Title: On real zeros of holomorphic Hecke cusp forms
Abstract: In this talk we are going to explore so-called “real” zeros of holomorphic Hecke cusp of large weight on the modular surface. Gosh and Sarnak established that the number of real zeros tends to infinity as the weight $k$ goes to infinity. To do so, they studied the behavior of holomorphic Hecke cusp forms close to the cusp, i.e. for $z=x+iy$ with $y > k^{1/2}$. In this region the cusp form is well approximated by a single Fourier coefficient and the problem of finding real zeros boils down to studying sign changes of Fourier coefficients of holomorphic Hecke cusp forms. Low in the fundamental domain, say for $1 \leq y \leq 2$, investigating sign changes of Fourier coefficients is not sufficient and the problem is less understood. In this talk we explain ongoing work on detecting real zeros low in the fundamental domain on average over a large Hecke basis. The averaging allows us to establish an almost sharp quantitative bound for the number of real zeros, compared to what is predicted by a random model for holomorphic Hecke cusp forms.
