Zeros of certain Drinfeld modular functions

For every positive integer m  , there is a unique Drinfeld modular function, holomorphic on the Drinfeld upper-half plane, jm(z)jm(z) with the following t-expansionjm(z)=1tm(q−1)+∑i=1∞cm(i)ti(q−1). These functions are analogs of certain modular functions from the classical theory that have many fascinating properties. For example, they are used to prove the famous denominator formula for the Monster Lie algebra. Here we prove that (as in the classical case) the zeros of jm(z)jm(z) in the fundamental domain FF of the Drinfeld upper-half plane Ω   for Γ:=GL2(Fq[T])Γ:=GL2(Fq[T])F:={z∈Ω:|z|=inf{|z−a|:a ∈Fq[T]}⩾1}, are on the unit circle |z|=1|z|=1. Moreover, if q   is odd, the zeros are transcendental over Fq(T)Fq(T).