Hilbert Eisenstein series as Doi-Naganuma lift

TL;DR

This paper demonstrates that incoherent Hilbert Eisenstein series for real quadratic fields can be represented as Doi-Naganuma lifts of Eisenstein series over rationals, with applications to non-integrality of Borcherds product values at Heegner points.

Contribution

It establishes a new connection between Hilbert Eisenstein series and Doi-Naganuma lifts, extending previous results to higher levels and providing explicit descriptions of related L-functions and Weil representations.

Findings

Incoherent Hilbert Eisenstein series can be expressed as Doi-Naganuma lifts.

Values of Borcherds products at Heegner points are not integral units for certain conditions.

Explicit description of Rankin-Selberg L-functions and Weil representations in this context.

Abstract

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.