# Drinfeld discriminant function and Fourier expansion of harmonic cochains

## TL;DR

This paper develops a Fourier expansion theory for harmonic cochains on Bruhat-Tits buildings, extending previous work, and applies it to analyze modular units and divisor groups on Drinfeld modular varieties.

## Contribution

It generalizes Fourier expansion techniques for harmonic cochains to higher ranks and applies these to study modular units and divisor groups in higher-dimensional Drinfeld spaces.

## Key findings

- Established Fourier expansion theory for harmonic cochains on $ ext{PGL}_r$ buildings.
- Analyzed modular units on Drinfeld symmetric spaces.
- Derived higher-dimensional analogues of classical modular curve results.

## Abstract

Let $F_\infty=\mathbb{F}_q(\!(1/T)\!)$ be the completion of $\mathbb{F}_q(T)$ at $1/T$. We develop a theory of Fourier expansions for harmonic cochains on the edges of the Bruhat-Tits building of $\mathrm{PGL}_r(F_\infty)$, $r\geq 2$, generalizing an earlier construction of Gekeler for $r=2$. We then apply this theory to study modular units on the Drinfeld symmetric space $\Omega^r$ over $F_\infty$, and the cuspidal divisor groups of Satake compactifications of certain Drinfeld modular varieties. In particular, we obtain a higher dimensional analogue of a result of Ogg for classical modular curves $X_0(p)$ of prime level.