Meromorphic analogues of modular forms generating the kernel of Shintani's lift

TL;DR

This paper introduces meromorphic modular forms constructed from quadratic polynomials with negative discriminant, analyzing their Fourier coefficients and revealing their decomposition into algebraic and cusp form components.

Contribution

It extends the theory of modular forms by defining and studying meromorphic analogues related to negative discriminants, including explicit Fourier coefficient computations.

Findings

Fourier coefficients split into algebraic and cusp form parts

Explicit formulas for Fourier coefficients of meromorphic modular forms

Connection to classical holomorphic modular forms and lifts

Abstract

We study the meromorphic modular forms defined as sums of -k (k>1) powers of integral quadratic polynomials with negative discriminant. These functions can be viewed as meromorphic analogues of the holomorphic modular forms defined in the same way with positive discriminant, first investigated by Zagier in connection with the Doi-Naganuma map and then by Kohnen and Zagier in connection with the Shimura-Shintani lifts. We compute the Fourier coefficients of these meromorphic modular forms and we show that they split into the sum of a meromorphic modular form with computable algebraic Fourier coefficients and a holomorphic cusp form.