On the Drinfeld discriminant function

Abstract

The discriminant function Δ is a certain rigid analytic modularform defined on Drinfeld’s upper half-plane Ο. Its absolutevalue ❘Δ❘ may be considered as a function on theassociated Bruhat–Tits tree T. We compare log ❘Δ❘ with the conditionally convergent complex-valued Eisenstein series Edefined on T and thereby obtain results about the growth of ❘Δ❘ and of some related modular forms. We further determine to what extent roots may be extracted of Δ(z)/Δ(nz),regarded as a holomorphic function on Ο. In some cases, this enables us to calculate cuspidal divisor class groups of modular curves.