Drinfeld discriminant function and Fourier expansion of harmonic cochains

Let (Formula presented.) be the completion of (Formula presented.) at 1/T. We develop a theory of Fourier expansions for harmonic cochains on the edges of the Bruhat–Tits building of (Formula presented.), (Formula presented.), generalizing an earlier construction of Gekeler for (Formula presented.). We then apply this theory to study modular units on the Drinfeld symmetric space (Formula presented.) over (Formula presented.), 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 (Formula presented.) of prime level.