to $1$ modulo $4$ and let $F={\mathbb{Q}}(\sqrt{p})$. We write ${\mathcal{O}}_F$ for the ring of integers and $\partial_F$ for the different of $F$. Considering the lattice $L^0={\mathbb{Z}}^2\oplus {\mathcal{O}}_F$ in the rational quadratic space $L^0\otimes_{\mathbb{Q}}{\mathbb{Q}}$ of signature $(2,2)$, one obtains from Borcherds’ result a lifting from weakly holomorphic elliptic modular forms of weight zero for the group $\Gamma_0(p)$ with Nebentypus character $\epsilon_p=(\frac{\cdot}{p})$ to meromorphic Hilbert modular forms for the Hilbert modular group $\Gamma={\operatorname{SL}}_2({\mathcal{O}}_F)$ [@Br1; @BB]. This lifting can be viewed as a multiplicative analogue of the Naganuma lift from holomorphic modular forms of weight $k$ for $\Gamma_0(p)$ with Nebentypus $\epsilon_p$ to holomorphic Hilbert modular forms of weight $k$ for $\Gamma$ [@Na; @Za].

There is another lifting from holomorphic elliptic modular forms to Hilbert modular forms, namely the celebrated Doi-Naganuma lift which maps holomorphic modular forms of weight $k$ for ${\operatorname{SL}}_2({\mathbb{Z}})$ to holomorphic Hilbert modular forms of weight $k$ for $\Gamma$ [@DN]. It was pointed out by Zagier that this lifting should have a multiplicative analogue as well [@Za3]. Moreover, Zagier stated several properties of such a multiplicative lifting and suggested that a proof could probably be given following the argument of [@Br1]. One purpose of the present paper is to work out 