On the zeros of a class of modular 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.