---
abstract: 'We construct a natural family of rational functions $\tilde\Psi_m$ on a Hilbert modular surface from the classical $j$-invariant and its Hecke translates. These functions are obtained by means of a multiplicative analogue of the Doi-Naganuma lifting and can be viewed as twisted Borcherds products. We then study when the value of $\tilde\Psi_m$ at a CM point associated to a non-biquadratic quartic CM field generates the ‘CM class field’ of the reflex field. For the real quadratic field ${\mathbb{Q}}(\sqrt{5})$, we factorize the norm of some of these CM values to ${\mathbb{Q}}(\sqrt 5)$ numerically.'
address:
- 'Mathematisches Institut, Universität zu Köln, Weyertal 86–90, D-50931 Köln, Germany'
- 'Department of Mathematics, University of Wisconsin Madison, Van Vleck Hall, Madison, WI 53706, USA'
author:
- Jan Hendrik Bruinier and Tonghai Yang
title: Twisted Borcherds products on Hilbert modular surfaces and their CM values
---

[^1]

Introduction {#sect:1}
============

In his papers [@Bo1] and [@Bo2] Borcherds constructed a lifting from certain weakly holomorphic elliptic modular forms of weight $1-n/2$ to meromorphic modular forms on the orthogonal group of a rational quadratic space of signature $(2,n)$. Here we consider a related construction in the particular case that $n=2$.

Let $p$ be a prime congruent 