I am trying to follow Iwaniec's book on automorphic forms and am getting hung up on the basic definitions. He seems to construct modular forms from elliptic functions and I can't really follow the reasoning in the details.
Here is the setup: let $f(u;w_1,w_2)$ be a homogeneous elliptic function on the lattice $w_1\mathbb{Z}+w_2\mathbb{Z}\subset\mathbb{C}$. That is $f$ is invariant under translation by this lattice. $u\in\mathbb{C}$. By homogeneous we mean that $f(\lambda u; \lambda w_1, \lambda w_2)= \lambda^{-k}f(u;w_1,w_2)$. Thus we have $$ f(u;w_1,w_2) = (w_2)^{-k}f\left(\frac{u}{w_2}; \frac{w_1}{w_2},1\right) $$ By possibly interchanging $w_1$ and $w_2$ we have $z=w_1/w_2\in\mathbb{H}$ (the upper half plane). Thus we can write $$ f(u;w_1,w_2) = (w_2)^{-k}f\left(\frac{u}{w_2}; \frac{w_1}{w_2}\right) $$ Where $f(v;z)$ is now a function from $\mathbb{C}\times\mathbb{H}$. I think maybe I am missing something in this transition because in my head I am just thinking of $f(v;z)$ as shorthand for $f(v;z,1)$. Anyway, by transforming $w_1$ and $w_2$ by a matrix $\gamma\in SL_2(\mathbb{Z})$ we get a new basis for the same lattice so the $f$ is invariant. Their ratio naturally is transformed by the fractional linear transform $z\mapsto \frac{az+b}{cz+d}$. Now the identity I am struggling with is he says: "As a function of $z$, $f$ has the property that $$f(\gamma z)=(cz+d)^kf(z)$$" When I try to work that out I get $$ f(v;\gamma z)= f\left(v;\frac{az+b}{cz+d},1\right)=(cz+d)^kf(v(cz+d);az+b,cz+d)=(cz+d)^kf(v(cz+d);z,1)$$ This is almost what we want but $v$ has a factor of $(cz+d)$ which seems to contradict the identity $f(\gamma z)=(cz+d)^kf(z)$ if we were to leave $v$ constent unless we choose $v$ to be zero in order to get $f$ as a pure function of $z$. What am I missing? Thank you.