The <Klein j-invariant>
$$
j(\tau)=\frac{E_4(\tau)^3}{\Delta(\tau)}
$$
is a weight-zero modular function for $\Gamma(1)$, holomorphic on $\mathfrak h$ and meromorphic at infinity. Let
$$
\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma_1(2).
$$
Since $c$ is even,
$$
\gamma'=\begin{pmatrix}a&2b\\c/2&d\end{pmatrix}\in\Gamma(1),
\qquad
2\gamma\tau=\gamma'(2\tau).
$$
The modular invariance of $j$ therefore gives
$$
\frac{j(\gamma\tau)}{j(2\gamma\tau)}
=\frac{j(\tau)}{j(2\tau)}.
$$
The quotient is meromorphic on $\mathfrak h$ and at the cusps, so it is a weight-zero modular function of level $\Gamma_1(2)$. This is the <Level-two modular ratio of Klein j-invariants>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 137 / 3 / b / Solution

{c}
{title2=$j(\tau)$}

The Klein j-invariant is the weight-zero level-one modular function
$$
j(\tau)=\frac{E_4(\tau)^3}{\Delta(\tau)}
=q^{-1}+744+O(q).
$$
It is holomorphic on the upper half-plane and has a simple pole at infinity.


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