Compositio Mathematica



On the Drinfeld discriminant function


ERNST-ULRICH GEKELER a1
a1 Fachbereich 9 Mathematik Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken, Germany; e-mail: gekeler@math.uni-sb.de

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Abstract

The discriminant function $\Delta$ is a certain rigid analytic modular form defined on Drinfeld's upper half-plane $\Omega$. Its absolute value $\vert \Delta\vert$ may be considered as a function on the associated Bruhat–Tits tree ${\cal T}$. We compare $\log \vert \Delta\vert$ with the conditionally convergent complex-valued Eisenstein series $E$ defined on ${\cal T}$ and thereby obtain results about the growth of $\vert \Delta$ and of some related modular forms. We further determine to what extent roots may be extracted of $\Delta(z)/\Delta(nz)$, regarded as a holomorphic function on $\Omega$. In some cases, this enables us to calculate cuspidal divisor class groups of modular curves.


Key Words: Drinfeld upper half-plane; improper Eisenstein series; modular units; cuspidal divisor class group.