regularized theta lifting for the dual pair ${\operatorname{SL}}_2({\mathbb{R}})$, ${\operatorname{O}}(2,2)$ as in [@Bo2; @Br2], and by considering suitable “twists” of Siegel theta functions as kernel functions. It would actually be very interesting to describe such twisted Borcherds liftings in greater generality for ${\operatorname{O}}(2,n)$. However, it seems not quite clear what the right “twists” of Siegel theta functions should be in general. We have not pursued this approach in the present paper, because the proof using automorphic Green functions leads to the result in a direct way. Moreover, the construction of the Green functions should be of independent interest.

If $m$ is a positive integer, we write $\tilde\Psi_m$ for the symmetric Hilbert modular function of weight $0$ which is the twisted Borcherds lift via Theorem \[hilbertintro\] of the unique weakly holomorphic modular form $$J_m=q^{-m}+O(q)\in {\mathbb{Z}}[j].$$ In this way we obtain a ‘canonical’ family of rational functions on the Hilbert modular surface associated to $\Gamma$. It seems natural to ask for their arithmetic properties. For instance, one can ask whether $\tilde \Psi_1$, the lifting of the $j$-function, has a nice moduli interpretation as an invariant of abelian surfaces with ${\mathcal{O}}_F$-multiplication and $\partial_F^{-1}$-polarization.

It is a well-known and beautiful fact that $j(\frac{D+\sqrt{-D}}2)$ generates the 