Drinfeld discriminant function and Fourier expansion of harmonic cochains

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 ${\mathrm{PGL}}_r(F_{\infty })$, $r\ge 2$, 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.

The paper has been significantly revised and extended