Title: On the real zeros of depth 1 quasimodular forms Show Chinese title

Title: 关于深度1拟模形式的实零点

Abstract: We discuss the critical points of modular forms, or more generally the zeros of quasimodular forms of depth $1$ for $\mathrm{PSL}_2(\mathbb Z)$. 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 $\{z \in \mathbb H: \Re(z)=1/2\}$, and (2) more than half of zeros of $f_k$ in the standard fundamental domain lie on the geodesic segment $\{z \in \mathbb H: \Re(z)=1/2\}$ for large enough $k$ with $k\equiv 0 \pmod{12}$. Show Chinese abstract

Abstract: 我们讨论模形式的临界点，或者更一般地，讨论深度为$1$的拟模形式在$\mathrm{PSL}_2(\mathbb Z)$处的零点。 特别是，我们考虑唯一权为$k$的模形式$f_k$的导数，这些导数在常数$1$之后具有最多连续的零傅里叶系数。 我们的主要结果表明，(1) 在标准基本区域中，Eisenstein 级数导数附近的所有深度$1$伪模形式的零点都位于测地线段$\{z \in \mathbb H: \Re(z)=1/2\}$上，(2) 对于足够大的$k$且满足$k\equiv 0 \pmod{12}$的情况，标准基本区域中$f_k$的零点超过一半位于测地线段$\{z \in \mathbb H: \Re(z)=1/2\}$上。