I have seen the following theorem in several places, but have yet to find a clear proof:
Any weight 0 modular function can be expressed as a rational function of the J invariant
I would love to try and figure how to prove this on my own, however I don't know where to start. I believe I have seen somewhere that the leading term of $\frac{1}{q}$ is the key as it allows polynomials of J to include poles of any order at $q=0$, but I don't know what else to do. Any help or links to resources appreciated.
It is the theory of compact Riemann surfaces.
The field of functions meromorphic on the first modular curve $X(1) =SL_2(Z) \setminus\mathcal{H}\cup i\infty$ is $\Bbb{C}(j)$ because $j$ has only one simple pole at $i\infty$
whence for $f$ meromorphic on $X(1)$ $$g(z) = f(z) \prod_{p \in SL_2(Z) \setminus H} (j(z)-j(p))^{\max(0,-v_p(f))}$$ is meromorphic on $X(1)$ with no poles on $\mathcal{H}$.
Let $g_0 = g, e = \max(0,-v_{i\infty}(g_l)), c_l = \lim_{z \to i\infty} e^{2i \pi (e-l) z} g_l(z), g_{l+1} = g_l -c_l j^{e-l} $ then $g_e$ is holomorphic on $X(1)$ thus (maximum modulus principle) it is constant $= g_e(i\infty)$ and $$f(z) = \frac{g_e(i\infty)+\sum_{l=0}^{e-1} c_l j(z)^{e-l}}{\prod_{p \in SL_2(Z) \setminus H} (j(z)-j(p))^{\max(0,-v_p(f))}}$$