Let $M_{12}$ be weight 12(integral weighted) entire modular form(i.e. holomorphic at cusps). So over $dim_C(M_{12})=2$. Now pick out any $f,g\in M_{12}$ linearly independent.
$\textbf{Q:}$ The book says $[f,g]:SL_2(Z)\backslash H\to CP_1$ is isomorphism where $H$ is the upper half plane union all cusps. Does the book mean identify $SL_2(Z)\backslash H$ not as an orbifold rather than simply a sphere without orbifold structure?(I could not see why this has to be biholomorphic if it is equipped with orbifold structure as there will be locally branched covering.) I can see this is obviously bijection but I do not see this as homeomorphism.
Ref. Zagier 1-2-3 Modular Forms pg 11 Cor 2.