} a& b \\ c & d\end{pmatrix} : N \mid c, ad-bc = 1\right\}.$$

The curve thus obtained is called $X_0(N)$ and is very important in the theory of modular forms. In particular one knows that it is of genus 0 if and only if $N$ is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, or 25, so a formula of the kind that you quote exists if and only if $N$ is in this list.

Answer 2

This is not a complete answer. The book "Elliptic Functions" by Serge Lang has some formulae (called the "modular equation") which are related. The idea is this. Any holomorphic function on the upper half plane which is $SL(2,{\mathbb Z})$ invariant and is meromorphic at the cusp at infinity is a rational function in $J$. Given $g\in SL(2,{\mathbb Q})$, the function $J(g\tau)$ is invariant under some congruence subgroup $\Delta $ (of finite index ) of $SL(2,{\mathbb Z})$; therefore, if $S$ denotes a full set of representatives of $SL(2,{\mathbb Z})/\Delta$ then the coefficients of the polynomial $$\prod _{\gamma \in S} (X-J(g\gamma \tau))=\sum a_i(J)X^i$$ are rational functions in $J(\tau)$. Taking '$g=\begin{pmatrix} 2 & 0 \cr 0 & 1 \end{pmatrix}$' we get that $J(2\tau)$ is a root of such a polynomial, and hence get the "duplication formula". That is, a root of a cubic equation may be expressed in terms of radicals involving the coefficients $a_i(J)$. I think the other formulae are similar but I not sure.