Meromorphic analogues of modular forms generating the kernel of Shimura’s lift

We study the meromorphic modular forms defined as sums of $-k$ $(k \geq 2)$ 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 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.

Keywords

modular forms, complex multiplication

2010 Mathematics Subject Classification

11F03, 11F11, 11F37

Full Text (PDF format)

Published 16 April 2015