Drinfeld discriminant function and Fourier expansion of harmonic cochains

I will discuss my joint work with Fu-Tsun Wei from Tsing Hua University in Taiwan.

Let $K$ be the completion of $\mathbb{F}_q(T)$ at $1/T$ and $r\geq 2$ be an integer. In an ongoing project, we study modular units on the Drinfeld symmetric space $\Omega^r$ over $K$, harmonic cochains on the edges of the Bruhat-Tits building of $PGL_r(K)$, and the cuspidal divisor groups of certain Drinfeld modular varieties of dimension $r-1$. In particular, we obtained a higher dimensional analogue of a well-known result of Ogg for classical modular curves $X_0(p)$ of prime level.