\{n\in {\mathbb{Z}};\; c(-n)\neq 0\}$. In particular, $\Psi(z, C) =1$ for any constant $C$.

3.  The lifting is multiplicative, i.e., if $f,g\in {\mathbb{Z}}[j]$, then $\Psi(f+g)=\Psi(f)\Psi(g)$.

We now briefly describe the idea of the proof. For a positive integer $m$, we define in Section \[sect:3\] a certain “twisted Hirzebruch-Zagier divisor” $\tilde
T_m$ on the Hilbert modular surface corresponding to $\Gamma$ (see also [@Za2] pp. 166). In Section \[sect:4\], following [@Br1], we construct an automorphic Green function $\tilde
\Phi_{mp^2}(z,s)$ for $\tilde T_m$. We study its main properties and compute its Fourier expansion (Theorems \[fourier1\] and \[fourier2\]). By means of an identity relating certain finite exponential sums to Kloosterman sums (Lemma \[GaHb\]), we find that the Fourier coefficients of $\tilde \Phi_{mp^2}(z,s)$ are closely related to the coefficients of non-holomorphic Poincaré series of weight zero for ${\operatorname{SL}}_2({\mathbb{Z}})$ (see Section \[sect:2\]). Using the fact that any weakly holomorphic modular form of weight zero can be uniquely written as a linear combination of the non-holomorphic Poincaré series, Theorem \[hilbertintro\] can be deduced. Here the main point is an identity expressing $\log|\Psi(z,f)|$ as a linear combination of the automorphic Green functions $\tilde\Phi_{mp^2}(z,1)$, see Theorem \[hilbert\] (iv).

An alternative proof could be given by interpreting the lifting as a 