Abstract:
It is known that a modular form of weight k has about k/12 zeros in the fundamental domain (and at infinity). A classical question in the analytic theory of modular forms is “can we locate the zeros of a distinguished family of modular forms?”. In 1970, F. Rankin and Swinnerton-Dyer proved that the zeros of the Eisenstein series all lie on the circular part of the boundary of the fundamental domain. In this talk, we will discuss recent results on the zeros and zero distribution of theta functions associated with a lattice, a special type of modular forms. In particular, we will discuss the zeros of a family of unimodular even lattices that are closely related to the E_8 lattice. As it turns out, the zeros in the fundamental domain all lie on the line Re(z)=1/2.
No prior knowledge of modular forms will be assumed.
Based on: https://arxiv.org/abs/2509.06128