On the real zeros of depth 1 quasimodular forms

Abstract

We discuss the critical points of modular forms, or more generally the zeros of quasimodular forms of depth 1 for ${\mathrm{PSL}}_2\left(Z\right)$. In particular, we consider the derivatives of the unique weight k modular forms $f_k$ with the maximal number of consecutive zero Fourier coefficients following the constant 1. Our main results state that (1) every zero of a depth 1 quasimodular form near the derivative of the Eisenstein series in the standard fundamental domain lies on the geodesic segment $\{z\in H:\Re (z)=1/2\}$, and (2) more than quarter of zeros of $f_k$ in the standard fundamental domain lie on the geodesic segment $\{z\in H:\Re (z)=1/2\}$ for large enough k with $k\equiv 0\left(\mathrm{mod}12\right)$.