Cycle integrals of meromorphic Hilbert modular forms

TL;DR

This paper proves a rationality result for cycle integrals of meromorphic Hilbert modular forms, providing explicit formulas and constructing related harmonic Maass forms, advancing understanding of Hilbert modular forms and their lifts.

Contribution

It introduces explicit formulas for cycle integrals of meromorphic Hilbert modular forms and constructs locally harmonic Hilbert-Maass forms, linking them via a new regularized theta lift.

Findings

Established rationality of cycle integrals for certain meromorphic forms.

Derived explicit formulas in terms of harmonic Maass form Fourier coefficients.

Constructed locally harmonic Hilbert-Maass forms and related theta lift.

Abstract

We establish a rationality result for linear combinations of traces of cycle integrals of certain meromorphic Hilbert modular forms. These are meromorphic counterparts to the Hilbert cusp forms $\omega_m(z_1,z_2)$, which Zagier investigated in the context of the Doi-Naganuma lift. We give an explicit formula for these cycle integrals, expressed in terms of the Fourier coefficients of harmonic Maass forms. A key element in our proof is the explicit construction of locally harmonic Hilbert-Maass forms on $\mathbb{H}^2$, which are analogous to the elliptic locally harmonic Maass forms examined by Bringmann, Kane, and Kohnen. Additionally, we introduce a regularized theta lift that maps elliptic harmonic Maass forms to locally harmonic Hilbert-Maass forms and is closely related to the Doi-Naganuma lift.