On the Zeros of a Class of Modular Functions

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 the following: f k ( A ; τ ) : = 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 ( τ ) in the fundamental domain of SL 2 ( 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.