Hilbert Eisenstein series as Doi-Naganuma lift

In this paper, we show that incoherent Hilbert Eisenstein series for a real quadratic fields can be expressed as the Doi-Naganums lift of an incoherent Eisenstein series over $\mathbb{Q}$. As an application, we show when $N$ is odd and square-free, the values at Heegner points of Borcherds product on $X_0(N)^2$ with effective divisors are not integral units when the discriminants are sufficiently large. This generalizes a result of the first author to higher levels. In the process, we explicitly describe the Rankin-Selberg type L-function that appeared in the work of Bruinier-Kudla-Yang when the quadratic space has signature (2, 2), and give a new construction of fundamental invariant vectors appearing in Weil representations of finite quadratic modules.