On computing quaternion quotient graphs for function fields

Abstract

Let Λ be a Fq[T ]-maximal order in a division quaternion algebra over Fq(T ) which is split at the place∞. The present article gives an algorithm to compute a fundamental domain for the action of the group of units Λ∗ on the Bruhat-Tits tree T associated to PGL2(Fq((1/T ))). 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 Λ∗ in terms of generators and relations. Moreover we determine an upper bound for its running time using that Λ∗\T is almost Ramanujan.