How are Dedikind$\eta$ quotients connected to modular curves? Im trying to understand the connection between the Dedekind $\eta$ function and its products/quotients and certain modular curves. Why is that the modular function 
$j_2 = \left( \frac{\eta(\tau)}{\eta(2\tau)} \right)^{24}$ given as an $\eta$-product generates the function field for the modular curve $\chi_0(2)$. I think the 24 power creates a $1/q$ in the $q$ expansion, (similar to the $j$-invariant, which generates the function field for $\chi(1)$.  
I've seen this stated but never with explanation. What is the connection between $\eta$ functions and $\chi_0(l)$ in general?
Thanks!
 A: I don't know how to prove all the details, but there is what I understand so far : 
Let $F$ be the function field of meromorphic functions $X_0(2) \to \mathbb{C}$ 
(where $X_0(2)$ is the compactification of $Y_0(2) = \Gamma_0(2) \setminus \mathcal{H}$ a fundamental domain for the action of $\Gamma_0(2)$ on the upper-half plane)


*

*$\eta^{24}(\tau) = e^{2i \pi \tau} \prod_{n=1}^\infty (1-e^{2i \pi n \tau})$ is a modular form of weight $12$ (you can see it by looking at $\frac{\eta'(\tau)}{\eta(\tau)}$ and expanding $\frac{2i \pi n}{1-e^{-2i \pi n \tau}}$ over its poles), 
so that $j_2(\tau) = \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = e^{-2i \pi  \tau}\prod_{n=1}^\infty \frac{1}{(1+e^{2i \pi n \tau})^{24}}$ is modular of weight $0$ for $\Gamma_0(2)$, i.e. it is meromorphic on $X_0(2)$.

*(looking at $\frac{j_2{}'(\tau)}{j_2(\tau)}$ showing that  $j_2{}'$ doesn't vanish) $j_2(\tau)$ is bijective $X_0(2) \to \mathbb{C}$.

*If $f \in F$ is holomorphic, then it is constant (by Liouville's theorem).

*Take any $f \in F$ and let $a_1,\ldots, a_K$ and $b_1, \ldots , b_M$ be its poles and zeros $\in Y_0(2)$, and look at 
$$f(\tau) \frac{\prod_{k=1}^K j_2(\tau)-j_2(a_k)}{\prod_{m=1}^M j_2(\tau)-j_2(b_m)}$$
it has no poles on $Y_0(2)$, and a little work shows that (since it doesn't have any zero on $Y_0(2)$) it doesn't have a pole at the cusp neither, thus it is holomorphic on $X_0(2)$, and it is constant. 
Hence $f \in \mathbb{C}(j_2)$ and $j_2$ generates the function field $F$.
