Zeros of certain modular functions and an application

Abstract

1 Main resultLet (T) be the classical elliptic modular iftvariant which is a holomorphic fuftctielt in the upper half-plane H is invariant under the action ofthe modular group SL2(Z) and has a simple pole at infinity Let op(J ) be the monic polynomial in (T) obtained from iT)744 by the action of the n-th Hecke operator (the precise definition of which will be recalled later)In this paper we prove the following THEoREM 1 For each n all the zeros of the poynomial opO ) are simple and lie in the interval (O 1728)