Hecke operators on the zeros of modular forms - Aricheta, Victor Manuel R.; Nemenzo, Fidel R. (Institute of Mathematics, College of Science, University of the Philippines Diliman, 2013)
The first major result in locating the zeros of modular forms is due to Rankin and Swinnerton-Dyer who proved that all zeros of the Eisenstein series in the fundamental domain all lie on the unit circle. A modification of their approach led Getz to construct a family of modular forms with this same property. In here, we study the effect of the Hecke operators Tn to the zeros of modular forms. Following the construction of Getz, we provide a family of weight K modular forms for which the zeros of Tnf in the fundamental domain all lie on the unit circle.Then, we investigate the relationship between the zeros of a modular form f and Tnf. We prove that if Tn has distinct eigenvalues, then f and Tnf have precisely the same zeros if and only if f is an eigenform. As a consequence, for weights less than or equal to 24, we give the form of f depending on the number of zeros shared by f and Tnf.