On Drinfeld modular forms of higher rank II

Abstract

We show that the absolute value $|f|$ of an invertible holomorphic function $f$ on the Drinfeld symmetric space $\OM^r$ $(r\ge 2)$ is constant on fibers of the building map to the Bruhat-Tits building $\MB\MT$. Its logarithm $\log |f|$ is an affine map on the realization of $\MB\MT$. These results are used to study the vanishing loci of modular forms (coefficient forms, Eisenstein series, para-Eisenstein series) and to determine their images in $\MB\MT$.