The Serre Derivatives and Zeros of Modular Forms

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.