The Serre Derivatives and Zeros of Modular Forms

TL;DR

This paper investigates the zeros of Serre derivatives of modular forms, showing that zeros on the lower boundary are preserved under differentiation.

Contribution

It proves that zeros of weakly holomorphic modular forms on the lower boundary remain on the boundary after applying Serre derivatives.

Findings

Zeros of Serre derivatives lie on the lower boundary if original zeros do.

Zeros of weakly holomorphic modular forms are preserved under Serre differentiation.

The property of zeros being on the boundary is maintained through Serre derivatives.

Abstract

Since the work of F. Rankin and Swinnerton-Dyer on the zeros of Eisenstein series, many results have been obtained concerning the zeros of modular forms. In this paper, we study the zeros of Serre derivatives of modular forms. In particular, we prove that if all the zeros of a weakly holomorphic modular form in the standard fundamental domain lie on the lower boundary, then the same property holds for its Serre derivative.