On the Divisibility Properties of the Fourier Coefficients of Meromorphic Hilbert Modular Forms

TL;DR

This paper investigates the rationality and divisibility of Fourier coefficients of meromorphic Hilbert modular forms over real quadratic fields, employing theta lifts and weak Maass forms to establish new divisibility conditions.

Contribution

It introduces new criteria for the rationality and divisibility of Fourier coefficients of meromorphic Hilbert modular forms, extending previous work by Zagier.

Findings

Fourier coefficients are rational with bounded denominators under certain conditions.

Divisibility properties are demonstrated for specific linear combinations of coefficients.

Conditions for rationality and divisibility are explicitly characterized.

Abstract

Following Zagier, this work studies the rationality and divisibility of Fourier coefficients of meromorphic Hilbert modular forms associated with real quadratic fields, using theta lifts and weak Maass forms. We establish conditions where these coefficients are rational with bounded denominators and demonstrate divisibility properties under suitable linear combinations.