On Drinfeld modular forms of higher rank II

TL;DR

This paper investigates properties of invertible holomorphic functions on Drinfeld symmetric spaces, showing their absolute value is constant on certain fibers and their logarithm is affine, aiding the study of modular forms' zeros and images.

Contribution

It establishes that the absolute value of invertible holomorphic functions on Drinfeld symmetric spaces is fiber-constant and their logs are affine, advancing understanding of modular forms' structure.

Findings

|f| is constant on fibers of the building map

log|f| is an affine map on the Bruhat-Tits building

Results help analyze zeros and images of modular forms

Abstract

We show that the absolute value $|f|$ of an invertible holomorphic function $f$ on the Drinfeld symmetric space $\OM^r$ $(r \geq 2)$ 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 $\MB\MT$.