On values of weakly holomorphic modular functions at divisors of meromorphic modular forms

TL;DR

This paper proves that certain weakly holomorphic modular functions take algebraic values at divisors of meromorphic modular forms with algebraic Fourier coefficients, extending classical results about zeros and poles.

Contribution

It establishes algebraicity of values of weakly holomorphic modular functions at divisors and extends Schneider's classical result to a broader class of modular forms.

Findings

Values of weakly holomorphic modular functions at divisors are algebraic.

Zeros or poles of non-zero meromorphic modular forms are either transcendental or imaginary quadratic irrational.

Extension of Schneider's classical result to forms with algebraic Fourier coefficients.

Abstract

We show that the values of a certain family of weakly holomorphic modular functions at points in the divisors of any meromorphic modular form with algebraic Fourier coefficients are algebraic. We use this to extend the classical result of Schneider by proving that zeros or poles of any non-zero meromorphic modular form with algebraic Fourier coefficients are either transcendental or imaginary quadratic irrational.