Why does the Weber function take on real values? In the middle of the proof, the author argues... "cleary it is real-valued." I don't follow the argument. Info about the Weber function $\mathfrak{f}_1$ is given at
http://en.wikipedia.org/wiki/Weber_modular_function
What about the general case, and what if $\tau=\sqrt{-14}$?
In general, as non-constant holomorphic functions, the Weber functions are not real-valued. However, the infinite products in the Weber functions take real values (at least) when $q = e^{2\pi i\tau}$ is a positive real number, which is the case whenever $\operatorname{Re} \tau \in \mathbb{Z}$, and and the functions themselves when the lone factor $q^{-1/48} = e^{-\pi i\tau/24}$ for $\mathfrak{f}$ and $\mathfrak{f}_1$ resp. $q^{1/24} = e^{\pi i\tau/12}$ for $\mathfrak{f}_2$ is also real, which is the case for $\operatorname{Re}\tau \in 24\mathbb{Z}$, resp. $\operatorname{Re}\tau \in 12\mathbb{Z}$ (they may also take real values for some values of $\tau$ with non-integral real part). In particular, for the purely imaginary $\tau = \sqrt{-14}$, all three Weber functions take a real value.