On the zeros of a class of modular functions

TL;DR

This paper studies zeros of a broad class of weakly holomorphic modular functions, showing they lie on the unit circle and are transcendental, extending previous results and connecting to quantum gravity models.

Contribution

It generalizes existing results on modular form zeros to a wider class of functions with specific q-expansion properties.

Findings

Zeros lie on the unit circle in the fundamental domain

Zeros are transcendental numbers

Includes special cases related to quantum gravity functions

Abstract

We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose $q$-expansions satisfy \[ f_k(A, \tau) \colon = q^{-k}(1+a(1)q+a(2)q^2+...) + O(q),\] where $a(n)$ are numbers satisfying a certain analytic condition. We show that the zeros of such $f_k(\tau)$ in the fundamental domain of $SL_2(\mathbb{Z})$ lie on $|\tau|=1$ and are transcendental. We recover as a special case earlier work of Witten on extremal "partition" functions $Z_k(\tau)$. These functions were originally conceived as possible generalizations of constructions in three-dimensional quantum gravity.