Let $\Lambda$ be a maximal $𝔽{}_q\left[T\right]$-order in a division quaternion algebra over $𝔽{}_q\left(T\right)$ which is split at the place $\infty$. The present article gives an algorithm to compute a fundamental domain for the action of the group of units ${\Lambda }^{*}$ on the Bruhat-Tits tree $𝒯$ associated to ${\mathrm{PGL}}_2\left(𝔽{}_q\left((1/T)\right)\right)$. This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group ${\Lambda }^{*}$ in terms of generators and relations. Moreover we determine an upper bound...

We study the integral model of the Drinfeld modular curve $X_1\left(n\right)$ for a prime $n\in {𝔽}_q\left[T\right]$. A function field analogue of the theory of Igusa curves is introduced to describe its reduction mod $n$. A result describing the universal deformation ring of a pair consisting of a supersingular Drinfeld module and a point of order $n$ in terms of the Hasse invariant of that Drinfeld module is proved. We then apply Jung-Hirzebruch resolution for arithmetic surfaces to produce a regular model of $X_1\left(n\right)$ which, after contractions in...