a proof along these lines.

Let ${\mathbb{H}}$ be the upper complex half plane, and put $q=e(\tau)=e^{2\pi
i\tau}$ for $\tau\in {\mathbb{H}}$. Recall that a weakly holomorphic modular form for ${\operatorname{SL}}_2({\mathbb{Z}})$ is a meromorphic modular form for ${\operatorname{SL}}_2({\mathbb{Z}})$ which is holomorphic outside the cusp $\infty$. In particular, every weakly holomorphic modular form of weight zero for ${\operatorname{SL}}_2({\mathbb{Z}})$ is a polynomial in the $j$-function (which we normalize such that $j(\tau)=q^{-1}+744+O(q)$). In Section \[sect:5\] we shall prove the following theorem. See Theorem \[hilbert\] for a more detailed statement.

\[hilbertintro\] Let $f=\sum_{n\gg -\infty}c(n)q^n\in {\mathbb{Z}}[j]$ be a weakly holomorphic modular form of weight $0$ for ${\operatorname{SL}}_2({\mathbb{Z}})$ with integral Fourier coefficients. Then there exists a symmetric meromorphic Hilbert modular function $\Psi(z,f)$ for $\Gamma$ (of weight $0$, with trivial multiplier system, defined over $F$) such that:

1.  The divisor of $\Psi(z,f)$ is determined by the polar part of $f$ at the cusp $\infty$. It equals $\sum_{n>0}  c(-n) \tilde T_{n}$, where $\tilde T_{n}$ denotes the twisted Hirzebruch-Zagier divisor of discriminant $n$ defined in Section \[sect:3\].

2.  The function $\Psi(z,f)$ has the Borcherds product expansion $$\begin{aligned}
    \Psi(z,f)&= \prod_{\substack{\nu\in\partial_F^{-1} \\
    \nu>0}}\prod_{b\,(p)} \left(1-e(\tfrac{b}{p}+\nu
    z_1+\nu'z_2)\right)^{\epsilon_p(b)c(p\nu\nu')},
    $$ which converges normally for all $z=(z_1,z_2)\in {\mathbb{H}}^2$ with $\Im(z_1)\Im(z_2)> Np$ outside the set of poles, where $N=\max
    