On the zeros of a class of modular functions

2018/07/11 by Sweeting, Naomi, Woo, Katharine
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1807.04310

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, τ) \colon = q^-k(1+a(1)q+a(2)q²+...) + O(q), where a(n) are numbers satisfying a certain analytic condition. We show that the zeros of such f_k(τ) in the fundamental domain of SL₂(ℤ) lie on |τ|=1 and are transcendental. We recover as a special case earlier work of Witten on extremal "partition" functions Z_k(τ). These functions were originally conceived as possible generalizations of constructions in three-dimensional quantum gravity.

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