Interlacing of zeros of weakly holomorphic modular forms

TL;DR

This paper proves that zeros of extremal weakly holomorphic modular forms interlace and demonstrates similar behavior for most basis forms on the lower boundary of the fundamental domain, addressing a question by Nozaki.

Contribution

It establishes the interlacing property of zeros for extremal and basis weakly holomorphic modular forms, advancing understanding of their zero distributions.

Findings

Zeros of extremal forms interlace

Zeros of most basis forms interlace on the lower boundary

Addresses a question posed by Nozaki

Abstract

We prove that the zeros of a family of extremal modular forms interlace, settling a question of Nozaki. Additionally, we show that the zeros of almost all forms in a basis for the space of weakly holomorphic modular forms of weight $k$ for $\SL_2(\mathbb{Z})$ interlace on most of the lower boundary of the fundamental domain.